Site Map | Polytopes | Dynkin Diagrams | Vertex Figures, etc. | Incidence Matrices | Index |
Sure, anything provided hereafter could be found already in the individual incidence matrix files, and sometimes also in some of the explanatory pages as well. None the less a missing link is that of dimensional analogy of the various members of a family of polytopes. Esp. for those generally existing cases.
In the followings some general dimensional series of polytopes get detailed.
These polytopes generally are self-dual. Further they are closely related to the pyramid product. In fact Sn here is nothing but the Sn-1 pyramid. Thence, by means of the lace prism notation, Sn = x3o...o3o (n nodes) can be described as well as ox3oo...oo3oo&#x (n-1 node positions).
Dimension | 1D | 2D | 3D | 4D | 5D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
x |
x3o |
x3o3o |
x3o3o3o |
x3o3o3o3o |
x3o...o3o |
Acronym |
line |
trig |
tet |
pen |
hix | n-simplex |
Vertex Count | 2 | 3 line | 4 trig | 5 tet | 6 pen | n+1 |
Facet Count | 3 line | 4 trig | 5 tet | 6 pen | n+1 | |
Circumradius |
1/2 0.5 |
1/sqrt(3) 0.577350 |
sqrt(3/8) 0.612372 |
sqrt(2/5) 0.632456 |
sqrt(5/12) 0.645497 | sqrt(n)/sqrt[2(n+1)] |
Inradius |
1/2 0.5 |
1/sqrt(12) 0.288675 |
1/sqrt(24) 0.204124 |
1/sqrt(40) 0.158114 |
1/sqrt(60) 0.129099 | 1/sqrt[2n(n+1)] |
Volume | 1 |
sqrt(3)/4 0.433013 |
sqrt(2)/12 0.117851 |
sqrt(5)/96 0.023292 |
sqrt(3)/480 0.0036084 | sqrt[(n+1)/(2n)]/n! |
Surface | 2 | 3 |
sqrt(3) 1.732051 |
5 sqrt(2)/12 0.589256 |
sqrt(5)/16 0.139754 | (n+1) sqrt[n/(2n-1)]/(n-1)! |
Dihedral angles | 0° | 60° |
arccos(1/3) 70.528779° |
arccos(1/4) 75.522488° |
arccos(1/5) 78.463041° | arccos(1/n) |
Dimension | 6D | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
x3o3o3o3o3o |
x3o3o3o3o3o3o |
x3o3o3o3o3o3o3o |
x3o3o3o3o3o3o3o3o |
x3o3o3o3o3o3o3o3o3o |
x3o...o3o |
Acronym |
hop |
oca |
ene |
day |
ux | n-simplex |
Vertex Count | 7 hix | 8 hop | 9 oca | 10 ene | 11 day | n+1 |
Facet Count | 7 hix | 8 hop | 9 oca | 10 ene | 11 day | n+1 |
Circumradius |
sqrt(3/7) 0.654654 |
sqrt(7)/4 0.661438 |
2/3 0.666667 |
sqrt(9/20) 0.670820 |
sqrt(5/11) 0.674200 | sqrt(n)/sqrt[2(n+1)] |
Inradius |
1/sqrt(84) 0.109109 |
1/sqrt(112) 0.094491 |
1/12 0.083333 |
1/sqrt(180) 0.074536 |
1/sqrt(220) 0.067420 | 1/sqrt[2n(n+1)] |
Volume |
sqrt(7)/5760 0.00045933 |
1/20160 0.000049603 |
1/215040 0.0000046503 |
sqrt(5)/5806080 0.00000038513 |
sqrt(11)/116121600 0.0000000028562 | sqrt[(n+1)/(2n)]/n! |
Surface |
7 sqrt(3)/480 0.025259 |
sqrt(7)/720 0.0036747 |
1/2240 0.00044643 |
1/21504 0.000046503 |
11 sqrt(5)/5806080 0.0000042364 | (n+1) sqrt[n/(2n-1)]/(n-1)! |
Dihedral angles |
arccos(1/6) 80.405932° |
arccos(1/7) 81.786789° |
arccos(1/8) 82.819244° |
arccos(1/9) 83.620630° |
arccos(1/10) 84.260830° | arccos(1/n) |
Within these polytopes rSn generally can be described as the segmentotope of the regular simplex Sn-1 atop the rectified simplex rSn-1. Thence, by means of the lace prism notation, rSn = o3x3o...o3o (n nodes) can be described as well as xo3ox3oo...oo3oo&#x (n-1 node positions).
Furthermore are rectified simplices special cases of the Coxeter-Elte-Gosset polytopes km,n, in fact those generally are clearly the ones of the form 0(n-1),1.
Dimension | 1D | 2D | 3D | 4D | 5D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
o3x |
o3x3o |
o3x3o3o |
o3x3o3o3o |
o3x3o...o3o | |
Acronym |
trig |
oct |
rap |
rix | rect. n-simplex | |
Vertex Count | 3 line | 6 square | 10 trip | 15 tepe | n(n+1)/2 | |
Facet Count rect. facets | 3 line | 4 trig | 5 oct | 6 rap | n+1 | |
Facet Count verf facets | 4 trig | 5 tet | 6 pen | n+1 | ||
Circumradius |
1/sqrt(3) 0.577350 |
1/sqrt(2) 0.707107 |
sqrt(3/5) 0.774597 |
sqrt(2/3) 0.816497 | sqrt[(n-1)/(n+1)] | |
Inradius wrt. rect. facets |
1/sqrt(6) 0.408248 |
1/sqrt(10) 0.316228 |
1/sqrt(15) 0.258199 | sqrt(2)/sqrt[n(n+1)] | ||
Inradius wrt. verf facets |
1/sqrt(12) 0.288675 |
1/sqrt(6) 0.408248 |
3/sqrt(40) 0.474342 |
2/sqrt(15) 0.516398 | (n-1)/sqrt[2n(n+1)] | |
Volume |
sqrt(3)/4 0.433013 |
sqrt(2)/3 0.471405 |
11 sqrt(5)/96 0.256216 |
13 sqrt(3)/240 0.093819 | (2n-n-1) sqrt[(n+1)/(2n)]/n! | |
Surface | 3 |
2 sqrt(3) 3.464102 |
25 sqrt(2)/12 2.946278 |
3 sqrt(5)/4 1.677051 | (n+1)(2n-1-n+1) sqrt[n/(2n-1)]/(n-1)! | |
Dihedral angles rect - rect |
60° verf - verf |
arccos(1/4) 75.522488° |
arccos(1/5) 78.463041° | arccos(1/n) | ||
Dihedral angles verf - rect |
arccos(-1/3) 109.471221° |
arccos(-1/4) 104.477512° |
arccos(-1/5) 101.536959° | arccos(-1/n) | ||
Dimension | 6D | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3x3o3o3o3o |
o3x3o3o3o3o3o |
o3x3o3o3o3o3o3o |
o3x3o3o3o3o3o3o3o |
o3x3o3o3o3o3o3o3o3o |
o3x3o...o3o |
Acronym |
ril |
roc |
rene |
reday |
ru | rect. n-simplex |
Vertex Count | 21 penp | 28 hixip | 36 hopip | 45 ocpe | 55 enep | n(n+1)/2 |
Facet Count rect. facets | 7 rix | 8 ril | 9 roc | 10 rene | 11 reday | n+1 |
Facet Count verf facets | 7 hix | 8 hop | 9 oca | 10 ene | 11 day | n+1 |
Circumradius |
sqrt(5/7) 0.845154 |
sqrt(3)/2 0.866025 |
sqrt(7)/3 0.881917 |
2/sqrt(5) 0.894427 |
3/sqrt(11) 0.904534 | sqrt[(n-1)/(n+1)] |
Inradius wrt. rect. facets |
1/sqrt(21) 0.218218 |
1/sqrt(28) 0.188982 |
1/6 0.166667 |
1/sqrt(45) 0.149071 |
1/sqrt(55) 0.134840 | sqrt(2)/sqrt[n(n+1)] |
Inradius wrt. verf facets |
5/sqrt(84) 0.545545 |
3/sqrt(28) 0.566947 |
7/12 0.583333 |
4/sqrt(45) 0.596285 |
9/sqrt(220) 0.606780 | (n-1)/sqrt[2n(n+1)] |
Volume |
19 sqrt(7)/1920 0.026182 |
1/168 0.0059524 |
247/215040 0.0011486 |
251 sqrt(5)/2903040 0.00019333 |
1013 sqrt(11)/116121600 0.000028933 | (2n-n-1) sqrt[(n+1)/(2n)]/n! |
Surface |
63 sqrt(3)/160 0.681995 |
29 sqrt(7)/360 0.213130 |
121/2240 0.054018 |
31/2688 0.011533 |
5533 sqrt(5)/5806080 0.0021309 | (n+1)(2n-1-n+1) sqrt[n/(2n-1)]/(n-1)! |
Dihedral angles rect - rect |
arccos(1/6) 80.405932° |
arccos(1/7) 81.786789° |
arccos(1/8) 82.819244° |
arccos(1/9) 83.620630° |
arccos(1/10) 84.260830° | arccos(1/n) |
Dihedral angles verf - rect |
arccos(-1/6) 99.594068° |
arccos(-1/7) 98.213211° |
arccos(-1/8) 97.180756° |
arccos(-1/9) 96.379370° |
arccos(-1/10) 95.739170° | arccos(-1/n) |
These non-convex polytopes frSn generally are facetings of the rectified simplex rSn.
According to the fact that the mere Wythoffian construction provides Grünbaumian polytopes only, it is the secondary operation of replacing those double covered facets by single covers instead, which breaks down their orientability for all odd dimensions. Thence there a volume cannot be calculated. For the even dimensional cases we observe that no hemifacets occur and that the facet types alternate between prograde and retrograde wrt. the increasing absolute values of their inradii.
From the above shown segmentotope representation of the rectified simplex rSn, it becomes obvious that the polytopes frSn likewise can be given as such, though non-covex for sure, being generally the stack of the simplex Sn-1 atop the facetorectified simplex frSn-1.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
hemi ( x3o3/2x ) |
hemi ( x3o3o3/2x ) |
hemi ( x3o3o3o3/2x ) |
hemi ( x3o3o3o3o3/2x ) |
hemi ( x3o...o3o3/2x ) |
Acronym |
thah |
firp |
firx |
firl | facetorect. n-simplex |
Vertex Count | 6 | 10 | 15 | 21 | n(n+1)/2 |
Facet Count simplex | 4 trig | 5 tet | 6 pen | 7 hix | n+1 |
Facet Count prism |
3 square (hemi) | 10 trip | 15 tepe | 21 penp | n(n+1)/2 |
Facet Count duoprism |
10 triddip (hemi) | 35 tratet | (n-1)n(n+1)/6 | ||
Circumradius |
1/sqrt(2) 0.707107 |
sqrt(3/5) 0.774597 |
sqrt(2/3) 0.816497 |
sqrt(5/7) 0.845154 | sqrt[(n-1)/(n+1)] |
Inradius wrt. simplex |
1/sqrt(6) 0.408248 |
− 3/sqrt(40) 0.474342 |
2/sqrt(15) 0.516398 |
+ 5/sqrt(84) 0.545545 | (n-1)/sqrt[2n(n+1)] |
Inradius wrt. prism | 0 |
+ 1/sqrt(60) 0.129099 |
1/sqrt(24) 0.204124 |
− 3/sqrt(140) 0.253546 | (n-3)/sqrt[4(n-1)(n+1)] |
Inradius wrt. duoprism | 0 |
+ 1/sqrt(168) 0.077152 | (n-5)/sqrt[6(n-2)(n+1)] | ||
Volume | - |
sqrt(5)/32 0.069877 | - |
sqrt(7)/576 0.0045933 | - / sqrt[(n+1)/2n+2]/((n/2)!)2 |
Surface |
3+sqrt(3) 4.732051 |
[5 sqrt(2)+30 sqrt(3)]/12 4.919383 |
[30+20 sqrt(2)+sqrt(5)]/16 3.782521 |
[7 sqrt(3)+105 sqrt(5) +350 sqrt(6)]/480 2.300485 | ? |
Dihedral angles simp. - (next) |
arccos[1/sqrt(3)] 54.735610° |
arccos[sqrt(3/8)] 52.238756° |
arccos[sqrt(2/5)] 50.768480° |
arccos[sqrt(5/12)] 49.797034° | arccos[sqrt((n-1)/2n)] |
Dihedral angles prism - (next) |
arccos(2/3) 48.189685° (prism - prism) | 45° | ? | ? | |
Dihedral angles duopr. - (next) |
? (duopr. - duopr.) | ? | |||
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
hemi ( x3o3o3o3o3o3/2x ) |
hemi ( x3o3o3o3o3o3o3/2x ) |
hemi ( x3o3o3o3o3o3o3o3/2x ) |
hemi ( x3o3o3o3o3o3o3o3o3/2x ) |
hemi ( x3o...o3o3/2x ) |
Acronym |
froc |
? |
? |
? | facetorect. n-simplex |
Vertex Count | 28 | 36 | 45 | 55 | n(n+1)/2 |
Facet Count simplex | 8 hop | 9 oca | 10 ene | 11 day | n+1 |
Facet Count prism | 28 hixip | 36 hopip | 45 ocpe | 55 enep | n(n+1)/2 |
Facet Count duoprism I | 56 trapen | 84 trahix | 120 trihop | 165 trioc | (n-1)n(n+1)/6 |
Facet Count duoprism II |
35 tetdip (hemi) | 126 tetpen | 210 tethix | 330 tethop | (n-2)(n-1)n(n+1)/24 |
Facet Count duoprism III |
126 pendip (hemi) | 462 penhix | (n-3)(n-2)(n-1)n(n+1)/120 | ||
Circumradius |
sqrt(3)/2 0.866025 |
sqrt(7)/3 0.881917 |
2/sqrt(5) 0.894427 |
3/sqrt(11) 0.904534 | sqrt[(n-1)/(n+1)] |
Inradius wrt. simplex |
3/sqrt(28) 0.566947 |
− 7/12 0.583333 |
4/sqrt(45) 0.596285 |
+ 9/sqrt(220) 0.606780 | (n-1)/sqrt[2n(n+1)] |
Inradius wrt. prism |
1/sqrt(12) 0.288675 |
+ 5/sqrt(252) 0.314970 |
3/sqrt(80) 0.335410 |
− 7/sqrt(396) 0.351763 | (n-3)/sqrt[4(n-1)(n+1)] |
Inradius wrt. duoprism I |
1/sqrt(60) 0.129099 |
− 1/6 0.166667 |
2/sqrt(105) 0.195180 |
+ 5/sqrt(528) 0.217597 | (n-5)/sqrt[6(n-2)(n+1)] |
Inradius wrt. douprism II | 0 |
+ 1/sqrt(360) 0.052705 |
1/sqrt(120) 0.091287 |
− 3/sqrt(616) 0.120873 | (n-7)/sqrt[8(n-3)(n+1)] |
Inradius wrt. douprism III | 0 |
+ 1/sqrt(660) 0.038925 | (n-9)/sqrt[10(n-4)(n+1)] | ||
Volume | - |
1/6144 0.00016276 | - |
sqrt(11)/921600 0.0000035988 | - / sqrt[(n+1)/2n+2]/((n/2)!)2 |
Surface | ? | ? | ? | ? | ? |
Dihedral angles simp. - (next) |
arccos[sqrt(3/7)] 49.106605° |
arccos[sqrt(7)/4] 48.590378° |
arccos(2/3) 48.189685° |
arccos[3/sqrt(20)] 47.869585° | arccos[sqrt((n-1)/2n)] |
Dihedral angles prism - (next) | ? | ? | ? | ? | ? |
Dihedral angles duopr. I - (next) | ? | ? | ? | ? | ? |
Dihedral angles duopr. II - (next) |
? (duopr. II - duopr. II) | ? | ? | ? | |
Dihedral angles duopr. III - (next) |
? (duopr. III - duopr. III) | ? |
Within these polytopes brSn generally can be described as the segmentotope of the rectified simplex rSn-1 atop the birectified simplex brSn-1. Thence, by means of the lace prism notation, brSn = o3o3x3o...o3o (n nodes) can be described as well as oo3xo3ox3oo...oo3oo&#x (n-1 node positions).
Furthermore are birectified simplices special cases of the Coxeter-Elte-Gosset polytopes km,n, in fact those generally are clearly the ones of the form 0(n-2),2.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
o3o3x |
o3o3x3o |
o3o3x3o3o |
o3o3x3o3o3o |
o3o3x3o...o3o |
Acronym |
tet |
rap |
dot |
bril | birect. n-simplex |
Vertex Count | 4 trig | 10 trip | 20 triddip | 35 tratet | (n-1)n(n+1)/6 |
Facet Count rect. fac. | 5 oct | 6 rap | 7 rix | n+1 | |
Facet Count birect. fac. | 4 trig | 5 tet | 6 rap | 7 dot | n+1 |
Circumradius |
sqrt(3/8) 0.612372 |
sqrt(3/5) 0.774597 |
sqrt(3)/2 0.866025 |
sqrt(6/7) 0.925820 | sqrt[(3n-6)/(2n+2)] |
Inradius wrt. rect. facets |
1/sqrt(24) 0.204124 |
1/sqrt(10) 0.316228 |
sqrt(3/20) 0.387298 |
2/sqrt(21) 0.436436 | (n-2)/sqrt[2n(n+1)] |
Inradius wrt. birect. facets |
3/sqrt(40) 0.474342 |
sqrt(3/20) 0.387298 |
sqrt(3/28) 0.327327 | 3/sqrt[2n(n+1)] | |
Volume |
sqrt(2)/12 0.117851 |
11 sqrt(5)/96 0.256216 |
11 sqrt(3)/80 0.238157 |
151 sqrt(7)/2880 0.138718 | [3n-(n+1) 2n+n(n+1)/2] sqrt[(n+1)/2n]/n! |
Surface |
sqrt(3) 1.732051 |
25 sqrt(2)/12 2.946278 |
11 sqrt(5)/8 3.074593 |
161 sqrt(3)/120 2.323835 | (n+1) [3n-1-(n-1) 2n-1+n(n-3)/2] sqrt[n/2n-1]/(n-1)! |
Dihedral angles rect. - rect. |
arccos(1/4) 75.522488° |
arccos(1/5) 78.463041° |
arccos(1/6) 80.405932° | arccos(1/n) | |
Dihedral angles rect. - birect. |
arccos(-1/4) 104.477512° |
arccos(-1/5) 101.536959° |
arccos(-1/6) 99.594068° | arccos(-1/n) | |
Dihedral angles birect. - birect. |
arccos(1/3) 70.528779° |
arccos(1/4) 75.522488° |
arccos(1/5) 78.463041° |
arccos(1/6) 80.405932° | arccos(1/n) |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3o3x3o3o3o3o |
o3o3x3o3o3o3o3o |
o3o3x3o3o3o3o3o3o |
o3o3x3o3o3o3o3o3o3o |
o3o3x3o...o3o |
Acronym |
broc |
brene |
breday |
bru | birect. n-simplex |
Vertex Count | 56 trapen | 84 trahix | 120 trahop | 165 traoc | (n-1)n(n+1)/6 |
Facet Count rect. facets | 8 ril | 9 roc | 10 rene | 11 reday | n+1 |
Facet Count birect. facets | 8 bril | 9 broc | 10 brene | 11 breday | n+1 |
Circumradius |
sqrt(15)/4 0.968246 | 1 |
sqrt(21/20) 1.024695 |
sqrt(12/11) 1.044466 | sqrt[(3n-6)/(2n+2)] |
Inradius wrt. rect. facets |
5/sqrt(112) 0.472456 |
1/2 0.5 |
7/sqrt(180) 0.521749 |
4/sqrt(55) 0.539360 | (n-2)/sqrt[2n(n+1)] |
Inradius wrt. birect. facets |
3/sqrt(112) 0.283473 |
1/4 0.25 |
1/sqrt(20) 0.223607 |
3/sqrt(220) 0.202260 | 3/sqrt[2n(n+1)] |
Volume |
397/6720 0.059077 |
1431/71680 0.019964 |
913 sqrt(5)/362880 0.0056259 |
299 sqrt(11)/725760 0.0013664 | [3n-(n+1) 2n+n(n+1)/2] sqrt[(n+1)/2n]/n! |
Surface |
359 sqrt(7)/720 1.319201 |
1311/2240 0.585268 |
1135/5376 0.211124 |
16621 sqrt(5)/580608 0.064012 | (n+1) [3n-1-(n-1) 2n-1+n(n-3)/2] sqrt[n/2n-1]/(n-1)! |
Dihedral angles rect. - rect. |
arccos(1/7) 81.786789° |
arccos(1/8) 82.819244° |
arccos(1/9) 83.620630° |
arccos(1/10) 84.260830° | arccos(1/n) |
Dihedral angles rect. - birect. |
arccos(-1/7) 98.213211° |
arccos(-1/8) 97.180756° |
arccos(-1/9) 96.379370° |
arccos(-1/10) 95.739170° | arccos(-1/n) |
Dihedral angles birect. - birect. |
arccos(1/7) 81.786789° |
arccos(1/8) 82.819244° |
arccos(1/9) 83.620630° |
arccos(1/10) 84.260830° | arccos(1/n) |
Within these polytopes tSn generally can be described as the bistratic lace tower of the regular simplex Sn-1 atop an u-scaled Sn-1 atop the truncated simplex tSn-1. Thence, by means of the lace tower notation, tSn = x3x3o...o3o (n nodes) can be described as well as xux3oox3ooo...ooo3ooo&#xt (n-1 node positions). As such those also could be referred to as simplexial tutsatopes: in fact tutsatopes are quite similarily defined as the ursatopes, just that the part that there was played (within 4D) by the lacing teddies here now is taken by according tuts.
Dimension | 1D | 2D | 3D | 4D | 5D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
x3x |
x3x3o |
x3x3o3o |
x3x3o3o3o |
x3x3o...o3o | |
Acronym |
hig |
tut |
tip |
tix | trunc. n-simplex | |
Vertex Count | 6 | 12 | 20 | 30 | n(n+1) | |
Facet Count trunc. facets | 3 line | 4 hig | 5 tut | 6 tip | n+1 | |
Facet Count verf facets | 3 line | 4 trig | 5 tet | 6 pen | n+1 | |
Circumradius | 1 |
sqrt(11/8) 1.172604 |
sqrt(8/5) 1.264911 |
sqrt(7)/2 1.322876 | sqrt[(5n-4)/(2n+2)] | |
Inradius wrt. trunc. facets |
sqrt(3)/2 0.866025 |
sqrt(3/8) 0.612372 |
3/sqrt(40) 0.474342 |
sqrt(3/20) 0.387298 | 3/sqrt[2n(n+1)] | |
Inradius wrt. verf facets |
sqrt(3)/2 0.866025 |
5/sqrt(24) 1.020621 |
7/sqrt(40) 1.106797 |
sqrt(27/20) 1.161895 | (2n-1)/sqrt[2n(n+1)] | |
Volume |
3 sqrt(3)/2 2.598076 |
23 sqrt(2)/12 2.710576 |
19 sqrt(5)/24 1.770220 |
79 sqrt(3)/160 0.855200 | (3n-n-1) sqrt[(n+1)/(2n)]/n! | |
Surface | 6 |
7 sqrt(3) 12.124356 |
10 sqrt(2) 14.142136 |
77 sqrt(5)/16 10.761077 | (n+1)(3n-1-n+1) sqrt[n/(2n-1)]/(n-1)! | |
Dihedral angles trunc - trunc |
arccos(1/4) 75.522488° |
arccos(1/5) 78.463041° | arccos(1/n) | |||
Dihedral angles verf - trunc | 120° |
arccos(-1/3) 109.471221° |
arccos(-1/4) 104.477512° |
arccos(-1/5) 101.536959° | arccos(-1/n) | |
Dimension | 6D | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
x3x3o3o3o3o |
x3x3o3o3o3o3o |
x3x3o3o3o3o3o3o |
x3x3o3o3o3o3o3o3o |
x3x3o3o3o3o3o3o3o3o |
x3x3o...o3o |
Acronym |
til |
toc |
tene |
teday |
tu | trunc. n-simplex |
Vertex Count | 42 | 56 | 72 | 90 | 110 | n(n+1) |
Facet Count trunc. facets | 7 tix | 8 til | 9 toc | 10 tene | 11 teday | n+1 |
Facet Count verf facets | 7 hix | 8 hop | 9 oca | 10 ene | 11 day | n+1 |
Circumradius |
sqrt(13/7) 1.362771 |
sqrt(31)/4 1.391941 |
sqrt(2) 1.414214 |
sqrt(41/20) 1.431782 |
sqrt(23/11) 1.445998 | sqrt[(5n-4)/(2n+2)] |
Inradius wrt. trunc. facets |
sqrt(3/28) 0.327327 |
3/sqrt(112) 0.283473 |
1/4 0.25 |
1/sqrt(20) 0.223607 |
3/sqrt(220) 0.202260 | 3/sqrt[2n(n+1)] |
Inradius wrt. verf facets |
11/sqrt(84) 1.200198 |
13/sqrt(112) 1.228385 |
5/4 1.25 |
17/sqrt(180) 1.267105 |
19/sqrt(220) 1.280980 | (2n-1)/sqrt[2n(n+1)] |
Volume |
361 sqrt(7)/2880 0.331638 |
2179/20160 0.108085 |
39/1280 0.030469 |
19673 sqrt(5)/5806080 0.0075766 |
4217 sqrt(11)/8294400 0.0016862 | (3n-n-1) sqrt[(n+1)/(2n)]/n! |
Surface |
833 sqrt(3)/240 6.011660 |
241 sqrt(7)/240 2.656775 |
109/112 0.973214 |
6553/21504 0.304734 |
12023 sqrt(5)/322560 0.083346 | (n+1)(3n-1-n+1) sqrt[n/(2n-1)]/(n-1)! |
Dihedral angles trunc - trunc |
arccos(1/6) 80.405932° |
arccos(1/7) 81.786789° |
arccos(1/8) 82.819244° |
arccos(1/9) 83.620630° |
arccos(1/10) 84.260830° | arccos(1/n) |
Dihedral angles verf - trunc |
arccos(-1/6) 99.594068° |
arccos(-1/7) 98.213211° |
arccos(-1/8) 97.180756° |
arccos(-1/9) 96.379370° |
arccos(-1/10) 95.739170° | arccos(-1/n) |
Within these polytopes btSn for n>3 can be described as the bistratic lace tower of the truncated simplex tSn-1 atop an u-scaled rectified simplex rSn-1 atop the bitruncated simplex btSn-1. Thence, by means of the lace tower notation, btSn = o3x3x3o...o3o (n nodes) can be described as well as xoo3xux3oox3ooo...ooo3ooo&#xt (n-1 node positions). A posteriori that latter lace tower then applies even for n=3 too, thereby quite similarily just reducing to its first 2 node positions.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
o3x3x |
o3x3x3o |
o3x3x3o3o |
o3x3x3o3o3o |
o3x3x3o...o3o |
Acronym |
tut |
deca |
bittix |
batal | bitrunc. n-simplex |
Vertex Count | 12 | 30 | 60 | 105 | (n+1)n(n-1)/2 |
Facet Count bitrunc. fac. | 4 trig | 5 tut | 6 deca | 7 bittix | n+1 |
Facet Count trunc. fac. | 4 hig | 5 tut | 6 tip | 7 tix | n+1 |
Circumradius |
sqrt(11/8) 1.172604 |
sqrt(2) 1.414214 |
sqrt(29/12) 1.554563 |
sqrt(19/7) 1.647509 | sqrt[(9n-16)/(2n+2)] |
Inradius bitrunc. fac. |
5/sqrt(24) 1.020621 |
sqrt(5/8) 0.790569 |
sqrt(5/12) 0.645497 |
5/sqrt(84) 0.545545 | 5/sqrt[2n(n+1)] |
Inradius trunc. fac. |
sqrt(3/8) 0.612372 |
sqrt(5/8) 0.790569 |
7/sqrt(60) 0.903696 |
sqrt(27/28) 0.981981 | (2n-3)/sqrt[2n(n+1)] |
Volume |
23 sqrt(2)/12 2.710576 |
115 sqrt(5)/48 5.357246 |
841 sqrt(3)/240 6.069395 | ? | ? |
Surface |
7 sqrt(3) 12.124356 |
115 sqrt(2)/6 27.105760 |
153 sqrt(5)/8 42.764800 | ? | ? |
Dihedral angles bitrunc - bitrunc |
arccos(1/4) 75.522488° |
arccos(1/5) 78.463041° |
arccos(1/6) 80.405932° | arccos(1/n) | |
Dihedral angles bitrunc - trunc |
arccos(-1/3) 109.471221° |
arccos(-1/4) 104.477512° |
arccos(-1/5) 101.536959° |
arccos(-1/6) 99.594068° | arccos(-1/n) |
Dihedral angles trunc - trunc |
arccos(1/3) 70.528779° |
arccos(1/4) 75.522488° |
arccos(1/5) 78.463041° |
arccos(1/6) 80.405932° | arccos(1/n) |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3x3x3o3o3o3o |
o3x3x3o3o3o3o3o |
o3x3x3o3o3o3o3o3o |
o3x3x3o3o3o3o3o3o3o |
o3x3x3o...o3o |
Acronym |
bittoc |
batene |
? |
? | bitrunc. n-simplex |
Vertex Count | 168 | 252 | 360 | 495 | (n+1)n(n-1)/2 |
Facet Count bitrunc. fac. | 8 batal | 9 bittoc | 10 batene | 11 ? | n+1 |
Facet Count trunc. fac. | 8 til | 9 toc | 10 tene | 11 teday | n+1 |
Circumradius |
sqrt(47)/4 1.713914 |
sqrt(29)/3 1.795055 |
sqrt(13)/2 1.802776 |
sqrt(37/11) 1.834022 | sqrt[(9n-16)/(2n+2)] |
Inradius bitrunc. fac. |
5/sqrt(112) 0.472456 |
5/12 0.416667 |
sqrt(5)/6 0.372678 |
sqrt(5/44) 0.337100 | 5/sqrt[2n(n+1)] |
Inradius trunc. fac. |
11/sqrt(112) 1.039402 |
13/12 1.083333 |
sqrt(45)/6 1.118034 |
17/sqrt(220) 1.146140 | (2n-3)/sqrt[2n(n+1)] |
Volume | ? | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles bitrunc - bitrunc |
arccos(1/7) 81.786789° |
arccos(1/8) 82.819244° |
arccos(1/9) 83.620630° |
arccos(1/10) 84.260830° | arccos(1/n) |
Dihedral angles bitrunc - trunc |
arccos(-1/7) 98.213211° |
arccos(-1/8) 97.180756° |
arccos(-1/9) 96.379370° |
arccos(-1/10) 95.739170° | arccos(-1/n) |
Dihedral angles trunc - trunc |
arccos(1/7) 81.786789° |
arccos(1/8) 82.819244° |
arccos(1/9) 83.620630° |
arccos(1/10) 84.260830° | arccos(1/n) |
This case applies to odd dimensions only. These also occur (scaled down) as intersection kernels of facet-regular bi-simplex compounds. Further they occur (again scaled) as equatorial mid-sections of the vertex-first oriented (then even-dimensional) hypercube Cn+1.
Note that those can be generally provided too as next-to-center rectified simplex alterprisms oo3oo3...xo3ox...3oo3oo&#x (n-1 node positions).
Dimension | 1D | 3D | 5D | 7D | 9D |
nD (2k+1)D |
---|---|---|---|---|---|---|
Dynkin diagram |
x |
o3x3o |
o3o3x3o3o |
o3o3o3x3o3o3o |
o3o3o3o3x3o3o3o3o |
o3o...o3x3o...o3o |
Acronym |
line |
oct |
dot |
he |
icoy | mid-rect. n-simplex |
Vertex Count | 2 | 6 square | 20 triddip | 70 tetdip | 252 pendip |
(n+1)!/[((n+1)/2)!]2 (2(k+1))!/((k+1)!)2 |
Facet Count | 4+4 trig | 6+6 rap | 8+8 bril | 10+10 trene |
2(n+1) 4(k+1) | |
Circumradius |
1/2 0.5 |
1/sqrt(2) 0.707107 |
sqrt(3)/2 0.866025 | 1 |
sqrt(5)/2 1.118034 |
sqrt[(n+1)/8] sqrt(k+1)/2 |
Inradius |
1/2 0.5 |
1/sqrt(6) 0.408248 |
sqrt(3/20) 0.387298 |
1/sqrt(7) 0.377964 |
sqrt(5)/6 0.372678 |
sqrt[(n+1)/(8n)] sqrt[(k+1)/(8k+4)] |
Volume | 1 |
sqrt(2)/3 0.471405 |
11 sqrt(3)/80 0.238157 |
151/1260 0.119841 | ? | ? |
Surface | 2 |
2 sqrt(3) 3.464102 |
11 sqrt(5)/8 3.074593 |
151 sqrt(7)/180 2.219491 | ? | ? |
Dihedral angles wrt. mid-rect margin | 0° |
arccos(-1/3) 109.471221° |
arccos(-1/5) 101.536959° |
arccos(-1/7) 98.213211° |
arccos(-1/9) 96.379370° | arccos(-1/n) |
Dihedral angles wrt. offset margin |
arccos(1/5) 78.463041° |
arccos(1/7) 81.786789° |
arccos(1/9) 83.620630° | arccos(1/n) |
This case applies to even dimensions only. These also occur (scaled down) as intersection kernels of facet-regular bi-simplex compounds. Further they occur (again scaled) as equatorial mid-sections of the vertex-first oriented (then odd-dimensional) hypercube Cn+1.
Dimension | 2D | 4D | 6D | 8D | 10D |
nD (2k)D |
---|---|---|---|---|---|---|
Dynkin diagram |
x3x |
o3x3x3o |
o3o3x3x3o3o |
o3o3o3x3x3o3o3o |
o3o3o3o3x3x3o3o3o3o |
o3o...o3x3x3o...o3o |
Acronym |
hig |
deca |
fe |
be |
? | mid-trunc. n-simplex |
Vertex Count | 6 | 30 | 140 | 630 | 2772 |
(n+1)!/((n/2)!)2 (2k+1)!/(k!)2 |
Facet Count | 3+3 line | 5+5 tut | 7+7 bittix | 9+9 tattoc | 11+11 ? |
2(n+1) 2(2k+1) |
Circumradius | 1 |
sqrt(2) 1.414214 |
sqrt(3) 1.732051 | 2 |
sqrt(5) 2.236068 |
sqrt(n/2) sqrt(k) |
Inradius |
sqrt(3)/2 0.866025 |
sqrt(5/8) 0.790569 |
sqrt(7/12) 0.763763 |
3/4 0.75 |
sqrt(11/20) 0.741620 |
sqrt[(n+1)/(2n)] sqrt[(2k+1)/(4k)] |
Volume |
3 sqrt(3)/2 2.598076 |
115 sqrt(5)/48 5.357246 |
5887 sqrt(7)/1440 10.816346 | ? | ? | ? |
Surface | 6 |
115 sqrt(2)/6 27.105760 |
5887 sqrt(3)/120 84.971526 | ? | ? | ? |
Dihedral angles wrt. mid-trunc margin | 120° |
arccos(-1/4) 104.477512° |
arccos(-1/6) 99.594068° |
arccos(-1/8) 97.180756° |
arccos(-1/10) 95.739170° | arccos(-1/n) |
Dihedral angles wrt. offset margin |
arccos(1/4) 75.522488° |
arccos(1/6) 80.405932° |
arccos(1/8) 82.819244° |
arccos(1/10) 84.260830° | arccos(1/n) |
The common unit circumradius of all these shows that they occur as vertex figure of an according dimensional honeycomb. In fact they are the hull-of-roots polytopes of the according dimensional root lattice An. Furthermore it forces that the facet-to-bodycenter pyramids all are CRF, i.e. that all these polytopes can be decomposed accordingly.
Within these polytopes eSn generally can be described as the bistratic lace tower of the regular simplex Sn-1 atop the maximal expanded simplex eSn-1 atop the dual regular simplex -Sn-1. Thence, by means of the lace tower notation, eSn = x3o...o3x (n nodes) can be described as well as xxo3ooo...ooo3oxx&#xt (n-1 node positions). Note that the midsection here is of the very same form eSn-1, just one dimension less. Therefore that mentioned unit circumradius property here simply follows by dimensional induction.
Dimension | 1D | 2D | 3D | 4D | 5D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
x3x |
x3o3x |
x3o3o3x |
x3o3o3o3x |
x3o3o...o3o3x | |
Acronym |
hig |
co |
spid |
scad | max-exp. n-simplex | |
Vertex Count | 6 | 12 | 20 | 30 | n(n+1) | |
Facet Count simplex | 3+3 line | 4+4 trig | 5+5 tet | 6+6 pen | n+1 per type | |
Facet Count prism | 6 square | 10+10 trip | 15+15 tepe | n(n+1)/2 per type | ||
Facet Count duoprism I | 20 triddip | (n+1)n(n-1)/6 per type | ||||
Circumradius | 1 | 1 | 1 | 1 | 1 | |
Inradius wrt. simplex facets |
sqrt(3)/2 0.866025 |
sqrt(2/3) 0.816497 |
sqrt(5/8) 0.790569 |
sqrt(3/5) 0.774597 | sqrt[(n+1)/2n] | |
Inradius wrt. prism facets |
1/sqrt(2) 0.707107 |
sqrt(5/12) 0.645497 |
sqrt(3/8) 0.612372 | sqrt[(n+1)/(4n-4)] | ||
Inradius wrt. d.pr. I fac. |
1/sqrt(3) 0.577350 | sqrt[(n+1)/(6n-12)] | ||||
Volume |
3 sqrt(3)/2 2.598076 |
5 sqrt(2)/3 2.357023 |
35 sqrt(5)/48 1.630466 |
21 sqrt(3)/40 0.909327 | (2n)! sqrt[(n+1)/(2n)]/(n!)3 | |
Surface | 6 |
6+2 sqrt(3) 9.464102 |
5 sqrt(2)/6+5 sqrt(3) 9.838765 |
(30+20 sqrt(2)+sqrt(5))/8 7.565042 | ? | |
Dihedral angles simplex - (next) | 120° |
arccos[-1/sqrt(3)] 125.264390° |
arccos(-sqrt(3/8)) 127.761244° |
arccos[-sqrt(2/5)] 129.231520° | arccos[-sqrt((n-1)/2n)] | |
Dihedral angles prism - (next) |
arccos(-2/3) 131.810315° | 135° | arccos[-sqrt((2n-4)/(3n-3)] | |||
Dimension | 6D | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
x3o3o3o3o3x |
x3o3o3o3o3o3x |
x3o3o3o3o3o3o3x |
x3o3o3o3o3o3o3o3x |
x3o3o3o3o3o3o3o3o3x |
x3o3o...o3o3x |
Acronym |
staf |
suph |
soxeb |
? |
? | max-exp. n-simplex |
Vertex Count | 42 | 56 | 72 | 90 | 110 | n(n+1) |
Facet Count simplex | 7+7 hix | 8+8 hop | 9+9 oca | 10+10 ene | 11+11 day | n+1 per type |
Facet Count prism | 21+21 penp | 28+28 hixip | 36+36 hopip | 45+45 ocpe | 55+55 enep | (n+1)n/2 per type |
Facet Count duoprism I | 35+35 tratet | 56+56 trapen | 84+84 trahix | 120+120 trihop | 165+165 trioc | (n+1)n(n-1)/6 per type |
Facet Count duoprism II | 70 tetdip | 126+126 tetpen | 210+210 tethix | 330+330 tethop | (n+1)n(n-1)(n-2)/24 per type | |
Facet Count duoprism III | 252 pendip | 462+462 penhix | (n+1)n(n-1)(n-2)(n-3)/120 per type | |||
Circumradius | 1 | 1 | 1 | 1 | 1 | 1 |
Inradius wrt. simplex facets |
sqrt(7/12) 0.763763 |
2/sqrt(7) 0.755929 |
3/4 0.75 |
sqrt(5)/3 0.745356 |
sqrt(11/20) 0.741620 | sqrt[(n+1)/2n] |
Inradius wrt. prism facets |
sqrt(7/20) 0.591608 |
1/sqrt(3) 0.577350 |
3/sqrt(28) 0.566947 |
sqrt(5)/4 0.559017 |
sqrt(11)/6 0.552771 | sqrt[(n+1)/(4n-4)] |
Inradius wrt. d.pr. I fac. |
sqrt(7/24) 0.540062 |
2/sqrt(15) 0.516398 |
1/2 0.5 |
sqrt(5/21) 0.487950 |
sqrt(11/48) 0.478714 | sqrt[(n+1)/(6n-12)] |
Inradius wrt. d.pr. II fac. |
1/2 0.5 |
3/sqrt(40) 0.474342 |
sqrt(5/24) 0.456435 |
sqrt(11/56) 0.443203 | sqrt[(n+1)/(8n-24)] | |
Inradius wrt. d.pr. III fac. |
1/sqrt(5) 0.447214 |
sqrt(11/60) 0.428174 | sqrt[(n+1)/(10n-40)] | |||
Volume |
77 sqrt(7)/480 0.424423 |
143/840 0.170238 |
429/7168 0.059849 |
2431 sqrt(5)/290304 0.018725 |
46189 sqrt(11)/29030400 0.0052769 | (2n)! sqrt[(n+1)/(2n)]/(n!)3 |
Surface |
7[sqrt(3)+15 sqrt(5)+50 sqrt(6)]/240 4.600970 |
[350+42 sqrt(3)+sqrt(7)+105 sqrt(15)]/360 2.311264 | ? | ? | ? | ? |
Dihedral angles simplex - (next) |
arccos[-sqrt(5/12)] 130.202966° |
arccos[-sqrt(3/7)] 130.893395° |
arccos[-sqrt(7)/4] 131.409622° |
arccos(-2/3) 131.810315° |
arccos[-3/sqrt(20)] 132.130415° | arccos[-sqrt((n-1)/2n)] |
Dihedral angles prism - (next) |
arccos[-sqrt(8/15)] 136.911277° |
arccos[-sqrt(5)/3] 138.189685° |
arccos[-2/sqrt(7)] 139.106605° |
arccos[-sqrt(7/12)] 139.797034° |
arccos[-4/sqrt(27)] 140.335965° | arccos[-sqrt((2n-4)/(3n-3))] |
Dihedral angles d.pr. I - (next) | ? | ? | ? | ? | ? | ? |
Dihedral angles d.pr. II - (next) | ? | ? | ? | ? | ||
Dihedral angles d.pr. III - (next) | ? | ? |
Interestingly this class belongs to an even wider class of (then mostly hyperbolic) polytopes which all have that common property that the nD (or rather: rank n) representant occurs as ridge faceting midsection within the (n+1)D case (for the finite cases) resp. as a ridge faceting subspace within the rank n+1 case (for the infinite cases). This then is the general maximal-expanded class xPo3o...o3oPx, or, even more general, the class of cyclotruncated xPo3o...o3oPxQ*a. Below is a small enlisting thereof.
xPo3o...o3oPxQ*a | |||||
P = 3 | P = 4 | P = 5 | P = 6 | ||
Q = 2 |
r = 1
x3o3x - co x3o3o3x - spid x3o3o3o3x - scad x3o3o3o3o3x - staf x3o3o3o3o3o3x - suph x3o3o3o3o3o3o3x - soxeb ... |
r = ∞
x4o4x - squat x4o3o4x - chon x4o3o3o4x - test x4o3o3o3o4x - penth x4o3o3o3o3o4x - axh x4o3o3o3o3o3o4x - hepth ... |
r = sqrt[-(1+sqrt(5))/2] = 1.272020 i
x5o5x - tepet x5o3o5x - spidded x5o3o3o5x ... |
r = sqrt(-1) = 1 i
x6o6x - tehat x6o3o6x - spiddihexah ... | |
Q = 3 |
r = ∞
x3o3x3*a - that x3o3o3x3*a - batatoh x3o3o3o3x3*a - cytopit x3o3o3o3o3x3*a - cytaxh x3o3o3o3o3o3x3*a - cytloh ... |
r = sqrt(-1) = 1 i
x4o4x3*a - tehat x4o3o4x3*a - cytoch x4o3o3o4x3*a ... |
r = sqrt[-(sqrt(5)-1)/2] = 0.786151 i
x5o5x3*a - phat x5o3o5x3*a ... |
r = 1/sqrt(-2) = 0.707107 i
x6o6x3*a - shexat ... | |
Q = 4 |
r = sqrt(-1) = 1 i
x3o3x4*a - tehat x3o3o3x4*a - cyticth x3o3o3o3x4*a ... |
r = 1/sqrt[-sqrt(2)] = 0.840896 i
x4o4x4*a - teoct x4o3o4x4*a ... |
r = sqrt[(3+sqrt(2)-sqrt(5)-sqrt(10))/2] = 0.701474 i x5o5x4*a ... |
r = sqrt[-(sqrt(2)-1)] = 0.643594 i
x6o6x4*a ... |
These non-convex polytopes reSn generally are facetings of the maximal expanded simplex eSn.
While one third of the facets always are hemifacets and the second third could be considered to be prograde throughout, the remaining one would alternate wrt. its retrogradeness. Thence for the odd dimensional series members the volume always results in zero.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
o3x3x3/2*a |
o3x3x3/2*a3o |
o3x3x3/2*a3o3o |
o3x3x3/2*a3o3o3o |
o3x3x3/2*a3o...o3o |
Acronym |
oho |
duhd |
dehad |
fohaf | retroexp. n-simplex |
Vertex Count | 12 | 20 | 30 | 42 | n(n+1) |
Facet Count simplex | 4+4 trig | 5+5 tet | 6+6 pen | 7+7 hix | n+1 (each) |
Facet Count retroexp. simp. | 4 hig | 5 oho | 6 duhd | 7 dehad | n+1 |
Circumradius | 1 | 1 | 1 | 1 | 1 |
Inradius wrt. simplex facets |
sqrt(2/3) 0.816497 |
sqrt(5/8) 0.790569 |
sqrt(3/5) 0.774597 |
sqrt(7/12) 0.763763 | sqrt[(n+1)/2n] |
Inradius wrt. retroexp. simp. | 0 | 0 | 0 | 0 | 0 |
Volume | 0 |
5 sqrt(5)/8 0.232924 | 0 |
7 sqrt(7)/2880 0.0064306 | 0 / sqrt[(n+1)3/2n-2]/n! |
Surface |
8 sqrt(3) 13.856406 |
5 sqrt(2)/6 1.178511 |
31 sqrt(5)/8 8.664763 |
7 sqrt(3)/240 0.050518 | ? |
Dihedral angles sim. - r.exp. sim. |
arccos(1/3) 70.528779° |
arccos(1/4) 75.522488° |
arccos(1/5) 78.463041° |
arccos(1/6) 80.405932° | arccos(1/n) |
Dihedral angles r.exp. s. - r.exp. s. |
arccos(1/4) 75.522488° |
arccos(1/5) 78.463041° |
arccos(1/6) 80.405932° | arccos(1/n) | |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3x3x3/2*a3o3o3o3o |
o3x3x3/2*a3o3o3o3o3o |
o3x3x3/2*a3o3o3o3o3o3o |
o3x3x3/2*a3o3o3o3o3o3o3o |
o3x3x3/2*a3o...o3o |
Acronym |
hehah |
? |
? |
? | retroexp. n-simplex |
Vertex Count | 56 | 72 | 90 | 110 | n(n+1) |
Facet Count simplex | 8+8 hop | 9+9 oca | 10+10 ene | 11+11 day | n+1 per type |
Facet Count prism | 8 fohaf | 9 hehah | 10 ? | 11 ? | n+1 |
Circumradius | 1 | 1 | 1 | 1 | 1 |
Inradius wrt. simplex |
2/sqrt(7) 0.755929 |
3/4 0.75 |
sqrt(5)/3 0.745356 |
sqrt(11/20) 0.741620 | sqrt[(n+1)/2n] |
Inradius wrt. retroexp. simp. | 0 | 0 | 0 | 0 | 0 |
Volume | 0 |
3/35840 0.000083705 | 0 |
11 sqrt(11)/58060800 0.00000062836 | 0 / sqrt[(n+1)3/2n-2]/n! |
Surface | ? | ? | ? | ? | ? |
Dihedral angles sim. - r.exp. sim. |
arccos(1/7) 81.786789° |
arccos(1/8) 82.819244° |
arccos(1/9) 83.620630° |
arccos(1/10) 84.260830° | arccos(1/n) |
Dihedral angles r.exp. sim. - r.exp. sim. |
arccos(1/7) 81.786789° |
arccos(1/8) 82.819244° |
arccos(1/9) 83.620630° |
arccos(1/10) 84.260830° | arccos(1/n) |
These polytopes otSn also are known as permutotopes Pn+1 and in fact the set of their vertices each can be found to be in one-to-one correspondence, that is being mapped from and therefore being labeled by the permutations of the first n+1 natural numbers in such a way that the edges will represent the set of transpositions (permutations of any 2 elements only).
This very labeling moreover shows that each otSn also can be represented within an n-dimensional subspace of the (n+1)-dimensional space, where that labeling just is given by the respective all-integer coordinates. In fact this representation then is nothing but a sqrt(2)-scaled version of otSn when being used as one of the facets of the also sqrt(2)-scaled otSn+1.
It further should be mentioned that otSn generally is also the Voronoi cell of the root lattice An*. It therefore always allows for a noble periodic continuation as a euclidean honeycomb, the Voronoi complex V(An*) = x3x3x3...x3*a.
Dimension | 1D | 2D | 3D | 4D | 5D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
x |
x3x |
x3x3x |
x3x3x3x |
x3x3x3x3x |
x3x...x3x |
Acronym |
dyad |
hig |
toe |
gippid |
gocad | omnitr. n-simplex |
Vertex Count | 2 | 6 | 24 | 120 | 720 | (n+1)! |
Facet Count wrt. type 1 | 2 vertex | 3 line | 4 hig | 5 toe | 6 gippid |
n+1 (n+1)!/[n! 1!] |
Facet Count wrt. type 2 | 3 line | 6 square | 10 hip | 15 tope |
n(n+1)/2 (n+1)!/[(n-1)! 2!] | |
Facet Count wrt. type 3 | 4 hig | 10 hip | 20 hiddip | (n+1)!/[(n-2)! 3!] | ||
Facet Count wrt. type 4 | 5 toe | 15 tope | (n+1)!/[(n-3)! 4!] | |||
Facet Count wrt. type 5 | 6 gippid | (n+1)!/[(n-4)! 5!] | ||||
Circumradius |
1/2 0.5 | 1 |
sqrt(5/2) 1.581139 |
sqrt(5) 2.236068 |
sqrt(35)/2 2.958040 | sqrt[(n+2)!/((n-1)! 4!)] |
Inradius wrt. facet type 1 |
1/2 0.5 |
sqrt(3)/2 0.866025 |
sqrt(3/2) 1.224745 |
sqrt(5/2) 1.581139 |
sqrt(15)/2 1.936492 | sqrt[(n2+n)/8] |
Inradius wrt. facet type 2 |
sqrt(3)/2 0.866025 |
sqrt(2) 1.414214 |
sqrt(15)/2 1.936492 |
sqrt(6) 2.449490 | sqrt(n2-1)/2 | |
Inradius wrt. facet type 3 |
sqrt(3/2) 1.224745 |
sqrt(15)/2 1.936492 |
sqrt(27)/2 2.598076 | sqrt[3(n2-n-2)/8] | ||
Inradius wrt. facet type 4 |
sqrt(5/2) 1.581139 |
sqrt(6) 2.449490 | sqrt[(n2-2n-3)/2] | |||
Inradius wrt. facet type 5 |
sqrt(15)/2 1.936492 | sqrt[5(n2-3n-4)/8] | ||||
Volume | 1 |
3 sqrt(3)/2 2.598076 |
8 sqrt(2) 11.313708 |
125 sqrt(5)/4 69.877124 |
324 sqrt(3) 561.184462 | (n+1)n-1 sqrt[(n+1)/(2n)] |
Surface | 2 | 6 |
6+12 sqrt(3) 26.784610 | ? | ? | ? |
Dihedral angles types 1 - 2 | 120° |
arccos[-1/sqrt(3)] 125.264390° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-sqrt(2/5)] 129.231520° | arccos[-sqrt((n-1)/2n)] | |
Dihedral angles types 1 - 3 |
arccos(-1/3) 109.471221° |
arccos[-1/sqrt(6)] 114.094843° |
arccos[-1/sqrt(5)] 116.565051° | ? | ||
Dihedral angles types 1 - 4 |
arccos(-1/4) 104.477512° |
arccos[-1/sqrt(10)] 108.434949° | ? | |||
Dihedral angles types 1 - 5 |
arccos(-1/5) 101.536959° | ? | ||||
Dihedral angles types 2 - 3 |
arccos[-1/sqrt(3)] 125.264390° |
arccos(-2/3) 131.810315° | 135° | arccos[-sqrt([2(n-2)]/[3(n-1)])] | ||
Dihedral angles types 2 - 4 |
arccos[-1/sqrt(6)] 114.094843° | 120° | ? | |||
Dihedral angles types 2 - 5 |
arccos[-1/sqrt(10)] 108.434949° | ? | ||||
Dihedral angles types 3 - 4 |
arccos[-sqrt(3/8)] 127.761244° | 135° | arccos[-sqrt([3(n-3)]/[4(n-2)])] | |||
Dihedral angles types 3 - 5 |
arccos[-1/sqrt(5)] 116.565051° | ? | ||||
Dihedral angles types 4 - 5 |
arccos[-sqrt(2/5)] 129.231520° | arccos[-sqrt([4(n-4)]/[5(n-3)])] | ||||
Dimension | 6D | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
x3x3x3x3x3x |
x3x3x3x3x3x3x |
x3x3x3x3x3x3x3x |
x3x3x3x3x3x3x3x3x |
x3x3x3x3x3x3x3x3x3x |
x3x...x3x |
Acronym |
gotaf |
guph |
goxeb |
? |
? | omnitr. n-simplex |
Vertex Count | 5040 | 40320 | 362880 | 3628800 | 39916800 | (n+1)! |
Facet Count wrt. type 1 | 7 gocad | 8 gotaf | 9 guph | 10 goxeb | 11 ? |
n+1 (n+1)!/[n! 1!] |
Facet Count wrt. type 2 | 21 gippiddip | 28 gocadip | 36 gotafip | 45 guphip | 55 ? |
n(n+1)/2 (n+1)!/[(n-1)! 2!] |
Facet Count wrt. type 3 | 35 hatoe | 56 hagippid | 84 hagocad | 120 hagotaf | 165 haguph | (n+1)!/[(n-2)! 3!] |
Facet Count wrt. type 4 | 35 hatoe | 70 toedip | 126 toegippid | 210 toegocad | 330 toegotaf | (n+1)!/[(n-3)! 4!] |
Facet Count wrt. type 5 | 21 gippiddip | 56 hagippid | 126 toegippid | 252 ? | 462 ? | (n+1)!/[(n-4)! 5!] |
Facet Count wrt. type 6 | 7 gocad | 28 gocadip | 84 hagocad | 210 toegocad | 462 ? | (n+1)!/[(n-5)! 6!] |
Facet Count wrt. type 7 | 8 gotaf | 36 gotafip | 120 hagotaf | 330 toegotaf | (n+1)!/[(n-6)! 7!] | |
Facet Count wrt. type 8 | 9 guph | 45 guphip | 165 haguph | (n+1)!/[(n-7)! 8!] | ||
Facet Count wrt. type 9 | 10 goxeb | 55 ? | (n+1)!/[(n-8)! 9!] | |||
Facet Count wrt. type 10 | 11 ? | (n+1)!/[(n-9)! 10!] | ||||
Circumradius |
sqrt(14) 3.741657 |
sqrt(21) 4.582576 |
sqrt(30) 5.477226 |
sqrt(165)/2 6.422616 |
sqrt(55) 7.416198 | sqrt[(n+2)!/((n-1)! 4!)] |
Inradius wrt. facet type 1 |
sqrt(21)/2 2.291288 |
sqrt(7) 2.645751 | 3 |
3 sqrt(5)/2 3.354102 |
sqrt(55)/2 3.708099 | sqrt[(n2+n)/8] |
Inradius wrt. facet type 2 |
sqrt(35)/2 2.958040 |
2 sqrt(3) 3.464102 |
sqrt(63)/2 3.968627 |
2 sqrt(5) 4.472136 |
3 sqrt(11)/2 4.974937 | sqrt(n2-1)/2 |
Inradius wrt. facet type 3 |
sqrt(21/2) 3.240370 |
sqrt(15) 3.872983 |
9/2 4.5 |
sqrt(105)/2 5.123475 |
sqrt(33) 5.744563 | sqrt[3(n2-n-2)/8] |
Inradius wrt. facet type 4 |
sqrt(21/2) 3.240370 | 4 |
3 sqrt(5/2) 4.743416 |
sqrt(30) 5.477226 |
sqrt(77/2) 6.204837 | sqrt[(n2-2n-3)/2] |
Inradius wrt. facet type 5 |
sqrt(35)/2 2.958040 |
sqrt(15) 3.872983 |
3 sqrt(5/2) 4.743416 |
5 sqrt(5)/2 5.590170 |
sqrt(165)/2 6.422616 | sqrt[5(n2-3n-4)/8] |
Inradius wrt. facet type 6 |
sqrt(21)/2 2.291288 |
2 sqrt(3) 3.464102 |
9/2 4.5 |
sqrt(30) 5.477226 |
sqrt(165)/2 6.422616 | sqrt[3(n2-4n-5)]/2 |
Inradius wrt. facet type 7 |
sqrt(7) 2.645751 |
sqrt(63)/2 3.968627 |
sqrt(105)/2 5.123475 |
sqrt(77/2) 6.204837 | sqrt[7(n2-5n-6)/8] | |
Inradius wrt. facet type 8 | 3 |
2 sqrt(5) 4.472136 |
sqrt(33) 5.744563 | sqrt(n2-6n-7) | ||
Inradius wrt. facet type 9 |
3 sqrt(5)/2 3.354102 |
3 sqrt(11)/2 4.974937 | 3 sqrt[(n2-7n-8)/8] | |||
Inradius wrt. facet type 10 |
sqrt(55)/2 3.708099 | sqrt[5(n2-8n-9)]/2 | ||||
Volume |
16807 sqrt(7)/8 5558.392786 | 65536 |
14348907/16 896806.6875 | ? | ? | (n+1)n-1 sqrt[(n+1)/(2n)] |
Surface | ? | ? | ? | ? | ? | ? |
Dihedral angles types 1 - 2 |
arccos[-sqrt(5/12)] 130.202966° |
arccos[-sqrt(3/7)] 130.893395° |
arccos[-sqrt(7)/4] 131.409622° |
arccos(-2/3) 131.810315° |
arccos[-3/sqrt(20)] 132.130415° | arccos[-sqrt((n-1)/2n)] |
Dihedral angles types 1 - 3 | ? | ? | ? | ? | ? | ? |
Dihedral angles types 1 - 4 | ? | ? | ? | ? | ? | ? |
Dihedral angles types 1 - 5 | ? | ? | ? | ? | ? | ? |
Dihedral angles types 1 - 6 | ? | ? | ? | ? | ? | ? |
Dihedral angles types 1 - 7 | ? | ? | ? | ? | ? | |
Dihedral angles types 1 - 8 | ? | ? | ? | ? | ||
Dihedral angles types 1 - 9 | ? | ? | ? | |||
Dihedral angles types 1 - 10 | ? | ? | ||||
Dihedral angles types 2 - 3 |
arccos[-sqrt(8/15)] 136.911277° |
arccos[-sqrt(5)/3] 138.189685° |
arccos[-2/sqrt(7)] 139.106605° |
arccos[-sqrt(7/12)] 139.797034° |
arccos[-4/sqrt(27)] 140.335965° | arccos[-sqrt([2(n-2)]/[3(n-1)])] |
Dihedral angles types 2 - 4 | ? | ? | ? | ? | ? | ? |
Dihedral angles types 2 - 5 | ? | ? | ? | ? | ? | ? |
Dihedral angles types 2 - 6 | ? | ? | ? | ? | ? | ? |
Dihedral angles types 2 - 7 | ? | ? | ? | ? | ? | |
Dihedral angles types 2 - 8 | ? | ? | ? | ? | ||
Dihedral angles types 2 - 9 | ? | ? | ? | |||
Dihedral angles types 2 - 10 | ? | ? | ||||
Dihedral angles types 3 - 4 |
arccos(-3/4) 138.590378° |
arccos[-sqrt(3/5)] 140.768480° |
arccos[-sqrt(5/8)] 142.238756° |
arccos[-3/sqrt(14)] 143.300775° |
arccos[-sqrt(21/32)] 144.104978° | arccos[-sqrt([3(n-3)]/[4(n-2)])] |
Dihedral angles types 3 - 5 | ? | ? | ? | ? | ? | ? |
Dihedral angles types 3 - 6 | ? | ? | ? | ? | ? | ? |
Dihedral angles types 3 - 7 | ? | ? | ? | ? | ? | |
Dihedral angles types 3 - 8 | ? | ? | ? | ? | ||
Dihedral angles types 3 - 9 | ? | ? | ? | |||
Dihedral angles types 3 - 10 | ? | ? | ||||
Dihedral angles types 4 - 5 |
arccos[-sqrt(8/15)] 136.911277° |
arccos[-sqrt(3/5)] 140.768480° |
arccos(-4/5) 143.130102° |
arccos[-sqrt(2/3)] 144.735610° |
arccos[-sqrt(24/35)] 145.901874° | arccos[-sqrt([4(n-4)]/[5(n-3)])] |
Dihedral angles types 4 - 6 | ? | ? | ? | ? | ? | ? |
Dihedral angles types 4 - 7 | ? | ? | ? | ? | ? | |
Dihedral angles types 4 - 8 | ? | ? | ? | ? | ||
Dihedral angles types 4 - 9 | ? | ? | ? | |||
Dihedral angles types 4 - 10 | ? | ? | ||||
Dihedral angles types 5 - 6 |
arccos[-sqrt(5/12)] 130.202966° |
arccos[-sqrt(5)/3] 138.189685° |
arccos[-sqrt(5/8)] 142.238756° |
arccos[-sqrt(2/3)] 144.735610° |
arccos(-5/6) 146.442690° | arccos[-sqrt([5(n-5)]/[6(n-4)])] |
Dihedral angles types 5 - 7 | ? | ? | ? | ? | ? | |
Dihedral angles types 5 - 8 | ? | ? | ? | ? | ||
Dihedral angles types 5 - 9 | ? | ? | ? | |||
Dihedral angles types 5 - 10 | ? | ? | ||||
Dihedral angles types 6 - 7 |
arccos[-sqrt(3/7)] 130.893395° |
arccos[-2/sqrt(7)] 139.106605° |
arccos[-3/sqrt(14)] 143.300775° |
arccos[-sqrt(24/35)] 145.901874° | arccos[-sqrt([6(n-6)]/[7(n-5)])] | |
Dihedral angles types 6 - 8 | ? | ? | ? | ? | ||
Dihedral angles types 6 - 9 | ? | ? | ? | |||
Dihedral angles types 6 - 10 | ? | ? | ||||
Dihedral angles types 7 - 8 |
arccos[-sqrt(7)/4] 131.409622° |
arccos[-sqrt(7/12)] 139.797034° |
arccos[-sqrt(21/32)] 144.104978° | arccos[-sqrt([7(n-7)]/[8(n-6)])] | ||
Dihedral angles types 7 - 9 | ? | ? | ? | |||
Dihedral angles types 7 - 10 | ? | ? | ||||
Dihedral angles types 8 - 9 |
arccos(-2/3) 131.810315° |
arccos[-4/sqrt(27)] 140.335965° | arccos[-sqrt([8(n-8)]/[9(n-7)])] | |||
Dihedral angles types 8 - 10 | ? | ? | ||||
Dihedral angles types 9 - 10 |
arccos[-3/sqrt(20)] 132.130415° | arccos[-sqrt([9(n-9)]/[10(n-8)])] |
These polytopes are closely related to the tegum product. In fact On here is nothing but the On-1 bipyramid. Thence, by means of the tegum sum notation, On = x3o...o3o4o (n nodes) can be described as well as qo ox3oo...oo3oo4oo&#zx (n node positions).
On the other hand these polytopes On generally can also be described as the segmentotope of the regular simplex Sn-1 atop the dual simplex -Sn-1. Thence, by means of the lace prism notation, On = x3o...o3o4o (n nodes) can be described as well as xo3oo...oo3ox&#x (n-1 node positions).
The regular Orthoplex On generally is the dual of the regular hypercube Cn.
Dimension | 1D | 2D | 3D | 4D | 5D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
q |
x4o |
x3o4o |
x3o3o4o |
x3o3o3o4o |
x3o...o3o4o |
Acronym |
q-line |
square |
oct |
hex |
tac | n-orthoplex |
Vertex Count | 2 | 4 q-line | 6 square | 8 oct | 10 hex | 2n |
Facet Count | 4 line | 8 trig | 16 tet | 32 pen | 2n | |
Circumradius |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
Inradius |
1/sqrt(2) 0.707107 |
1/2 0.5 |
1/sqrt(6) 0.408248 |
1/sqrt(8) 0.353553 |
1/sqrt(10) 0.316228 | 1/sqrt(2n) |
Volume |
sqrt(2) 1.414214 | 1 |
sqrt(2)/3 0.471405 |
1/6 0.166667 |
sqrt(2)/30 0.047140 | sqrt(2n)/n! |
Surface | 2 | 4 |
2 sqrt(3) 3.464102 |
4 sqrt(2)/3 1.885618 |
sqrt(5)/3 0.745356 | 2 sqrt[2n-1 n]/(n-1)! |
Dihedral angles | 0° | 90° |
arccos(-1/3) 109.471221° | 120° |
arccos(-3/5) 126.869898° | arccos(2/n - 1) |
Dimension | 6D | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
x3o3o3o3o4o |
x3o3o3o3o3o4o |
x3o3o3o3o3o3o4o |
x3o3o3o3o3o3o3o4o |
x3o3o3o3o3o3o3o3o4o |
x3o...o3o4o |
Acronym |
gee |
zee |
ek |
vee |
ka | n-simplex |
Vertex Count | 12 tac | 14 gee | 16 zee | 18 ek | 20 vee | 2n |
Facet Count | 64 hix | 128 hop | 256 oca | 512 ene | 1024 day | 2n |
Circumradius |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
Inradius |
1/sqrt(12) 0.288675 |
1/sqrt(14) 0.267261 |
1/4 0.25 |
1/sqrt(18) 0.235702 |
1/sqrt(20) 0.223607 | 1/sqrt(2n) |
Volume |
1/90 0.011111 |
sqrt(2)/630 0.0022448 |
1/2520 0.00039683 |
sqrt(2)/22680 0.000062355 |
1/113400 0.0000088183 | sqrt(2n)/n! |
Surface |
2 sqrt(3)/15 0.230940 |
sqrt(7)/45 0.058794 |
4/315 0.012698 |
1/420 0.0023810 |
sqrt(5)/5670 0.00039437 | 2 sqrt[2n-1 n]/(n-1)! |
Dihedral angles |
arccos(-2/3) 131.810315° |
arccos(-5/7) 135.584691° |
arccos(-3/4) 138.590378° |
arccos(-7/9) 141.057559° |
arccos(-4/5) 143.130102° | arccos(2/n - 1) |
Interestingly this class belongs to an even wider class of (then mostly hyperbolic) polytopes which all have that common property that the nD (or rather: rank n) representant occurs as ridge faceting midsection within the (n+1)D case (for the finite cases) resp. as a ridge faceting subspace within the rank n+1 case (for the infinite cases). This then is the general regular class of the regular xPo3o...o3o4o. Within this class it happens moreover generally that this subspace additionally acts as a true mirror of symmetry. Below is a small enlisting thereof.
xPo3o...o3o4o | |||
P = 3 | P = 4 | P = 5 | P = 6 |
---|---|---|---|
r = 1/sqrt(2) = 0.707107
x3o4o - oct x3o3o4o - hex x3o3o3o4o - tac x3o3o3o3o4o - gee x3o3o3o3o3o4o - zee x3o3o3o3o3o3o4o - ek x3o3o3o3o3o3o3o4o - vee x3o3o3o3o3o3o3o3o4o - ka ... |
r = ∞
x4o4o - squat x4o3o4o - chon x4o3o3o4o - test x4o3o3o3o4o - penth x4o3o3o3o3o4o - axh x4o3o3o3o3o3o4o - hepth ... |
r = sqrt[-1-sqrt(5)]/2 = 0.899454 i
x5o4o - peat x5o3o4o - doehon x5o3o3o4o - shitte ... |
r = 1/sqrt(-2) = 0.707107 i
x6o4o - shexat x6o3o4o - shexah ... |
These polytopes are closely related to the prism product. In fact Cn generally can be described as the Cn-1-prism, i.e. the segmentotope of the regular hypercube Cn-1 atop the (identical) hypercube Cn-1. Thence, by means of the lace prism notation, Cn = o3o...o3o4x (n nodes) can be described as well as oo3oo...oo3oo4xx&#x (n-1 node positions).
The regular hypercube Cn generally is the dual of the regular orthoplex On.
Dimension | 1D | 2D | 3D | 4D | 5D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
x |
o4x |
o3o4x |
o3o3o4x |
o3o3o3o4x |
o3o...o3o4x |
Acronym |
line |
square |
cube |
tes |
pent | n-hypercube |
Vertex Count | 2 | 4 q-line | 8 q-trig | 16 q-tet | 32 q-pen | 2n |
Facet Count | 4 line | 6 square | 8 cube | 10 tes | 2n | |
Circumradius |
1/2 0.5 |
1/sqrt(2) 0.707107 |
sqrt(3)/2 0.866025 | 1 |
sqrt(5)/2 1.118034 | sqrt(n)/2 |
Inradius |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
Volume | 1 | 1 | 1 | 1 | 1 | 1 |
Surface | 2 | 4 | 6 | 8 | 10 | 2n |
Dihedral angles | 0° | 90° | 90° | 90° | 90° | 90° |
Dimension | 6D | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3o3o3o3o4x |
o3o3o3o3o3o4x |
o3o3o3o3o3o3o4x |
o3o3o3o3o3o3o3o4x |
o3o3o3o3o3o3o3o3o4x |
o3o...o3o4x |
Acronym |
ax |
hept |
octo |
enne |
deker | n-hypercube |
Vertex Count | 64 q-hix | 128 q-hop | 256 q-oca | 512 q-ene | 1024 q-day | 2n |
Facet Count | 12 pent | 14 ax | 16 hept | 18 octo | 20 enne | 2n |
Circumradius |
sqrt(3/2) 1.224745 |
sqrt(7)/2 1.322876 |
sqrt(2) 1.414214 |
3/2 1.5 |
sqrt(5/2) 1.581139 | sqrt(n)/2 |
Inradius |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
Volume | 1 | 1 | 1 | 1 | 1 | 1 |
Surface | 12 | 14 | 16 | 18 | 20 | 2n |
Dihedral angles | 90° | 90° | 90° | 90° | 90° | 90° |
The common unit circumradius of all these shows that they occur as vertex figure of an according dimensional honeycomb. In fact they are the hull-of-large-roots polytopes of the according dimensional root lattice Cn (or equivalently the hull-of-small-roots polytopes of the according dimensional root lattice Bn). Furthermore it forces that the facet-to-bodycenter pyramids all are CRF, i.e. that all these polytopes can be decomposed accordingly.
Within these polytopes rOn generally can be described as the bistratic lace tower of the regular orthoplex On-1 atop the rectified orthoplex rOn-1 atop the regular orthoplex On-1. Thence, by means of the lace tower notation, rOn = o3x3o...o3o4o (n nodes) can be described as well as xox3oxo3ooo...ooo3ooo4ooo&#xt (n-1 node positions).
On the other hand these polytopes rOn generally can also be described within a different orientation as the bistratic lace tower of the rectified simplex rSn-1 atop the maximal-expanded simplex eSn-1 atop the inverted rectified simplex -rSn-1. Thence, by means of the lace tower notation, rOn = o3x3o...o3o4o (n nodes) can be described as well as oxo3xoo3ooo...ooo3oox3oxo&#xt (n-1 node positions). As the according midsection therefore generally is eSn-1, and those polytopes already where mentioned to have this unit circumradius property, it becomes apparent that this property here applies as well.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
o3x4o |
o3x3o4o |
o3x3o3o4o |
o3x3o3o3o4o |
o3x3o...o3o4o |
Acronym |
co |
ico |
rat |
rag | rect. n-orthoplex |
Vertex Count | 12 x2q | 24 cube | 40 ope | 60 hexip | 2n(n-1) |
Facet Count rect. facets | 8 trig | 16 oct | 32 rap | 64 rix | 2n |
Facet Count verf facets | 6 square | 8 oct | 10 hex | 12 tac | 2n |
Circumradius | 1 | 1 | 1 | 1 | 1 |
Inradius wrt. rect. facets |
sqrt(2/3) 0.816497 |
1/sqrt(2) 0.707107 |
sqrt(2/5) 0.632456 |
1/sqrt(3) 0.577350 | sqrt(2/n) |
Inradius wrt. verf facets |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
Volume |
5 sqrt(2)/3 2.357023 | 2 |
9 sqrt(2)/10 1.272792 |
29/45 0.644444 | (2n-n) sqrt(2n)/n! |
Surface |
6+2 sqrt(3) 9.464102 |
8 sqrt(2) 11.313708 |
(5+11 sqrt(5))/3 9.865583 |
(6 sqrt(2)+52 sqrt(3))/15 6.570128 | 2 [n+(2n-1-n) sqrt(n)] sqrt(2n-1)/(n-1)! |
Dihedral angles rect. - orthopl. |
arccos[-1/sqrt(3)] 125.264390° | 120° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(6)] 114.094843° | arccos[-1/sqrt(n)] |
Dihedral angles rect. - rect. |
arccos(-3/5) 126.869898° |
arccos(-2/3) 131.810315° | arccos(2/n - 1) | ||
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3x3o3o3o3o4o |
o3x3o3o3o3o3o4o |
o3x3o3o3o3o3o3o4o |
o3x3o3o3o3o3o3o3o4o |
o3x3o...o3o4o |
Acronym |
rez |
rek |
riv |
rake | rect. n-orthoplex |
Vertex Count | 84 taccup | 112 geep | 144 zeep | 180 ekip | 2n(n-1) |
Facet Count rect. facets | 128 ril | 256 roc | 512 rene | 1024 reday | 2n |
Facet Count verf facets | 14 gee | 16 zee | 18 ek | 20 vee | 2n |
Circumradius | 1 | 1 | 1 | 1 | 1 |
Inradius wrt. rect. facets |
sqrt(2/7) 0.534522 |
1/2 0.5 |
sqrt(2)/3 0.471405 |
1/sqrt(5) 0.447214 | sqrt(2/n) |
Inradius wrt. verf facets |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
Volume |
121 sqrt(2)/630 0.271619 |
31/315 0.098413 |
503 sqrt(2)/22680 0.031365 |
169/18900 0.0089418 | (2n-n) sqrt(2n)/n! |
Surface |
(7+57 sqrt(7))/45 3.506841 |
(480+8 sqrt(2))/315 1.559726 |
25/42 0.595238 |
[5 sqrt(2)+502 sqrt(5)]/5760 0.199220 | 2 [n+(2n-1-n) sqrt(n)] sqrt(2n-1)/(n-1)! |
Dihedral angles rect. - orthopl. |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos(-1/3) 109.471221° |
arccos[-1/sqrt(10)] 108.434949° | arccos[-1/sqrt(n)] |
Dihedral angles rect. - rect. |
arccos(-5/7) 135.584691° |
arccos(-3/4) 138.590378° |
arccos(-7/9) 141.057559° |
arccos(-4/5) 143.130102° | arccos(2/n - 1) |
Within these polytopes rCn generally can be described as the bistratic lace tower of the rectified hypercube rCn-1 atop the q-scaled hypercube Cn-1 atop the (alike oriented) rectified hypercube rCn-1. Thence, by means of the lace tower notation, rCn = o3o...o3x4o (n nodes) can be described as well as ooo3ooo...ooo3xox4oqo&#xt (n-1 node positions).
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
o3x4o |
o3o3x4o |
o3o3o3x4o |
o3o3o3o3x4o |
o3o...o3x4o |
Acronym |
co |
rit |
rin |
rax | rect. n-hypercube |
Vertex Count | 12 x q | 32 o3x q | 80 o3o3x q | 192 o3o3o3x q | n 2n-1 |
Facet Count rect. facets | 6 square | 8 co | 10 rit | 12 rin | 2n |
Facet Count verf facets | 8 trig | 16 tet | 32 pen | 64 hix | 2n |
Circumradius | 1 |
sqrt(3/2) 1.224745 |
sqrt(2) 1.414214 |
sqrt(5/2) 1.581139 | sqrt[(n-1)/2] |
Inradius wrt. rect. facets |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
Inradius wrt. verf facets |
sqrt(2/3) 0.816497 |
3/sqrt(8) 1.060660 |
sqrt(8/5) 1.264911 |
5/sqrt(12) 1.443376 | (n-1)/sqrt(2n) |
Volume |
5 sqrt(2)/3 2.357023 |
23/6 3.833333 |
119 sqrt(2)/30 5.609714 |
719/90 7.988889 | (n!-1) sqrt(2n)/n! |
Surface |
6+2 sqrt(3) 9.464102 |
44 sqrt(2)/3 20.741799 |
(115+sqrt(5))/3 39.078689 |
(714 sqrt(2)+2 sqrt(3))/15 67.547506 | [n!-n+sqrt(n)] sqrt(2n+1)/(n-1)! |
Dihedral angles rect. - simplex |
arccos[-1/sqrt(3)] 125.264390° | 120° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(6)] 114.094843° | arccos[-1/sqrt(n)] |
Dihedral angles rect. - rect. | 90° | 90° | 90° | 90° | |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3o3o3o3o3x4o |
o3o3o3o3o3o3x4o |
o3o3o3o3o3o3o3x4o |
o3o3o3o3o3o3o3o3x4o |
o3o...o3x4o |
Acronym |
rasa |
recto |
ren |
rade | rect. n-hypercube |
Vertex Count | 448 | 1024 | 2304 | 5120 | n 2n-1 |
Facet Count rect. facets | 14 rax | 16 rasa | 18 recto | 20 ren | 2n |
Facet Count verf facets | 128 hop | 256 oca | 512 ene | 1024 day | 2n |
Circumradius |
sqrt(3) 1.732051 |
sqrt(7/2) 1.870829 | 2 |
3/sqrt(2) 2.121320 | sqrt[(n-1)/2] |
Inradius wrt. rect. facets |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
Inradius wrt. verf facets |
6/sqrt(14) 1.603567 |
7/4 1.75 |
8/sqrt(18) 1.885618 |
9/sqrt(20) 2.012461 | (n-1)/sqrt(2n) |
Volume |
5039 sqrt(2)/630 11.311464 |
40319/2520 15.999603 |
362879 sqrt(2)/22680 22.627355 |
3628799/113400 31.999991 | (n!-1) sqrt(2n)/n! |
Surface |
(5033+sqrt(7)/45 111.903239 |
(4+40312 sqrt(2))/315 180.996118 |
60479/210 287.995238 |
(1814395 sqrt(2)+sqrt(5))/5670 452.547487 | [n!-n+sqrt(n)] sqrt(2n+1)/(n-1)! |
Dihedral angles rect. - orthopl. |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos(-1/3) 109.471221° |
arccos[-1/sqrt(10)] 108.434949° | arccos[-1/sqrt(n)] |
Dihedral angles rect. - rect. | 90° | 90° | 90° | 90° | 90° |
These non-convex polytopes frCn generally are facetings of the rectified hypercube rCn.
Facets here always come within pairs – except for the hemifacets, which occur for the odd dimensional series members. Subsequent ones always alternate between prograde and retrograde. Thence for these odd dimensional series members the volume always results in zero, as the facet pyramids of those hemifacets clearly are degenerate, while the other ones cancel out by means of those pairings, then using a prograde and a retrograde base respectively. For the even dimensional series members however, due to the missing hemifacets, those pairings will be either both pro- or both retrograde.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
x3x3/2o3*a |
x3x3/2o3o3*a |
x3x3/2o3o3o3*a |
x3x3/2o3o3o3o3*a |
x3x3/2o3o...o3*a |
Acronym |
oho |
firt |
firn |
forx | facetorect. n-hyp.c. |
Vertex Count | 12 | 32 | 80 | 192 | n 2n-1 |
Facet Count simplex | 4+4 trig | 8+8 tet | 16+16 pen | 32+32 hix | 2n-1 (each) |
Facet Count trunc. simp. | 4 hig | 8+8 tut | 16+16 tip | 32+32 tix | 2n-1 (each) |
Facet Count bitrunc. simp. | 16 deca | 32+32 bittix | 2n-1 (each) | ||
Circumradius | 1 |
sqrt(3/2) 1.224745 |
sqrt(2) 1.414214 |
sqrt(5/2) 1.581139 | sqrt[(n-1)/2] |
Inradius wrt. simplex |
+/− sqrt(2/3) 0.816497 |
−/− 3/sqrt(8) 1.060660 |
+/− sqrt(8/5) 1.264911 |
+/+ 5/sqrt(12) 1.443376 | (n-1)/sqrt(2n) |
Inradius wrt. trunc. simp. | 0 |
+/+ 1/sqrt(8) 0.353553 |
−/+ sqrt(2/5) 0.632456 |
−/− sqrt(3)/2 0.866025 | (n-3)/sqrt(2n) |
Inradius wrt. bitrunc. simp. | 0 |
+/+ 1/sqrt(12) 0.288675 | (n-5)/sqrt(2n) | ||
Volume | 0 |
10/3 3.333333 | 0 |
488/45 10.844444 | 0 / ? |
Surface |
8 sqrt(3) 13.856406 |
32 sqrt(2) 45.254834 |
64 sqrt(5) 143.108351 |
256 sqrt(3) 443.405007 | sqrt(n 8n-1) |
Dihedral angles sim. - trunc.sim. |
arccos(1/3) 70.528779° | 60° |
arccos(3/5) 53.130102° |
arccos(2/3) 48.189685° | arccos[(n-2)/n] |
Dihedral angles tr.sim. - bitr.sim. |
arccos(3/5) 53.130102° |
arccos(2/3) 48.189685° | arccos[(n-2)/n] | ||
Dihedral angles k-tr.s. - k-tr.s. | 60° |
arccos(2/3) 48.189685° |
arccos[(n-2)/n] n 2(k+1) | ||
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
x3x3/2o3o3o3o3o3*a |
x3x3/2o3o3o3o3o3o3*a |
x3x3/2o3o3o3o3o3o3o3*a |
x3x3/2o3o3o3o3o3o3o3o3*a |
x3x3/2o3o...o3*a |
Acronym |
frasa |
fro |
fren |
frade | facetorect. n-hyp.c. |
Vertex Count | 448 | 1024 | 2304 | 5120 | n 2n-1 |
Facet Count simplex | 64+64 hop | 128+128 oca | 256+256 ene | 512+512 day | 2n-1 (each) |
Facet Count trunc. simp. | 64+64 til | 128+128 toc | 256+256 tene | 512+512 teday | 2n-1 (each) |
Facet Count bitrunc. simp. | 64+64 batal | 128+128 bittoc | 256+256 batene | 512+512 biteday (?) | 2n-1 (each) |
Facet Count tritrunc. simp. | 64 fe | 128+128 tattoc | 256+256 tatene | 512+512 tatday (?) | 2n-1 (each) |
Facet Count quadritr. simp. | 256 be | 512+512 quatday (?) | 2n-1 (each) | ||
Circumradius |
sqrt(3) 1.732051 |
sqrt(7/2) 1.870829 | 2 |
3/sqrt(2) 2.121320 | sqrt[(n-1)/2] |
Inradius wrt. simplex |
+/− sqrt(18/7) 1.603567 |
−/− 7/4 1.75 |
+/− sqrt(32)/3 1.885618 |
+/+ 9/sqrt(20) 2.012461 | (n-1)/sqrt(2n) |
Inradius wrt. trunc. simpl. |
−/+ sqrt(8/7) 1.069045 |
+/+ 5/4 1.25 |
−/+ sqrt(2) 1.414214 |
−/− 7/sqrt(20) 1.565248 | (n-3)/sqrt(2n) |
Inradius wrt. bitrunc. simpl. |
+/− sqrt(2/7) 0.534522 |
−/− 3/4 0.75 |
+/− sqrt(8)/3 0.942809 |
+/+ sqrt(5)/2 1.118034 | (n-5)/sqrt(2n) |
Inradius wrt. tritrunc. simpl. | 0 |
+/+ 1/4 0.25 |
−/+ sqrt(2)/3 0.471405 |
−/− 3/sqrt(20) 0.670820 | (n-7)/sqrt(2n) |
Inradius wrt. quadritr. simpl. | 0 |
+/+ 1/sqrt(20) 0.223607 | (n-9)/sqrt(2n) | ||
Volume | 0 | ? | 0 | ? | 0 / ? |
Surface |
512 sqrt(7) 1354.624671 | 4096 | 12288 |
16384 sqrt(5) 36635.737743 | sqrt(n 8n-1) |
Dihedral angles sim. - trunc.sim. |
arccos(5/7) 44.415309° |
arccos(3/4) 41.409622° |
arccos(7/9) 38.942441° |
arccos(4/5) 36.869898° | arccos[(n-2)/n] |
Dihedral angles tr.sim. - bitr.sim. |
arccos(5/7) 44.415309° |
arccos(3/4) 41.409622° |
arccos(7/9) 38.942441° |
arccos(4/5) 36.869898° | arccos[(n-2)/n] |
Dihedral angles bitr.sim. - tritr.sim. |
arccos(5/7) 44.415309° |
arccos(3/4) 41.409622° |
arccos(7/9) 38.942441° |
arccos(4/5) 36.869898° | arccos[(n-2)/n] |
Dihedral angles tritr.s. - quadrit.s. |
arccos(7/9) 38.942441° |
arccos(4/5) 36.869898° | arccos[(n-2)/n] | ||
Dihedral angles k-tr.s. - k-tr.s. |
arccos(3/4) 41.409622° |
arccos(4/5) 36.869898° |
arccos[(n-2)/n] n 2(k+1) |
Within these polytopes brOn generally can be described as the bistratic lace tower of the rectified orthoplex rOn-1 atop the birectified orthoplex brOn-1 atop the rectified orthoplex rOn-1. Thence, by means of the lace tower notation, brOn = o3o3x3o...o3o4o (n nodes) can be described as well as ooo3xox3oxo3ooo...ooo3ooo4ooo&#xt (n-1 node positions).
On the other hand these polytopes brOn generally can also be described within a different orientation as a tristratic lace tower oooo3oxoo3ooxo3ooox3oooo...oooo3xooo3oxoo3ooxo3oooo&#xt (n-1 node positions), where the right hand decorations and the lefthand decorations for the smaller dimensions well might interlace, or in the extremal 3D case even overlay and run out of the other end: ouoo3oouo&#xt.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
o3o4q |
o3o3x4o |
o3o3x3o4o |
o3o3x3o3o4o |
o3o3x3o...o3o4o |
Acronym |
q-cube |
rit |
nit |
brag | birect. n-orthoplex |
Vertex Count | 8 u-trig | 32 o3x q | 80 tisdip | 160 troct | 4n(n-1)(n-2)/3 |
Facet Count birect. facets | 16 tet | 32 rap | 64 dot | 2n | |
Facet Count rect. facets | 6 q-square | 8 co | 10 ico | 12 rat | 2n |
Circumradius |
sqrt(3/2) 1.224745 |
sqrt(3/2) 1.224745 |
sqrt(3/2) 1.224745 |
sqrt(3/2) 1.224745 |
sqrt(3/2) 1.224745 |
Inradius wrt. birect. facets |
3/sqrt(8) 1.060660 |
3/sqrt(10) 0.948683 |
sqrt(3)/2 0.866025 | 3/sqrt(2n) | |
Inradius wrt. rect. facets |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
Volume |
sqrt(8) 2.828427 |
23/6 3.833333 |
31 sqrt(2)/10 4.384062 | 4 | (3n-n 2n+n(n-1)/2) sqrt(2n)/n! |
Surface | 12 |
44 sqrt(2)/3 20.741799 |
[60+11 sqrt(5)]/3 28.198916 |
(54 sqrt(2)+44 sqrt(3))/5 30.515554 | (3n-1 sqrt(n 2n+1)-n(sqrt(n)-1) sqrt(23n-1)+n(n-1)(sqrt(n)-2) sqrt(2n-1))/(n-1)! |
Dihedral angles birect. - birect. |
arccos(-3/5) 126.869898° |
arccos(-2/3) 131.810315° | arccos(2/n - 1) | ||
Dihedral angles birect. - rect. | 120° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(6)] 114.094843° | arccos[-1/sqrt(n)] | |
Dihedral angles rect. - rect. | 90° | 90° | 90° | 90° | 90° |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3o3x3o3o3o4o |
o3o3x3o3o3o3o4o |
o3o3x3o3o3o3o3o4o |
o3o3x3o3o3o3o3o3o4o |
o3o3x3o...o3o4o |
Acronym |
barz |
bark |
brav |
brake | birect. n-orthoplex |
Vertex Count | 280 trahex | 448 tratac | 672 trigee | 960 trizee | 4n(n-1)(n-2)/3 |
Facet Count rect. facets | 128 bril | 256 broc | 512 brene | 1024 breday | 2n |
Facet Count verf facets | 14 rag | 16 rez | 18 rek | 20 riv | 2n |
Circumradius |
sqrt(3/2) 1.224745 |
sqrt(3/2) 1.224745 |
sqrt(3/2) 1.224745 |
sqrt(3/2) 1.224745 |
sqrt(3/2) 1.224745 |
Inradius wrt. birect. facets |
3/sqrt(14) 0.801784 |
3/4 0.75 |
1/sqrt(2) 0.707107 |
3/sqrt(20) 0.670820 | 3/sqrt(2n) |
Inradius wrt. rect. facets |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
Volume |
656 sqrt(2)/315 2.945156 |
4541/2520 1.801984 |
1679 sqrt(2)/2520 0.942248 |
24427/56700 0.430811 | (3n-n 2n+n(n-1)/2) sqrt(2n)/n! |
Surface |
(406+302 sqrt(7))/45 26.778153 |
(4764+968 sqrt(2))/315 19.469710 |
1679/140 11.992857 |
(2515 sqrt(2)+14608 sqrt(5))/5670 6.388224 | (3n-1 sqrt(n 2n+1)-n(sqrt(n)-1) sqrt(23n-1)+n(n-1)(sqrt(n)-2) sqrt(2n-1))/(n-1)! |
Dihedral angles birect. - birect. |
arccos(-5/7) 135.584691° |
arccos(-3/4) 138.590378° |
arccos(-7/9) 141.057559° |
arccos(-4/5) 143.130102° | arccos(2/n - 1) |
Dihedral angles birect. - rect. |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos(-1/3) 109.471221° |
arccos[-1/sqrt(10)] 108.434949° | arccos[-1/sqrt(n)] |
Dihedral angles rect. - rect. | 90° | 90° | 90° | 90° | 90° |
Within these polytopes brCn generally can be described as the bistratic lace tower of the birectified hypercube brCn-1 atop the rectified hypercube rCn-1 atop the birectified hypercube brCn-1. Thence, by means of the lace tower notation, brCn = o3o...o3x3o4o (n nodes) can be described as well as ooo3ooo...ooo3xox3oxo4ooo&#xt (n-1 node positions).
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
x3o4o |
o3x3o4o |
o3o3x3o4o |
o3o3o3x3o4o |
o3o...o3x3o4o |
Acronym |
oct |
ico |
nit |
brox | birect. n-hypercube |
Vertex Count | 6 square | 24 cube | 80 tisdip | 240 squatet | n(n-2) 2n-3 |
Facet Count rect. simplex | 8 trig | 16 oct | 32 rap | 64 rix | 2n |
Facet Count birect. h.cube | 8 oct | 10 ico | 12 nit | 2n | |
Circumradius |
1/sqrt(2) 0.707107 | 1 |
sqrt(3/2) 1.224745 |
sqrt(2) 1.414214 | sqrt[(n-2)/2] |
Inradius wrt. rect. simplex |
1/sqrt(6) 0.408248 |
1/sqrt(2) 0.707107 |
3/sqrt(10) 0.948683 |
2/sqrt(3) 1.154701 | (n-2)/sqrt(2n) |
Inradius wrt. birect. h.cube |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 | |
Volume |
sqrt(2)/3 0.471405 | 2 |
31 sqrt(2)/10 4.384062 |
331/45 14.711111 | ? |
Surface |
2 sqrt(3) 3.464102 |
8 sqrt(2) 11.313708 |
[60+11 sqrt(5)]/3 28.198916 |
122 sqrt(2)/3 57.511352 | ? |
Dihedral angles rect. - rect. |
arccos(-1/3) 109.471221° | 120° |
arccos(-3/5) 126.869898° |
arccos(-2/3) 131.810315° | arccos(2/n-1) |
Dihedral angles rect. - birect. | 120° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(6)] 114.094843° | arccos[-1/sqrt(n)] | |
Dihedral angles birect. - birect. | 90° | 90° | 90° | ||
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3o3o3o3x3o4o |
o3o3o3o3o3x3o4o |
o3o3o3o3o3o3x3o4o |
o3o3o3o3o3o3o3x3o4o |
o3o...o3x3o4o |
Acronym |
bersa |
bro |
barn |
brade | birect. n-hypercube |
Vertex Count | 672 squapen | 1792 squahix | 4608 squahop | 11520 squoc | n(n-2) 2n-3 |
Facet Count rect. simplex | 128 ril | 256 roc | 512 rene | 1024 reday | 2n |
Facet Count birect. h.cube | 14 brox | 16 bersa | 18 bro | 20 barn | 2n |
Circumradius |
sqrt(5/2) 1.581139 |
sqrt(3) 1.732051 |
sqrt(7/2) 1.870829 | 2 | sqrt[(n-2)/2] |
Inradius wrt. rect. simplex |
5/sqrt(14) 1.336306 |
3/2 1.5 |
7/sqrt(18) 1.649916 |
4/sqrt(5) 1.788854 | (n-2)/sqrt(2n) |
Inradius wrt. birect. h.cube |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
Volume |
4919 sqrt(2)/630 11.042090 | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles rect. - rect. |
arccos(-5/7) 135.584691° |
arccos(-3/4) 138.590378° |
arccos(-7/9) 141.057559° |
arccos(-4/5) 143.130102° | arccos(2/n - 1) |
Dihedral angles rect. - birect. |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos(-1/3) 109.471221° |
arccos[-1/sqrt(10)] 108.434949° | arccos[-1/sqrt(n)] |
Dihedral angles birect. - birect. | 90° | 90° | 90° | 90° | 90° |
Within these polytopes tOn generally can be described as the tetrastratic lace tower of the regular orthoplex On-1 atop the u-scaled regular orthoplex On-1 atop the truncated orthoplex tOn-1 atop the u-scaled regular orthoplex On-1 atop the regular orthoplex On-1. Thence, by means of the lace tower notation, tOn = x3x3o...o3o4o (n nodes) can be described as well as xuxux3ooxoo3ooooo...ooooo3ooooo4ooooo&#xt (n-1 node positions).
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
x3x4o |
x3x3o4o |
x3x3o3o4o |
x3x3o3o3o4o |
x3x3o...o3o4o |
Acronym |
toe |
thex |
tot |
tag | trunc. n-orthoplex |
Vertex Count | 24 | 48 | 80 | 120 | 4n(n-1) |
Facet Count trunc. facets | 8 hig | 16 tut | 32 tip | 64 tix | 2n |
Facet Count verf facets | 6 square | 8 oct | 10 hex | 12 tac | 2n |
Circumradius |
sqrt(5/2) 1.581139 |
sqrt(5/2) 1.581139 |
sqrt(5/2) 1.581139 |
sqrt(5/2) 1.581139 |
sqrt(5/2) 1.581139 |
Inradius wrt. trunc. facets |
sqrt(3/2) 1.224745 |
3/sqrt(8) 1.060660 |
3/sqrt(10) 0.948683 |
sqrt(3)/2 0.866025 | 3/sqrt(2n) |
Inradius wrt. verf facets |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
Volume |
8 sqrt(2) 11.313708 |
77/6 12.833333 |
119 sqrt(2)/15 11.219428 |
241/30 8.033333 | (3n-n) sqrt(2n)/n! |
Surface |
6+12 sqrt(3) 26.784610 |
100 sqrt(2)/3 47.140452 |
(5+76 sqrt(5))/3 58.313722 |
(2 sqrt(2)+158 sqrt(3))/5 55.298491 | 2[n+(3n-1-n) sqrt(n)] sqrt(2n-1)/(n-1)! |
Dihedral angles trunc. - orthopl. |
arccos[-1/sqrt(3)] 125.264390° | 120° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(6)] 114.094843° | arccos[-1/sqrt(n)] |
Dihedral angles trunc. - trunc. |
arccos(-1/3) 109.471221° | 120° |
arccos(-3/5) 126.869898° |
arccos(-2/3) 131.810315° | arccos(2/n - 1) |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
x3x3o3o3o3o4o |
x3x3o3o3o3o3o4o |
x3x3o3o3o3o3o3o4o |
x3x3o3o3o3o3o3o3o4o |
x3x3o...o3o4o |
Acronym |
taz |
tek |
tiv |
take | trunc. n-orthoplex |
Vertex Count | 168 | 224 | 288 | 360 | 4n(n-1) |
Facet Count trunc. facets | 128 til | 256 toc | 512 tene | 1024 teday | 2n |
Facet Count verf facets | 14 gee | 16 zee | 18 ek | 20 vee | 2n |
Circumradius |
sqrt(5/2) 1.581139 |
sqrt(5/2) 1.581139 |
sqrt(5/2) 1.581139 |
sqrt(5/2) 1.581139 |
sqrt(5/2) 1.581139 |
Inradius wrt. trunc. facets |
3/sqrt(14) 0.801784 |
3/4 0.75 |
1/sqrt(2) 0.707107 |
3/sqrt(20) 0.670820 | 3/sqrt(2n) |
Inradius wrt. verf facets |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
Volume |
218 sqrt(2)/63 4.893628 |
6553/2520 2.600397 |
1093 sqrt(2)/1260 1.226774 |
59039/113400 0.520626 | (3n-n) sqrt(2n)/n! |
Surface |
(7+722 sqrt(7))/45 42.605165 |
(8716+8 sqrt(2))/315 27.705758 |
437/28 15.607143 |
[5 sqrt(2)+19673 sqrt(5)]/5670 7.759654 | 2[n+(3n-1-n) sqrt(n)] sqrt(2n-1)/(n-1)! |
Dihedral angles trunc. - orthopl. |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos(-1/3) 109.471221° |
arccos[-1/sqrt(10)] 108.434949° | arccos[-1/sqrt(n)] |
Dihedral angles trunc. - trunc. |
arccos(-5/7) 135.584691° |
arccos(-3/4) 138.590378° |
arccos(-7/9) 141.057559° |
arccos(-4/5) 143.130102° | arccos(2/n - 1) |
Within these polytopes tCn generally can be described as the tristratic lace tower of the truncated hypercube tCn-1 atop the w-scaled regular hypercube Cn-1 atop a further w-scaled regular hypercube Cn-1 atop the truncated hypercube tCn-1. Thence, by means of the lace tower notation, tCn = o3o...o3x4x (n nodes) can be described as well as oooo3oooo...oooo3xoox4xwwx&#xt (n-1 node positions).
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
o3x4x |
o3o3x4x |
o3o3o3x4x |
o3o3o3o3x4x |
o3o...o3x4x |
Acronym |
tic |
tat |
tan |
tox | trunc. n-hypercube |
Vertex Count | 24 | 64 | 160 | 384 | n 2n |
Facet Count trunc. facets | 6 oc | 8 tic | 10 tat | 12 tan | 2n |
Facet Count verf facets | 8 trig | 16 tet | 32 pen | 64 hix | 2n |
Circumradius |
sqrt[7+4 sqrt(2)]/2 1.778824 |
sqrt[(5+3 sqrt(2))/2] 2.149726 |
sqrt[13+8 sqrt(2)]/2 2.465447 |
sqrt[(8+5 sqrt(2))/2] 2.745093 | sqrt[(3n-2)+(2n-2) sqrt(2)]/2 |
Inradius wrt. trunc. facets |
[1+sqrt(2)]/2 1.207107 |
[1+sqrt(2)]/2 1.207107 |
[1+sqrt(2)]/2 1.207107 |
[1+sqrt(2)]/2 1.207107 |
[1+sqrt(2)]/2 1.207107 |
Inradius wrt. verf facets |
(3+2 sqrt(2))/sqrt(12) 1.682522 |
(3+2 sqrt(2))/sqrt(8) 2.060660 |
(5+4 sqrt(2))/sqrt(20) 2.382945 |
(5+3 sqrt(2))/sqrt(12) 2.668121 | [n+(n-1) sqrt(2)]/sqrt(4n) |
Volume |
(21+14 sqrt(2))/3 13.599663 |
(101+72 sqrt(2))/6 33.803896 |
(1230+869 sqrt(2))/30 81.965053 |
(8909+6300 sqrt(2))/90 197.983838 | ? |
Surface |
12+12 sqrt(2)+2 sqrt(3) 32.434664 |
(168+116 sqrt(2))/3 110.682924 |
(505+360 sqrt(2)+sqrt(5))/3 338.784317 |
(7380+5214 sqrt(2)+2 sqrt(3))/15 983.811574 | ? |
Dihedral angles trunc. - simplex |
arccos[-1/sqrt(3)] 125.264390° | 120° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(6)] 114.094843° | arccos[-1/sqrt(n)] |
Dihedral angles trunc. - trunc. | 90° | 90° | 90° | 90° | 90° |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3o3o3o3o3x4x |
o3o3o3o3o3o3x4x |
o3o3o3o3o3o3o3x4x |
o3o3o3o3o3o3o3o3x4x |
o3o...o3x4x |
Acronym |
tasa |
tocto |
ten |
tade | trunc. n-hypercube |
Vertex Count | 896 | 2048 | 4608 | 10240 | n 2n |
Facet Count trunc. facets | 14 tox | 16 tasa | 18 tocto | 20 ten | 2n |
Facet Count verf facets | 128 hop | 256 oca | 512 ene | 1024 day | 2n |
Circumradius |
sqrt[19+12 sqrt(2)]/2 2.998773 |
sqrt[(11+7 sqrt(2))/2] 3.232607 |
sqrt[25+16 sqrt(2)]/2 3.450631 |
sqrt[(14+9 sqrt(2))/2] 3.655675 | sqrt[(3n-2)+(2n-2) sqrt(2)]/2 |
Inradius wrt. trunc. facets |
[1+sqrt(2)]/2 1.207107 |
[1+sqrt(2)]/2 1.207107 |
[1+sqrt(2)]/2 1.207107 |
[1+sqrt(2)]/2 1.207107 |
[1+sqrt(2)]/2 1.207107 |
Inradius wrt. verf facets |
(7+6 sqrt(2))/sqrt(28) 2.926443 |
(7+4 sqrt(2))/4 3.164214 |
(9+8 sqrt(2))/6 3.385618 |
(9+5 sqrt(2))/sqrt(20) 3.593600 | [n+(n-1) sqrt(2)]/sqrt(4n) |
Volume | ? | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles trunc. - simplex |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos(-1/3) 109.471221° |
arccos[-1/sqrt(10)] 108.434949° | arccos[-1/sqrt(n)] |
Dihedral angles trunc. - trunc. | 90° | 90° | 90° | 90° | 90° |
These non-convex polytopes qtCn generally are nothing but the conjugates of the truncated hypercube tCn.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
o3x4/3x |
o3o3x4/3x |
o3o3o3x4/3x |
o3o3o3o3x4/3x |
o3o...o3x4/3x |
Acronym |
quith |
quitit |
quittin |
quotox | quasitrunc. n-hypercube |
Vertex Count | 24 | 64 | 160 | 384 | n 2n |
Facet Count quasitr. fac. | 6 og | 8 quith | 10 quitit | 12 quittin | 2n |
Facet Count verf facets | 8 trig | 16 tet | 32 pen | 64 hix | 2n |
Circumradius |
sqrt[7-4 sqrt(2)]/2 0.579471 |
sqrt[(5-3 sqrt(2))/2] 0.615370 |
sqrt[13-8 sqrt(2)]/2 0.649286 |
sqrt[(8-5 sqrt(2))/2] 0.681517 | sqrt[(3n-2)-(2n-2) sqrt(2)]/2 |
Inradius wrt. quasitr. fac. |
[sqrt(2)-1]/2 0.207107 |
[sqrt(2)-1]/2 0.207107 |
[sqrt(2)-1]/2 0.207107 |
[sqrt(2)-1]/2 0.207107 |
[sqrt(2)-1]/2 0.207107 |
Inradius wrt. verf facets |
(3-2 sqrt(2))/sqrt(12) +0.049529 |
(4-3 sqrt(2))/4 -0.060660 |
(5-4 sqrt(2))/sqrt(20) -0.146877 |
(3 sqrt(2)-5)/sqrt(12) -0.218631 | [(n-1) sqrt(2)-n]/sqrt(4n) |
Volume |
(21-14 sqrt(2))/3 0.400337 |
(72 sqrt(2)-101)/6 0.137229 |
(1230-869 sqrt(2))/30 0.034947 |
(6300 sqrt(2)-8909)/90 0.0060605 | ? |
Surface |
-12+12 sqrt(2)+2 sqrt(3) 8.434664 |
56-36 sqrt(2) 5.088312 | ? | ? | ? |
Dihedral angles quasitr. - simpl. |
arccos[1/sqrt(3)] 54.735610° | 60° |
arccos[1/sqrt(5)] 63.434949° |
arccos[1/sqrt(6)] 65.905157° | arccos[1/sqrt(n)] |
Dihedral angles quasitr. - quasitr. | 90° | 90° | 90° | 90° | 90° |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3o3o3o3o3x4/3x |
o3o3o3o3o3o3x4/3x |
o3o3o3o3o3o3o3x4/3x |
o3o3o3o3o3o3o3o3x4/3x |
o3o...o3x4/3x |
Acronym |
quitasa |
queto |
quiten |
quitade | quasitrunc. n-hypercube |
Vertex Count | 896 | 2048 | 4608 | 10240 | n 2n |
Facet Count quasitr. fac. | 14 quotox | 16 quitasa | 18 queto | 20 quiten | 2n |
Facet Count verf facets | 128 hop | 256 oca | 512 ene | 1024 day | 2n |
Circumradius |
sqrt[19-12 sqrt(2)]/2 0.712292 |
sqrt[(11-7 sqrt(2))/2] 0.741790 |
sqrt[25-16 sqrt(2)]/2 0.770160 |
sqrt[(14-9 sqrt(2))/2] 0.797521 | sqrt[(3n-2)-(2n-2) sqrt(2)]/2 |
Inradius wrt. quasitr. fac. |
[sqrt(2)-1]/2 0.207107 |
[sqrt(2)-1]/2 0.207107 |
[sqrt(2)-1]/2 0.207107 |
[sqrt(2)-1]/2 0.207107 |
[sqrt(2)-1]/2 0.207107 |
Inradius wrt. verf facets |
(7-6 sqrt(2))/sqrt(28) -0.280692 |
(4 sqrt(2)-7)/4 -0.335786 |
(9-8 sqrt(2))/6 -0.385618 |
(5 sqrt(2)-9)/sqrt(20) -0.431322 | [n-(n-1) sqrt(2)]/sqrt(4n) |
Volume | ? | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles quasitr. - simpl. |
arccos[1/sqrt(7)] 67.792346° |
arccos[1/sqrt(8)] 69.295189° |
arccos(1/3) 70.528779° |
arccos[1/sqrt(10)] 71.565051° | arccos[1/sqrt(n)] |
Dihedral angles quasitr. - quasitr. | 90° | 90° | 90° | 90° | 90° |
Within these polytopes btCn generally can be described as a stack of the bitruncated hypercube btCn-1 atop u-scaled rectified hypercube rCn-1 atop an (x,q)-variant truncated hypercube tCn-1 atop u-scaled rectified hypercube rCn-1 (again) atop the opposite bitruncated hypercube btCn-1. Thence, by means of the lace tower notation, btCn = o3o...o3x3x4o (n nodes) can be described as well as ooooo3ooooo...ooooo3xooox3xuxux4ooqoo&#xt (n-1 node positions). This representation then shows up those right angles generally.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
x3x4o |
o3x3x4o |
o3o3x3x4o |
o3o3o3x3x4o |
o3o...o3x3x4o |
Acronym |
toe |
tah |
bittin |
botox | bitrunc. n-hypercube |
Vertex Count | 24 | 96 | 320 | 960 | n(n-1) 2n-1 |
Facet Count bitrunc. fac. | 6 square | 8 toe | 10 tah | 12 bittin | 2n |
Facet Count trunc. simpl. | 8 hig | 16 tut | 32 tip | 64 tix | 2n |
Circumradius |
sqrt(5/2) 1.581139 |
sqrt(9/2) 2.121320 |
sqrt(13/2) 2.549510 |
sqrt(17/2) 2.915476 | sqrt[(4n-7)/2] |
Inradius wrt. bitrunc. fac. |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
Inradius wrt. trunc. simpl. |
sqrt(3/2) 1.224745 |
5/sqrt(8) 1.767767 |
7/sqrt(10) 2.213594 |
sqrt(27)/2 2.598076 | (2n-3)/sqrt(2n) |
Volume |
8 sqrt(2) 11.313708 |
307/6 51.166667 |
1801 sqrt(2)/15 169.799908 | ? | ? |
Surface |
6+12 sqrt(3) 26.784610 | ? | ? | ? | ? |
Dihedral angles trunc. - trunc. |
arccos(-1/3) 109.471221° | 120° |
arccos(-3/5) 126.869898° |
arccos(-2/3) 131.810315° | arccos[-(n-2)/n] |
Dihedral angles trunc. - bitrunc |
arccos[-1/sqrt(3)] 125.264390° | 120° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(6)] 114.094843° | arccos[-1/sqrt(n)] |
Dihedral angles bitrunc. - bitrunc. | 90° | 90° | 90° | 90° | |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3o3o3o3x3x4o |
o3o3o3o3o3x3x4o |
o3o3o3o3o3o3x3x4o |
o3o3o3o3o3o3o3x3x4o |
o3o...o3x3x4o |
Acronym |
betsa |
bato |
? |
? | bitrunc. n-hypercube |
Vertex Count | 2688 | 7168 | 18432 | 46080 | n(n-1) 2n-1 |
Facet Count bitrunc. fac. | 14 botox | 16 betsa | 18 bato | 20 ? | 2n |
Facet Count trunc. simpl. | 128 til | 256 toc | 512 tene | 1024 teday | 2n |
Circumradius |
sqrt(21/2) 3.240370 |
5/sqrt(2) 3.535534 |
sqrt(29/2) 3.807887 |
sqrt(33/2) 4.062019 | sqrt[(4n-7)/2] |
Inradius wrt. bitrunc. fac. |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
sqrt(2) 1.414214 |
Inradius wrt. trunc. simpl. |
11/sqrt(14) 2.939874 |
13/4 3.25 |
5/sqrt(2) 3.535534 |
17/sqrt(20) 3.801316 | (2n-3)/sqrt(2n) |
Volume | ? | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles trunc. - trunc. |
arccos(-5/7) 135.584691° |
arccos(-3/4) 138.590378° |
arccos(-7/9) 141.057559° |
arccos(-4/5) 143.130102° | arccos[-(n-2)/n] |
Dihedral angles trunc. - bitrunc. |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos(-1/3) 109.471221° |
arccos[-1/sqrt(10)] 108.434949° | arccos[-1/sqrt(n)] |
Dihedral angles bitrunc. - bitrunc. | 90° | 90° | 90° | 90° | 90° |
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
x3o4x |
o3x3o4x |
o3o3x3o4x |
o3o3o3x3o4x |
o3o...o3x3o4x |
Acronym |
sirco |
srit |
sirn |
srox | rhomb. n-hypercube |
Vertex Count | 24 | 96 | 320 | 960 | n(n-1) 2n-1 |
Facet Count rect. simpl. | 8 trig | 16 oct | 32 rap | 64 rix | 2n |
Facet Count prism | 12 square | 32 trip | 80 tepe | 192 penp | n 2n-1 |
Facet Count rhomb. hyp.cube | 6 square | 8 sirco | 10 srit | 12 sirn | 2n |
Circumradius |
sqrt[5+2 sqrt(2)]/2 1.398966 |
sqrt[2+sqrt(2)] 1.847759 |
[3+sqrt(2)]/2 2.207107 |
sqrt[(7+4 sqrt(2))/2] 2.515637 | sqrt[3n-4+2(n-2) sqrt(2)]/2 |
Inradius wrt. rect. simp. facets |
[3+sqrt(2)]/sqrt(12) 1.274274 |
1+1/sqrt(2) 1.707107 |
sqrt[(43+30 sqrt(2))/20] 2.066717 |
sqrt[(17+12 sqrt(2))/6] 2.379445 | sqrt[(3n2-8n+8+2n(n-2) sqrt(2))/4n] |
Inradius wrt. prism facets |
(1+sqrt(2))/2 1.207107 |
sqrt[(17+12 sqrt(2))/12] 1.682522 |
sqrt[(17+12 sqrt(2))/8] 2.060660 |
sqrt[(57+40 sqrt(2))/20] 2.382945 | sqrt[(3n2-10n+9+2(n-1)(n-2) sqrt(2))/(4(n-1))] |
Inradius wrt. rh. hyp.cub. facets |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
Volume |
[12+10 sqrt(2)]/3 8.714045 |
[45+32 sqrt(2)]/3 30.084945 |
[1205+843 sqrt(2)]/30 79.906068 |
[4426+3141 sqrt(2)]/45 197.067662 | ? |
Surface |
18+2 sqrt(3) 21.464102 |
8[4+4 sqrt(2)+sqrt(3)] 91.111240 |
[450+340 sqrt(2)+11 sqrt(5)]/3 318.476453 | ? | ? |
Dihedral angles rect. - prism |
arccos[-sqrt(2/3)] 144.735610° | 150° |
arccos[-2/sqrt(5)] 153.434949° |
arccos[-sqrt(5/6)] 155.905157° | arccos[-sqrt((n-1)/n)] |
Dihedral angles rect. - rhomb. | 120° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(6)] 114.094843° | arccos[-1/sqrt(n)] | |
Dihedral angles prism - rhomb. | 135° |
arccos[-1/sqrt(3)] 125.264390° | 120° |
arccos[-1/sqrt(5)] 116.565051° | arccos[-1/sqrt(n-1)] |
Dihedral angles rhomb. - rhomb. | 90° | 90° | 90° | 90° | |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3o3o3o3x3o4x |
o3o3o3o3o3x3o4x |
o3o3o3o3o3o3x3o4x |
o3o3o3o3o3o3o3x3o4x |
o3o...o3x3o4x |
Acronym |
sersa |
soro |
? |
? | rhomb. n-hypercube |
Vertex Count | 2688 | 7168 | 18432 | 46080 | n(n-1) 2n-1 |
Facet Count rect. simpl. | 128 ril | 256 roc | 512 rene | 1024 reday | 2n |
Facet Count prism | 448 hixip | 1024 hopip | 2304 ocpe | 5120 enep | n 2n-1 |
Facet Count rhomb. hyp.cube | 14 srox | 16 sersa | 18 soro | 20 ? | 2n |
Circumradius |
sqrt[17+10 sqrt(2)]/2 2.790257 |
sqrt[5+3 sqrt(2)] 3.040171 |
sqrt[23+14 sqrt(2)]/2 3.271047 |
sqrt[(13+8 sqrt(2))/2] 3.486668 | sqrt[3n-4+2(n-2) sqrt(2)]/2 |
Inradius wrt. rect. simp. facets |
sqrt[(99+70 sqrt(2))/28] 2.659182 |
sqrt[17+12 sqrt(2)]/2 2.914214 |
sqrt[179+126 sqrt(2)]/6 3.149916 |
sqrt[(57+40 sqrt(2))/10] 3.369993 | sqrt[(3n2-8n+8+2n(n-2) sqrt(2))/4n] |
Inradius wrt. prism facets |
sqrt[(43+30 sqrt(2))/12] 2.668121 |
sqrt[(121+84 sqrt(2))/28] 2.926443 |
sqrt[81+56 sqrt(2)]/4 3.164214 |
sqrt[209+144 sqrt(2)]/6 3.385618 | sqrt[(3n2-10n+9+2(n-1)(n-2) sqrt(2))/(4(n-1))] |
Inradius wrt. rh. hyp.cub. facets |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
Volume | ? | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles rect. - prism |
arccos[-sqrt(6/7)] 157.792346° |
arccos[-sqrt(7/8)] 159.295189° |
arccos[-sqrt(8/9)] 160.528779° |
arccos[-sqrt(9/10)] 161.565051° | arccos[-sqrt((n-1)/n)] |
Dihedral angles rect. - rhomb. |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos[-1/3] 109.471221° |
arccos[-1/sqrt(10)] 108.434949° | arccos[-1/sqrt(n)] |
Dihedral angles prism - rhomb. |
arccos[-1/sqrt(6)] 114.094843° |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos[-1/3] 109.471221° | arccos[-1/sqrt(n-1)] |
Dihedral angles rhomb. - rhomb. | 90° | 90° | 90° | 90° | 90° |
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
x3o4/3x |
o3x3o4/3x |
o3o3x3o4/3x |
o3o3o3x3o4/3x |
o3o...o3x3o4/3x |
Acronym |
querco |
qrit |
quarn |
qrax | quasirhomb. n-hypercube |
Vertex Count | 24 | 96 | 320 | 960 | n(n-1) 2n-1 |
Facet Count rect. simpl. | 8 trig | 16 oct | 32 rap | 64 rix | 2n |
Facet Count prism | 12 square | 32 trip | 80 tepe | 192 penp | n 2n-1 |
Facet Count qu.rh. hyp.cube | 6 square | 8 querco | 10 qrit | 12 quarn | 2n |
Circumradius |
sqrt[5-2 sqrt(2)]/2 0.736813 |
sqrt[2-sqrt(2)] 0.765367 |
[3-sqrt(2)]/2 0.792893 |
sqrt[(7-4 sqrt(2))/2] 0.819496 | sqrt[3n-4-2(n-2) sqrt(2)]/2 |
Inradius wrt. rect. simp. facets |
[3-sqrt(2)]/sqrt(12) 0.457777 |
1-1/sqrt(2) 0.292893 |
sqrt[(43-30 sqrt(2))/20] 0.169351 |
sqrt[(17-12 sqrt(2))/6] 0.0700443 | sqrt[(3n2-8n+8-2n(n-2) sqrt(2))/4n] |
Inradius wrt. prism facets |
(sqrt(2)-1)/2 0.207107 |
sqrt[(17-12 sqrt(2))/12] 0.0495288 |
sqrt[(17-12 sqrt(2))/8] 0.0606602 |
sqrt[(57-40 sqrt(2))/20] 0.146877 | sqrt[(3n2-10n+9-2(n-1)(n-2) sqrt(2))/(4(n-1))] |
Inradius wrt. qrh. hyp.cub. facets |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
Volume |
[10 sqrt(2)-12]/3 0.714045 |
[32 sqrt(2)-45]/3 0.0849447 |
[1205-843 sqrt(2)]/30 0.427266 |
[3141 sqrt(2)-4426]/45 0.356551 | ? |
Surface |
18+2 sqrt(3) 21.464102 | ? | ? | ? | ? |
Dihedral angles rect. - prism |
arccos[sqrt(2/3)] 35.264390° | 30° |
arccos[2/sqrt(5)] 26.565051° |
arccos[sqrt(5/6)] 24.094843° | arccos[sqrt((n-1)/n)] |
Dihedral angles rect. - qu.rh. | 60° |
arccos[1/sqrt(5)] 63.434949° |
arccos[1/sqrt(6)] 65.905157° | arccos[1/sqrt(n)] | |
Dihedral angles prism - qu.rh. | 45° |
arccos[1/sqrt(3)] 54.735610° | 60° |
arccos[1/sqrt(5)] 63.434949° | arccos[1/sqrt(n-1)] |
Dihedral angles qu.rh. - qu.rh. | 90° | 90° | 90° | 90° | |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3o3o3o3x3o4/3x |
o3o3o3o3o3x3o4/3x |
o3o3o3o3o3o3x3o4/3x |
o3o3o3o3o3o3o3x3o4/3x |
o3o...o3x3o4/3x |
Acronym |
quersa |
qro |
? |
? | quasirhomb. n-hypercube |
Vertex Count | 2688 | 7168 | 18432 | 46080 | n(n-1) 2n-1 |
Facet Count rect. simpl. | 128 ril | 256 roc | 512 rene | 1024 reday | 2n |
Facet Count prism | 448 hixip | 1024 hopip | 2304 ocpe | 5120 enep | n 2n-1 |
Facet Count qu.rh. hyp.cube | 14 qrax | 16 quersa | 18 qro | 20 ? | 2n |
Circumradius |
sqrt[17-10 sqrt(2)]/2 0.845261 |
sqrt[5-3 sqrt(2)] 0.870264 |
sqrt[23-14 sqrt(2)]/2 0.894568 |
sqrt[(13-8 sqrt(2))/2] 0.918230 | sqrt[3n-4-2(n-2) sqrt(2)]/2 |
Inradius wrt. rect. simp. facets |
sqrt[(99-70 sqrt(2))/28] 0.0134306 |
sqrt[17-12 sqrt(2)]/2 0.0857864 |
sqrt[179-126 sqrt(2)]/6 0.149916 |
sqrt[(57-40 sqrt(2))/10] 0.207716 | sqrt[(3n2-8n+8-2n(n-2) sqrt(2))/4n] |
Inradius wrt. prism facets |
sqrt[(43-30 sqrt(2))/12] 0.218631 |
sqrt[(121-84 sqrt(2))/28] 0.280692 |
sqrt[81-56 sqrt(2)]/4 0.335786 |
sqrt[209-144 sqrt(2)]/6 0.385618 | sqrt[(3n2-10n+9-2(n-1)(n-2) sqrt(2))/(4(n-1))] |
Inradius wrt. qrh. hyp.cub. facets |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
Volume | ? | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles rect. - prism |
arccos[sqrt(6/7)] 22.207654° |
arccos[sqrt(7/8)] 20.704811° |
arccos[sqrt(8/9)] 19.471221° |
arccos[sqrt(9/10)] 18.434949° | arccos[sqrt((n-1)/n)] |
Dihedral angles rect. - qu.rh. |
arccos[1/sqrt(7)] 67.792346° |
arccos[1/sqrt(8)] 69.295189° |
arccos[1/3] 70.528779° |
arccos[1/sqrt(10)] 84.260830° | arccos[1/sqrt(n)] |
Dihedral angles prism - qu.rh. |
arccos[1/sqrt(6)] 65.905157° |
arccos[1/sqrt(7)] 67.792346° |
arccos[1/sqrt(8)] 69.295189° |
arccos[1/3] 70.528779° | arccos[1/sqrt(n-1)] |
Dihedral angles qu.rh. - qu.rh. | 90° | 90° | 90° | 90° | 90° |
Within these polytopes eCn generally can be described as the tristratic lace tower of the regular hypercube Cn-1 atop the maximal expanded hypercube eCn-1 atop a further maximal expanded hypercube eCn-1 atop the regular hypercube Cn-1. Thence, by means of the lace tower notation, eCn = x3o...o3o4x (n nodes) can be described as well as oxxo3oooo...oooo3oooo4xxxx&#xt (n-1 node positions).
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
x3o4x |
x3o3o4x |
x3o3o3o4x |
x3o3o3o3o4x |
x3o...o3o4x |
Acronym |
sirco |
sidpith |
scant |
stoxog | max-exp. n-hypercube |
Vertex Count | 24 | 64 | 160 | 384 | n 2n |
Facet Count simplex | 8 trig | 16 tet | 32 pen | 64 hix |
2n n! 2n-0/[(n-0)!0!] |
Facet Count prism I | 12 square | 32 trip | 80 tepe | 192 penp |
n 2n-1 n! 2n-1/[(n-1)!1!] |
Facet Count duoprism I | 80 tisdip | 240 squatet |
n(n-1) 2n-3 n! 2n-2/[(n-2)!2!] | ||
Facet Count duoprism II | 160 tracube |
n(n-1)(n-2) 2n-4/3 n! 2n-3/[(n-3)!3!] | |||
Facet Count prism II | 24 cube | 40 tes | 60 pent |
2n(n-1) n! 22/[2!(n-2)!] | |
Facet Count hypercube | 6 square | 8 cube | 10 tes | 12 pent |
2n n! 21/[1!(n-1)!] |
Circumradius |
sqrt[5+2 sqrt(2)]/2 1.398966 |
sqrt[(3+sqrt(2))/2] 1.485633 |
sqrt[7+2 sqrt(2)]/2 1.567516 |
sqrt[2+1/sqrt(2)] 1.645329 | sqrt[(n+2)+sqrt(8)]/2 |
Inradius wrt. simplex facets |
[3+sqrt(2)]/sqrt(12) 1.274274 |
[1+2 sqrt(2)]/sqrt(8) 1.353553 |
[5+sqrt(2)]/sqrt(20) 1.434262 |
[1+3 sqrt(2)]/sqrt(12) 1.513420 | [n+sqrt(2)]/sqrt(4n) |
Inradius wrt. prism I facets |
(1+sqrt(2))/2 1.207107 |
[3+sqrt(2)]/sqrt(12) 1.274274 |
[1+2 sqrt(2)]/sqrt(8) 1.353553 |
[5+sqrt(2)]/sqrt(20) 1.434262 | [(n-1)+sqrt(2)]/sqrt[4(n-1)] |
Inradius wrt. d.pr. I fac. |
[3+sqrt(2)]/sqrt(12) 1.274274 |
[1+2 sqrt(2)]/sqrt(8) 1.353553 | [(n-2)+sqrt(2)]/sqrt[4(n-2)] | ||
Inradius wrt. d.pr. II fac. |
[3+sqrt(2)]/sqrt(12) 1.274274 | [(n-3)+sqrt(2)]/sqrt[4(n-3)] | |||
Inradius wrt. prism II facets |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 | |
Inradius wrt. hyp.cube fac. |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
Volume |
[12+10 sqrt(2)]/3 8.714045 |
[43+32 sqrt(2)]/6 14.709139 |
[355+251 sqrt(2)]/30 23.665587 |
[833+579 sqrt(2)]/45 36.707326 | ? |
Surface |
18+2 sqrt(3) 21.464102 |
[96+4 sqrt(2)+24 sqrt(3)]/3 47.742025 |
[150+20 sqrt(2)+60 sqrt(3)+sqrt(5)]/3 94.814463 |
[1080+300 sqrt(2)+601 sqrt(3)+30 sqrt(5)]/15 174.153910 | ? |
Dihedral angles simplex - (next) |
arccos[-sqrt(2/3)] 144.735610° | 150° |
arccos[-2/sqrt(5)] 153.434949° |
arccos[-sqrt(5/6)] 155.905157° | arccos[-sqrt((n-1)/n)] |
Dihedral angles prism I - (next) | 135° |
arccos[-sqrt(2/3)] 144.735610° | 150° |
arccos[-2/sqrt(5)] 153.434949° | arccos[-sqrt((n-2)/(n-1))] |
Dihedral angles d.pr. I - (next) |
arccos[-sqrt(2/3)] 144.735610° | 150° | arccos[-sqrt((n-3)/(n-2))] | ||
Dihedral angles d.pr. II - (next) |
arccos[-sqrt(2/3)] 144.735610° | arccos[-sqrt((n-4)/(n-3))] | |||
Dihedral angles prism II - hyp.cube | 135° | 135° | 135° | 135° | |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
x3o3o3o3o3o4x |
x3o3o3o3o3o3o4x |
x3o3o3o3o3o3o3o4x |
x3o3o3o3o3o3o3o3o4x |
x3o...o3o4x |
Acronym |
suposaz |
saxoke |
? |
? | max-exp. n-hypercube |
Vertex Count | 896 | 2048 | 4608 | 10240 | n 2n |
Facet Count simplex | 128 hop | 256 oca | 512 ene | 1024 day |
2n n! 2n-0/[(n-0)!0!] |
Facet Count prism I | 448 hixip | 1024 hopip | 2304 ocpe | 5120 enep |
n 2n-1 n! 2n-1/[(n-1)!1!] |
Facet Count duoprism I | 672 squapen | 1792 squahix | 4608 squahop | 11520 squoc |
n(n-1) 2n-3 n! 2n-2/[(n-2)!2!] |
Facet Count duoprism II | 560 tetcube | 1792 cubpen | 5376 cubhix | 15360 cubhop |
n(n-1)(n-2) 2n-4/3 n! 2n-3/[(n-3)!3!] |
Facet Count duoprism III | 280 tratess | 1120 tettes | 4032 pentes | 13440 teshix | n! 2n-4/[(n-4)!4!] |
Facet Count duoprism IV | 448 trapent | 2016 tetpent | 8064 penpent | n! 2n-5/[(n-5)!5!] | |
Facet Count duoprism V | 672 triax | 3360 tetax | n! 2n-6/[(n-6)!6!] | ||
Facet Count duoprism VI | 960 tetax | n! 2n-7/[(n-7)!7!] | |||
Facet Count prism II | 84 ax | 112 hept | 144 octo | 180 enne |
2n(n-1) n! 22/[2!(n-2)!] |
Facet Count hypercube | 14 ax | 16 hept | 18 octo | 20 enne |
2n n! 21/[1!(n-1)!] |
Circumradius |
sqrt[9+2 sqrt(2)]/2 1.719624 |
sqrt[(5+sqrt(2))/2] 1.790840 |
sqrt[11+2 sqrt(2)]/2 1.859330 |
sqrt[(6+sqrt(2))/2] 1.925385 | sqrt[(n+2)+sqrt(8)]/2 |
Inradius wrt. simplex facets |
[7+sqrt(2)]/sqrt(28) 1.590137 |
[1+4 sqrt(2)]/4 1.664214 |
[9+sqrt(2)]/6 1.735702 |
[1+5 sqrt(2)]/sqrt(20) 1.804746 | [n+sqrt(2)]/sqrt(4n) |
Inradius wrt. prism I facets |
[1+3 sqrt(2)]/sqrt(12) 1.513420 |
[7+sqrt(2)]/sqrt(28) 1.590137 |
[1+4 sqrt(2)]/4 1.664214 |
[9+sqrt(2)]/6 1.735702 | [(n-1)+sqrt(2)]/sqrt[4(n-1)] |
Inradius wrt. d.pr. I fac. |
[5+sqrt(2)]/sqrt(20) 1.434262 |
[1+3 sqrt(2)]/sqrt(12) 1.513420 |
[7+sqrt(2)]/sqrt(28) 1.590137 |
[1+4 sqrt(2)]/4 1.664214 | [(n-2)+sqrt(2)]/sqrt[4(n-2)] |
Inradius wrt. d.pr. II fac. |
[1+2 sqrt(2)]/sqrt(8) 1.353553 |
[5+sqrt(2)]/sqrt(20) 1.434262 |
[1+3 sqrt(2)]/sqrt(12) 1.513420 |
[7+sqrt(2)]/sqrt(28) 1.590137 | [(n-3)+sqrt(2)]/sqrt[4(n-3)] |
Inradius wrt. d.pr. III fac. |
[3+sqrt(2)]/sqrt(12) 1.274274 |
[1+2 sqrt(2)]/sqrt(8) 1.353553 |
[5+sqrt(2)]/sqrt(20) 1.434262 |
[1+3 sqrt(2)]/sqrt(12) 1.513420 | [(n-4)+sqrt(2)]/sqrt[4(n-4)] |
Inradius wrt. d.pr. IV fac. |
[3+sqrt(2)]/sqrt(12) 1.274274 |
[1+2 sqrt(2)]/sqrt(8) 1.353553 |
[5+sqrt(2)]/sqrt(20) 1.434262 | [(n-5)+sqrt(2)]/sqrt[4(n-5)] | |
Inradius wrt. d.pr. V fac. |
[3+sqrt(2)]/sqrt(12) 1.274274 |
[1+2 sqrt(2)]/sqrt(8) 1.353553 | [(n-6)+sqrt(2)]/sqrt[4(n-6)] | ||
Inradius wrt. d.pr. VI fac. |
[3+sqrt(2)]/sqrt(12) 1.274274 | [(n-7)+sqrt(2)]/sqrt[4(n-7)] | |||
Inradius wrt. prism II facets |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
Inradius wrt. hyp.cube fac. |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
Volume |
[8792+6101 sqrt(2)]/315 55.301959 | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles simplex - (next) |
arccos[-sqrt(6/7)] 157.792346° |
arccos[-sqrt(7/8)] 159.295189° |
arccos[-sqrt(8)/3] 160.528779° |
arccos[-3/sqrt(10)] 161.565051° | arccos[-sqrt((n-1)/n)] |
Dihedral angles prism - (next) |
arccos[-sqrt(5/6)] 155.905157° |
arccos[-sqrt(6/7)] 157.792346° |
arccos[-sqrt(7/8)] 159.295189° |
arccos[-sqrt(8)/3] 160.528779° | arccos[-sqrt((n-2)/(n-1))] |
Dihedral angles d.pr. I - (next) |
arccos[-2/sqrt(5)] 153.434949° |
arccos[-sqrt(5/6)] 155.905157° |
arccos[-sqrt(6/7)] 157.792346° |
arccos[-sqrt(7/8)] 159.295189° | arccos[-sqrt((n-3)/(n-2))] |
Dihedral angles d.pr. II - (next) | 150° |
arccos[-2/sqrt(5)] 153.434949° |
arccos[-sqrt(5/6)] 155.905157° |
arccos[-sqrt(6/7)] 157.792346° | arccos[-sqrt((n-4)/(n-3))] |
Dihedral angles d.pr. III - (next) |
arccos[-sqrt(2/3)] 144.735610° | 150° |
arccos[-2/sqrt(5)] 153.434949° |
arccos[-sqrt(5/6)] 155.905157° | arccos[-sqrt((n-5)/(n-4))] |
Dihedral angles d.pr. IV - (next) |
arccos[-sqrt(2/3)] 144.735610° | 150° |
arccos[-2/sqrt(5)] 153.434949° | arccos[-sqrt((n-6)/(n-5))] | |
Dihedral angles d.pr. V - (next) |
arccos[-sqrt(2/3)] 144.735610° | 150° | arccos[-sqrt((n-7)/(n-6))] | ||
Dihedral angles d.pr. VI - (next) |
arccos[-sqrt(2/3)] 144.735610° | arccos[-sqrt((n-8)/(n-7))] | |||
Dihedral angles prism II - hyp.cube | 135° | 135° | 135° | 135° | 135° |
These non-convex polytopes qeCn generally are nothing but the conjugates of the maximal expanded hypercube eCn.
Note that the pattern of retrogradeness, which is required for the correct conjugacy of the volume terms, has an interruption between the fourth and fifth dimension. This simply is because elsewise the volume values themselves would become negative, that is the choice of retrogradenesses just got reversed thereafter.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
x3o4/3x |
x3o3o4/3x |
x3o3o3o4/3x |
x3o3o3o3o4/3x |
x3o...o3o4/3x |
Acronym |
querco |
quidpith |
quacant |
quitoxog | quasiexp. n-hypercube |
Vertex Count | 24 | 64 | 160 | 384 | n 2n |
Facet Count simplex | 8 trig | 16 tet | 32 pen | 64 hix |
2n n! 2n-0/[(n-0)!0!] |
Facet Count prism I | 12 square | 32 trip | 80 tepe | 192 penp |
n 2n-1 n! 2n-1/[(n-1)!1!] |
Facet Count duoprism I | 80 tisdip | 240 squatet |
n(n-1) 2n-3 n! 2n-2/[(n-2)!2!] | ||
Facet Count duoprism II | 160 tracube |
n(n-1)(n-2) 2n-4/3 n! 2n-3/[(n-3)!3!] | |||
Facet Count prism II | 24 cube | 40 tes | 60 pent |
2n(n-1) n! 22/[2!(n-2)!] | |
Facet Count hypercube | 6 square | 8 cube | 10 tes | 12 pent |
2n n! 21/[1!(n-1)!] |
Circumradius |
sqrt[5-2 sqrt(2)]/2 0.736813 |
sqrt[(3-sqrt(2))/2] 0.890446 |
sqrt[7-2 sqrt(2)]/2 1.021221 |
sqrt[2-1/sqrt(2)] 1.137055 | sqrt[(n+2)-sqrt(8)]/2 |
Inradius wrt. simplex facets |
-[3-sqrt(2)]/sqrt(12) -0.457777 |
[2 sqrt(2)-1]/sqrt(8) 0.646447 |
[5-sqrt(2)]/sqrt(20) 0.801806 |
-[3 sqrt(2)-1]/sqrt(12) -0.936070 | [n-sqrt(2)]/sqrt(4n) |
Inradius wrt. prism I facets |
(sqrt(2)-1)/2 0.207107 |
-[3-sqrt(2)]/sqrt(12) -0.457777 |
-[2 sqrt(2)-1]/sqrt(8) -0.646447 |
[5-sqrt(2)]/sqrt(20) 0.801806 | [(n-1)-sqrt(2)]/sqrt[4(n-1)] |
Inradius wrt. d.pr. I fac. |
[3-sqrt(2)]/sqrt(12) 0.457777 |
-[2 sqrt(2)-1]/sqrt(8) -0.646447 | [(n-2)-sqrt(2)]/sqrt[4(n-2)] | ||
Inradius wrt. d.pr. II fac. |
[3-sqrt(2)]/sqrt(12) 0.457777 | [(n-3)-sqrt(2)]/sqrt[4(n-3)] | |||
Inradius wrt. prism II facets |
(sqrt(2)-1)/2 0.207107 |
-(sqrt(2)-1)/2 -0.207107 |
-(sqrt(2)-1)/2 -0.207107 |
+/− (sqrt(2)-1)/2 0.207107 | |
Inradius wrt. hyp.cube fac. |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
-(sqrt(2)-1)/2 -0.207107 |
-(sqrt(2)-1)/2 -0.207107 |
+/− (sqrt(2)-1)/2 0.207107 |
Volume |
[10 sqrt(2)-12]/3 0.392837 |
[32 sqrt(2)-43]/6 0.375806 |
[355-251 sqrt(2)]/30 0.0010799 |
[833-579 sqrt(2)]/45 0.314897 | ? |
Surface |
18+2 sqrt(3) 21.464102 |
[96+4 sqrt(2)+24 sqrt(3)]/3 47.742025 |
[150+20 sqrt(2)+60 sqrt(3)+sqrt(5)]/3 94.814463 |
[1080+300 sqrt(2)+601 sqrt(3)+30 sqrt(5)]/15 174.153910 | ? |
Dihedral angles simplex - (next) |
arccos[sqrt(2/3)] 35.264390° | 30° |
arccos[2/sqrt(5)] 26.565051° |
arccos[sqrt(5/6)] 24.094843° | arccos[sqrt((n-1)/n)] |
Dihedral angles prism I - (next) | 45° |
arccos[sqrt(2/3)] 35.264390° | 30° |
arccos[2/sqrt(5)] 26.565051° | arccos[sqrt((n-2)/(n-1))] |
Dihedral angles d.pr. I - (next) |
arccos[sqrt(2/3)] 35.264390° | 30° | arccos[sqrt((n-3)/(n-2))] | ||
Dihedral angles d.pr. II - (next) |
arccos[sqrt(2/3)] 35.264390° | arccos[sqrt((n-4)/(n-3))] | |||
Dihedral angles prism II - hyp.cube | 45° | 45° | 45° | 45° | |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
x3o3o3o3o3o4/3x |
x3o3o3o3o3o3o4/3x |
x3o3o3o3o3o3o3o4/3x |
x3o3o3o3o3o3o3o3o4/3x |
x3o...o3o4/3x |
Acronym |
quiposaz |
quaxoke |
? |
? | quasiexp. n-hypercube |
Vertex Count | 896 | 2048 | 4608 | 10240 | n 2n |
Facet Count simplex | 128 hop | 256 oca | 512 ene | 1024 day |
2n n! 2n-0/[(n-0)!0!] |
Facet Count prism I | 448 hixip | 1024 hopip | 2304 ocpe | 5120 enep |
n 2n-1 n! 2n-1/[(n-1)!1!] |
Facet Count duoprism I | 672 squapen | 1792 squahix | 4608 squahop | 11520 squoc |
n(n-1) 2n-3 n! 2n-2/[(n-2)!2!] |
Facet Count duoprism II | 560 tetcube | 1792 cubpen | 5376 cubhix | 15360 cubhop |
n(n-1)(n-2) 2n-4/3 n! 2n-3/[(n-3)!3!] |
Facet Count duoprism III | 280 tratess | 1120 tettes | 4032 pentes | 13440 teshix | n! 2n-4/[(n-4)!4!] |
Facet Count duoprism IV | 448 trapent | 2016 tetpent | 8064 penpent | n! 2n-5/[(n-5)!5!] | |
Facet Count duoprism V | 672 triax | 3360 tetax | n! 2n-6/[(n-6)!6!] | ||
Facet Count duoprism VI | 960 tetax | n! 2n-7/[(n-7)!7!] | |||
Facet Count prism II | 84 ax | 112 hept | 144 octo | 180 enne |
2n(n-1) n! 22/[2!(n-2)!] |
Facet Count hypercube | 14 ax | 16 hept | 18 octo | 20 enne |
2n n! 21/[1!(n-1)!] |
Circumradius |
sqrt[9-2 sqrt(2)]/2 1.242133 |
sqrt[(5-sqrt(2))/2] 1.338990 |
sqrt[11-2 sqrt(2)]/2 1.429298 |
sqrt[(6-sqrt(2))/2] 1.514230 | sqrt[(n+2)-sqrt(8)]/2 |
Inradius wrt. simplex facets |
[7-sqrt(2)]/sqrt(28) 1.055614 |
-[4 sqrt(2)-1]/4 -1.164214 |
[9-sqrt(2)]/6 1.264298 |
-[5 sqrt(2)-1]/sqrt(20) -1.357532 | [n-sqrt(2)]/sqrt(4n) |
Inradius wrt. prism I facets |
-[3 sqrt(2)-1]/sqrt(12) -0.936070 |
[7-sqrt(2)]/sqrt(28) 1.055614 |
-[4 sqrt(2)-1]/4 -1.164214 |
[9-sqrt(2)]/6 1.264298 | [(n-1)-sqrt(2)]/sqrt[4(n-1)] |
Inradius wrt. d.pr. I fac. |
[5-sqrt(2)]/sqrt(20) 0.801806 |
-[3 sqrt(2)-1]/sqrt(12) -0.936070 |
[7-sqrt(2)]/sqrt(28) 1.055614 |
-[4 sqrt(2)-1]/4 -1.164214 | [(n-2)-sqrt(2)]/sqrt[4(n-2)] |
Inradius wrt. d.pr. II fac. |
-[2 sqrt(2)-1]/sqrt(8) -0.646447 |
[5-sqrt(2)]/sqrt(20) 0.801806 |
-[3 sqrt(2)-1]/sqrt(12) -0.936070 |
[7-sqrt(2)]/sqrt(28) 1.055614 | [(n-3)-sqrt(2)]/sqrt[4(n-3)] |
Inradius wrt. d.pr. III fac. |
[3-sqrt(2)]/sqrt(12) 0.457777 |
-[2 sqrt(2)-1]/sqrt(8) -0.646447 |
[5-sqrt(2)]/sqrt(20) 0.801806 |
-[3 sqrt(2)-1]/sqrt(12) -0.936070 | [(n-4)-sqrt(2)]/sqrt[4(n-4)] |
Inradius wrt. d.pr. IV fac. |
[3-sqrt(2)]/sqrt(12) 0.457777 |
-[2 sqrt(2)-1]/sqrt(8) -0.646447 |
[5-sqrt(2)]/sqrt(20) 0.801806 | [(n-5)-sqrt(2)]/sqrt[4(n-5)] | |
Inradius wrt. d.pr. V fac. |
[3-sqrt(2)]/sqrt(12) 0.457777 |
-[2 sqrt(2)-1]/sqrt(8) -0.646447 | [(n-6)-sqrt(2)]/sqrt[4(n-6)] | ||
Inradius wrt. d.pr. VI fac. |
[3-sqrt(2)]/sqrt(12) 0.457777 | [(n-7)-sqrt(2)]/sqrt[4(n-7)] | |||
Inradius wrt. prism II facets |
-(sqrt(2)-1)/2 -0.207107 |
-(sqrt(2)-1)/2 -0.207107 |
-(sqrt(2)-1)/2 -0.207107 |
-(sqrt(2)-1)/2 -0.207107 |
-(sqrt(2)-1)/2 -0.207107 |
Inradius wrt. hyp.cube fac. |
-(sqrt(2)-1)/2 -0.207107 |
-(sqrt(2)-1)/2 -0.207107 |
-(sqrt(2)-1)/2 -0.207107 |
-(sqrt(2)-1)/2 -0.207107 |
-(sqrt(2)-1)/2 -0.207107 |
Volume |
[8792-6101 sqrt(2)]/315 0.520264 | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles simplex - (next) |
arccos[sqrt(6/7)] 22.207654° |
arccos[sqrt(7/8)] 20.704811° |
arccos[sqrt(8)/3] 19.471221° |
arccos[3/sqrt(10)] 18.434949° | arccos[sqrt((n-1)/n)] |
Dihedral angles prism - (next) |
arccos[sqrt(5/6)] 24.094843° |
arccos[sqrt(6/7)] 22.207654° |
arccos[sqrt(7/8)] 20.704811° |
arccos[sqrt(8)/3] 19.471221° | arccos[sqrt((n-2)/(n-1))] |
Dihedral angles d.pr. I - (next) |
arccos[2/sqrt(5)] 26.565051° |
arccos[sqrt(5/6)] 24.094843° |
arccos[sqrt(6/7)] 22.207654° |
arccos[sqrt(7/8)] 20.704811° | arccos[sqrt((n-3)/(n-2))] |
Dihedral angles d.pr. II - (next) | 30° |
arccos[2/sqrt(5)] 26.565051° |
arccos[sqrt(5/6)] 24.094843° |
arccos[sqrt(6/7)] 22.207654° | arccos[sqrt((n-4)/(n-3))] |
Dihedral angles d.pr. III - (next) |
arccos[sqrt(2/3)] 35.264390° | 30° |
arccos[2/sqrt(5)] 26.565051° |
arccos[sqrt(5/6)] 24.094843° | arccos[sqrt((n-5)/(n-4))] |
Dihedral angles d.pr. IV - (next) |
arccos[sqrt(2/3)] 35.264390° | 30° |
arccos[2/sqrt(5)] 26.565051° | arccos[sqrt((n-6)/(n-5))] | |
Dihedral angles d.pr. V - (next) |
arccos[sqrt(2/3)] 35.264390° | 30° | arccos[sqrt((n-7)/(n-6))] | ||
Dihedral angles d.pr. VI - (next) |
arccos[sqrt(2/3)] 35.264390° | arccos[sqrt((n-8)/(n-7))] | |||
Dihedral angles prism II - hyp.cube | 45° | 45° | 45° | 45° | 45° |
These non-convex polytopes reCn (a.k.a. socco series) generally are facetings of the maximal expanded hypercube eCn.
All facets within each member of this series here are obviously prograde, except for the simplices. Those however alternate within their retrogradeness wrt. the dimensional n quite similarily as they did for the facetorectified hypercubes frCn.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
o3x4x4/3*a |
o3x4x4/3*a3o |
o3x4x4/3*a3o3o |
o3x4x4/3*a3o3o3o |
o3x4x4/3*a3o...o3o |
Acronym |
socco |
steth |
sinnont |
soxaxog | retroexp. hypercube |
Vertex Count | 24 | 64 | 160 | 384 | n 2n |
Facet Count simplex | 8 trig | 16 tet | 32 pen | 64 hix | 2n |
Facet Count hypercube | 6 square | 8 cube | 10 tes | 12 pent | 2n |
Facet Count soc. ser. mem. | 6 oc | 8 socco | 10 steth | 12 sinnont | 2n |
Circumradius |
sqrt[5+2 sqrt(2)]/2 1.398966 |
sqrt[(3+sqrt(2))/2] 1.485633 |
sqrt[7+2 sqrt(2)]/2 1.567516 |
sqrt[2+1/sqrt(2)] 1.645329 | sqrt[(n+2)+sqrt(8)]/2 |
Inradius wrt. simplex facets |
-[3+sqrt(2)]/sqrt(12) -1.274274 |
[1+2 sqrt(2)]/sqrt(8) 1.353553 |
-[5+sqrt(2)]/sqrt(20) -1.434262 |
[1+3 sqrt(2)]/sqrt(12) 1.513420 | +/− [n+sqrt(2)]/sqrt(4n) |
Inradius wrt. hyp.cube fac. |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
Inradius wrt. soc. ser. mem. |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
Volume |
[6+8 sqrt(2)]/3 5.771236 |
[19+24 sqrt(2)]/6 8.823521 |
[120+149 sqrt(2)]/30 11.023927 |
[451+540 sqrt(2)]/90 13.496392 | ? |
Surface |
18+12 sqrt(2)+2 sqrt(3) 38.434664 |
[72+68 sqrt(2)]/3 56.055507 |
[125+120 sqrt(2)+sqrt(5)]/3 98.980565 |
[900+894 sqrt(2)+2 sqrt(3)]/15 144.518068 | ? |
Dihedral angles simplex - soc.s.m. |
arccos[1/sqrt(3)] 54.735610° | 60° |
arccos[1/sqrt(5)] 63.434949° |
arccos[1/sqrt(6)] 65.905157° | arccos[1/sqrt(n)] |
Dihedral angles hyp.cub. - soc.s.m. | 90° | 90° | 90° | 90° | 90° |
Dihedral angles soc.s.m. - soc.s.m. | 90° | 90° | 90° | 90° | |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3x4x4/3*a3o3o3o3o |
o3x4x4/3*a3o3o3o3o3o |
o3x4x4/3*a3o3o3o3o3o3o |
o3x4x4/3*a3o3o3o3o3o3o3o |
o3x4x4/3*a3o...o3o |
Acronym |
sososaz |
sook |
? |
? | retroexp. hypercube |
Vertex Count | 896 | 2048 | 4608 | 10240 | n 2n |
Facet Count simplex | 128 hop | 256 oca | 512 ene | 1024 day | 2n |
Facet Count hypercube | 14 ax | 16 hept | 18 octo | 20 enne | 2n |
Facet Count soc. ser. mem. | 14 soxaxog | 16 sososaz | 18 sook | 20 ? | 2n |
Circumradius |
sqrt[9+2 sqrt(2)]/2 1.719624 |
sqrt[(5+sqrt(2))/2] 1.790840 |
sqrt[11+2 sqrt(2)]/2 1.859330 |
sqrt[(6+sqrt(2))/2] 1.925385 | sqrt[(n+2)+sqrt(8)]/2 |
Inradius wrt. simplex facets |
-[7+sqrt(2)]/sqrt(28) -1.590137 |
[1+4 sqrt(2)]/4 1.664214 |
-[9+sqrt(2)]/6 -1.735702 |
[1+5 sqrt(2)]/sqrt(20) 1.804746 | +/− [n+sqrt(2)]/sqrt(4n) |
Inradius wrt. hyp.cube fac. |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
(1+sqrt(2))/2 1.207107 |
Inradius wrt. soc. ser. mem. |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
Volume | ? | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles simplex - soc.s.m. |
arccos[1/sqrt(7)] 67.792346° |
arccos[1/sqrt(8)] 69.295189° |
arccos(1/3) 70.528779° |
arccos[1/sqrt(10)] 71.565051° | arccos[1/sqrt(n)] |
Dihedral angles hyp.cub. - soc.s.m. | 90° | 90° | 90° | 90° | 90° |
Dihedral angles soc.s.m. - soc.s.m. | 90° | 90° | 90° | 90° | 90° |
These non-convex polytopes qreCn (a.k.a. gocco series) generally are memberwise conjugates of the retroexpanded hypercube reCn. Thence they are related to the quasiexpanded hypercube qeCn in a very similar way as the former had been to the non-quasi variants, i.e. the maximal expanded hypercubes eCn. In fact the qeCn are facetings of those.
Moreover it happens that all facets within each member of this series are fully prograde, without any exception. In fact their vertex figure always is convex. Thence those also are said to be locally convex.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
o3x4/3x4*a |
o3x4/3x4*a3o |
o3x4/3x4*a3o3o |
o3x4/x43*a3o3o3o |
o3x4/3x4*a3o...o3o |
Acronym |
gocco |
gittith |
ginnont |
goxaxog | quasiretroexp. hyp.cube |
Vertex Count | 24 | 64 | 160 | 384 | n 2n |
Facet Count simplex | 8 trig | 16 tet | 32 pen | 64 hix | 2n |
Facet Count hypercube | 6 square | 8 cube | 10 tes | 12 pent | 2n |
Facet Count goc. ser. mem. | 6 og | 8 gocco | 10 gittith | 12 ginnont | 2n |
Circumradius |
sqrt[5-2 sqrt(2)]/2 0.736813 |
sqrt[(3-sqrt(2))/2] 0.890446 |
sqrt[7-2 sqrt(2)]/2 1.021221 |
sqrt[(4-sqrt(2))/2] 1.137055 | sqrt[(n+2)-sqrt(8)]/2 |
Inradius wrt. simplex facets |
[3-sqrt(2)]/sqrt(12) 0.457777 |
[2 sqrt(2)-1]/sqrt(8) 0.646447 |
[5-sqrt(2)]/sqrt(20) 0.801806 |
[3 sqrt(2)-1]/sqrt(12) 0.936070 | [n-sqrt(2)]/sqrt(4n) |
Inradius wrt. hyp.cube fac. |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
Inradius wrt. goc. ser. mem. |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
Volume |
[8 sqrt(2)-6]/3 1.771236 |
[24 sqrt(2)-19]/6 2.490188 |
[149 sqrt(2)-120]/30 3.023927 |
[540 sqrt(2)-451]/90 3.474170 | ? |
Surface |
-6+12 sqrt(2)+2 sqrt(3) 14.434664 |
[68 sqrt(2)-24]/3 24.055507 |
[-65+120 sqrt(2)+sqrt(5)]/3 35.647232 |
[-540+894 sqrt(2)+2 sqrt(3)]/15 48.518068 | ? |
Dihedral angles simplex - goc.s.m. |
arccos[-1/sqrt(3)] 125.264390° | 120° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(6)] 114.094843° | arccos[-1/sqrt(n)] |
Dihedral angles hyp.cub. - goc.s.m. | 90° | 90° | 90° | 90° | 90° |
Dihedral angles goc.s.m. - goc.s.m. | 90° | 90° | 90° | 90° | |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
o3x4/3x4*a3o3o3o3o |
o3x4/3x4*a3o3o3o3o3o |
o3x4/3x4*a3o3o3o3o3o3o |
o3x4/3x4*a3o3o3o3o3o3o3o |
o3x4/3x4*a3o...o3o |
Acronym |
gososaz |
gook |
? |
? | quasiretroexp. hyp.cube |
Vertex Count | 896 | 2048 | 4608 | 10240 | n 2n |
Facet Count simplex | 128 hop | 256 oca | 512 ene | 1024 day | 2n |
Facet Count hypercube | 14 ax | 16 hept | 18 octo | 20 enne | 2n |
Facet Count goc. ser. mem. | 14 goxaxog | 16 gososaz | 18 gook | 20 ? | 2n |
Circumradius |
sqrt[9-2 sqrt(2)]/2 1.242133 |
sqrt[(5-sqrt(2))/2] 1.338990 |
sqrt[11-2 sqrt(2)]/2 1.429298 |
sqrt[(6-sqrt(2))/2] 1.514230 | sqrt[(n+2)-sqrt(8)]/2 |
Inradius wrt. simplex facets |
[7-sqrt(2)]/sqrt(28) 1.055614 |
[4 sqrt(2)-1]/4 1.164214 |
[9-sqrt(2)]/6 1.264298 |
[5 sqrt(2)-1]/sqrt(20) 1.357532 | [n-sqrt(2)]/sqrt(4n) |
Inradius wrt. hyp.cube fac. |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
(sqrt(2)-1)/2 0.207107 |
Inradius wrt. goc. ser. mem. |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
1/2 0.5 |
Volume | ? | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles simplex - goc.s.m. |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos(-1/3) 109.471221° |
arccos[-1/sqrt(10)] 108.434949° | arccos[-1/sqrt(n)] |
Dihedral angles hyp.cub. - goc.s.m. | 90° | 90° | 90° | 90° | 90° |
Dihedral angles goc.s.m. - goc.s.m. | 90° | 90° | 90° | 90° | 90° |
Dimension | 2D | 3D | 4D | 5D | nD |
---|---|---|---|---|---|
Dynkin diagram |
x4x |
x3x4x |
x3x3x4x |
x3x3x3x4x |
x3x...x3x4x |
Acronym |
oc |
girco |
gidpith |
gacnet | omnitr. n-hypercube |
Vertex Count | 8 | 48 | 384 | 3840 | 2n n! |
Facet Count wrt. type 1 | 4 line | 8 hig | 16 toe | 32 gippid | 2n |
Facet Count wrt. type 2 | 4 line | 12 square | 32 hip | 80 tope | 2n-1 n |
Facet Count wrt. type 3 | 6 oc | 24 op | 80 hodip | 2n-2 n!/[(n-2)! 2!] | |
Facet Count wrt. type 4 | 8 girco | 40 gircope | 2n-3 n!/[(n-3)! 3!] | ||
Facet Count wrt. type 5 | 10 gidpith | 2n-4 n!/[(n-4)! 4!] | |||
Circumradius |
sqrt[(2+sqrt(2))/2] 1.306563 |
sqrt[13+6 sqrt(2)]/2 2.317611 |
sqrt[8+3 sqrt(2)] 3.498949 |
sqrt[65+20 sqrt(2)]/2 4.829189 | sqrt[n(2n2-3n+4)/3 + n(n-1) sqrt(2)]/2 |
Inradius wrt. facet type 1 |
(1+sqrt(2))/2 1.207107 |
sqrt[9+6 sqrt(2)]/2 2.090770 |
(2+3 sqrt(2))/2 3.121320 |
sqrt[45+20 sqrt(2)]/2 4.280312 | sqrt[n(n2-2n+3)/2 + n(n-1) sqrt(2)]/2 |
Inradius wrt. facet type 2 |
(1+sqrt(2))/2 1.207107 |
(3+sqrt(2))/2 2.207107 |
sqrt[27+12 sqrt(2)]/2 3.315515 |
sqrt[(27+10 sqrt(2))/2] 4.535534 | sqrt[(n-1)(n2+2)/2 + n(n-1) sqrt(2)]/2 |
Inradius wrt. facet type 3 |
(1+2 sqrt(2))/2 1.914214 |
(5+sqrt(2))/2 3.207107 |
sqrt[57+18 sqrt(2)]/2 4.540260 | ? | |
Inradius wrt. facet type 4 |
sqrt[19+6 sqrt(2)]/2 2.621320 |
(7+sqrt(2))/2 4.207107 | ? | ||
Inradius wrt. facet type 5 |
sqrt[33+8 sqrt(2)]/2 3.328427 | ? | |||
Volume |
2[1+sqrt(2)] 4.828427 |
2[11+7 sqrt(2)] 41.798990 |
2[131+92 sqrt(2)] 522.215295 |
2[2053+1564 sqrt(2)] 8529.660023 | ? |
Surface | 8 |
12[2+sqrt(2)+sqrt(3)] 61.755172 | ? | ? | ? |
Dihedral angles types 1 - 2 | 135° |
arccos[-sqrt(2/3)] 144.735610° | 150° |
arccos[-2/sqrt(5)] 153.434949° | arccos[-sqrt((n-1)/n)] |
Dihedral angles types 1 - 3 |
arccos[-1/sqrt(3)] 125.264390° | 135° |
arccos[-sqrt(3/5)] 140.768480° | arccos[-sqrt((n-2)/n)] | |
Dihedral angles types 1 - 4 | 120° |
arccos[-sqrt(2/5)] 129.231520° | arccos[-sqrt((n-3)/n)] | ||
Dihedral angles types 1 - 5 |
arccos[-1/sqrt(5)] 116.565051° | arccos[-sqrt((n-4)/n)] | |||
Dihedral angles types 2 - 3 | 135° |
arccos[-sqrt(2/3)] 144.735610° | 150° | arccos[-sqrt((n-2)/(n-1))] | |
Dihedral angles types 2 - 4 |
arccos[-1/sqrt(3)] 125.264390° | 135° | arccos[-sqrt((n-3)/(n-1))] | ||
Dihedral angles types 2 - 5 | 120° | arccos[-sqrt((n-4)/(n-1))] | |||
Dihedral angles types 3 - 4 | 135° |
arccos[-sqrt(2/3)] 144.735610° | arccos[-sqrt((n-3)/(n-2))] | ||
Dihedral angles types 3 - 5 |
arccos[-1/sqrt(3)] 125.264390° | arccos[-sqrt((n-4)/(n-2))] | |||
Dihedral angles types 4 - 5 | 135° | arccos[-sqrt((n-4)/(n-3))] | |||
Dimension | 6D | 7D | 8D | 9D | nD |
Dynkin diagram |
x3x3x3x3x4x |
x3x3x3x3x3x4x |
x3x3x3x3x3x3x4x |
x3x3x3x3x3x3x3x4x |
x3x...x3x4x |
Acronym |
gotaxog |
guposaz |
gaxoke |
? | omnitr. n-hypercube |
Vertex Count | 46080 | 645120 | 10321920 | 185794560 | 2n n! |
Facet Count wrt. type 1 | 64 gocad | 128 gotaf | 256 guph | 512 goxeb | 2n |
Facet Count wrt. type 2 | 192 gippiddip | 448 gocadip | 1024 gotafip | 2304 guphip | 2n-1 n |
Facet Count wrt. type 3 | 240 otoe | 672 ogippid | 1792 ogocad | 4608 ogotaf | 2n-2 n!/[(n-2)! 2!] |
Facet Count wrt. type 4 | 160 hagirco | 560 toegirco | 1792 gircogippid | 5376 gircogocad | 2n-3 n!/[(n-3)! 3!] |
Facet Count wrt. type 5 | 60 gidpithip | 280 hagidpith | 1120 toegidpith | 4032 gippidgidpith | 2n-4 n!/[(n-4)! 4!] |
Facet Count wrt. type 6 | 12 gacnet | 84 gacnetip | 448 hagacnet | 2016 toegacnet | 2n-5 n!/[(n-5)! 5!] |
Facet Count wrt. type 7 | 14 gotaxog | 112 gotaxogip | 672 hagotaxog | 2n-6 n!/[(n-6)! 6!] | |
Facet Count wrt. type 8 | 16 guposaz | 144 guposazip | 2n-7 n!/[(n-7)! 7!] | ||
Facet Count wrt. type 9 | 18 gaxoke | 2n-8 n!/[(n-8)! 8!] | |||
Circumradius |
sqrt[(58+15 sqrt(2))/2] 6.293378 |
sqrt[189+42 sqrt(2)]/2 7.880307 |
sqrt[72+14 sqrt(2)] 9.581179 |
sqrt[417+72 sqrt(2)]/2 11.388847 | sqrt[n(2n2-3n+4)/3 + n(n-1) sqrt(2)]/2 |
Inradius wrt. facet type 1 |
sqrt[81+30 sqrt(2)]/2 5.554872 |
sqrt[133+42 sqrt(2)]/2 6.935362 |
7+sqrt(2) 8.414214 |
sqrt[297+72 sqrt(2)]/2 9.985281 | sqrt[n(n2-2n+3)/2 + n(n-1) sqrt(2)]/2 |
Inradius wrt. facet type 2 |
sqrt[95+30 sqrt(2)]/2 5.861450 |
sqrt[153+42 sqrt(2)]/2 7.286923 |
sqrt[231+56 sqrt(2)]/2 8.806190 |
9+sqrt(2) 10.414214 | sqrt[(n-1)(n2+2)/2 + n(n-1) sqrt(2)]/2 |
Inradius wrt. facet type 3 |
sqrt[(51+14 sqrt(2))/2] 5.949747 |
sqrt[165+40 sqrt(2)]/2 7.442589 |
sqrt[249+54 sqrt(2)]/2 9.018974 | ? | ? |
Inradius wrt. facet type 4 |
sqrt[99+24 sqrt(2)]/2 5.765005 |
sqrt[(83+18 sqrt(2))/2] 7.363961 |
sqrt[255+50 sqrt(2)]/2 9.023728 | ? | ? |
Inradius wrt. facet type 5 |
(9+sqrt(2))/2 5.207107 |
sqrt[153+30 sqrt(2)]/2 6.989750 |
sqrt[(123+22 sqrt(2))/2] 8.778175 | ? | ? |
Inradius wrt. facet type 6 |
sqrt[51+10 sqrt(2)]/2 4.035534 |
(11+sqrt(2))/2 6.207107 |
sqrt[219+36 sqrt(2)]/2 8.214495 | ? | ? |
Inradius wrt. facet type 7 |
sqrt[73+12 sqrt(2)]/2 4.742641 |
(13+sqrt(2))/2 7.207107 | ? | ? | |
Inradius wrt. facet type 8 |
sqrt[99+14 sqrt(2)]/2 5.449747 | ? | ? | ||
Inradius wrt. facet type 9 | ? | ? | |||
Volume | ? | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles types 1 - 2 |
arccos[-sqrt(5/6)] 155.905157° |
arccos[-sqrt(6/7)] 157.792346° |
arccos[-sqrt(7/8)] 159.295189° |
arccos[-sqrt(8)/3] 160.528779° | arccos[-sqrt((n-1)/n)] |
Dihedral angles types 1 - 3 |
arccos[-sqrt(2/3)] 144.735610° |
arccos[-sqrt(5/7)] 147.688467° | 150° |
arccos[-sqrt(7)/3] 151.874494° | arccos[-sqrt((n-2)/n)] |
Dihedral angles types 1 - 4 | 135° |
arccos[-sqrt(4/7)] 139.106605° |
arccos[-sqrt(5/8)] 142.238756° |
arccos[-sqrt(2/3)] 144.735610° | arccos[-sqrt((n-3)/n)] |
Dihedral angles types 1 - 5 |
arccos[-1/sqrt(3)] 125.264390° |
arccos[-sqrt(3/7)] 130.893395° | 135° |
arccos[-sqrt(5)/3] 138.189685° | arccos[-sqrt((n-4)/n)] |
Dihedral angles types 1 - 6 |
arccos[-1/sqrt(6)] 114.094843° |
arccos[-sqrt(2/7)] 122.311533° |
arccos[-sqrt(3/8)] 127.761244° |
arccos(-2/3) 131.810315° | arccos[-sqrt((n-5)/n)] |
Dihedral angles types 1 - 7 |
arccos[-1/sqrt(7)] 112.207654° | 120° |
arccos[-1/sqrt(3)] 125.264390° | arccos[-sqrt((n-6)/n)] | |
Dihedral angles types 1 - 8 |
arccos[-1/sqrt(8)] 110.704811° |
arccos[-sqrt(2)/3] 118.125506° | arccos[-sqrt((n-7)/n)] | ||
Dihedral angles types 1 - 9 |
arccos(-1/3) 109.471221° | arccos[-sqrt((n-8)/n)] | |||
Dihedral angles types 2 - 3 |
arccos[-2/sqrt(5)] 153.434949° |
arccos[-sqrt(5/6)] 155.905157° |
arccos[-sqrt(6/7)] 157.792346° |
arccos[-sqrt(7/8)] 159.295189° | arccos[-sqrt((n-2)/(n-1))] |
Dihedral angles types 2 - 4 |
arccos[-sqrt(3/5)] 140.768480° |
arccos[-sqrt(2/3)] 144.735610° |
arccos[-sqrt(5/7)] 147.688467° | 150° | arccos[-sqrt((n-3)/(n-1))] |
Dihedral angles types 2 - 5 |
arccos[-sqrt(2/5)] 129.231520° | 135° |
arccos[-sqrt(4/7)] 139.106605° |
arccos[-sqrt(5/8)] 142.238756° | arccos[-sqrt((n-4)/(n-1))] |
Dihedral angles types 2 - 6 |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(3)] 125.264390° |
arccos[-sqrt(3/7)] 130.893395° | 135° | arccos[-sqrt((n-5)/(n-1))] |
Dihedral angles types 2 - 7 |
arccos[-1/sqrt(6)] 114.094843° |
arccos[-sqrt(2/7)] 122.311533° |
arccos[-sqrt(3/8)] 127.761244° | arccos[-sqrt((n-6)/(n-1))] | |
Dihedral angles types 2 - 8 |
arccos[-1/sqrt(7)] 112.207654° | 120° | arccos[-sqrt((n-7)/(n-1))] | ||
Dihedral angles types 2 - 9 |
arccos[-1/sqrt(8)] 110.704811° | arccos[-sqrt((n-8)/(n-1))] | |||
Dihedral angles types 3 - 4 | 150° |
arccos[-2/sqrt(5)] 153.434949° |
arccos[-sqrt(5/6)] 155.905157° |
arccos[-sqrt(6/7)] 157.792346° | arccos[-sqrt((n-3)/(n-2))] |
Dihedral angles types 3 - 5 | 135° |
arccos[-sqrt(3/5)] 140.768480° |
arccos[-sqrt(2/3)] 144.735610° |
arccos[-sqrt(5/7)] 147.688467° | arccos[-sqrt((n-4)/(n-2))] |
Dihedral angles types 3 - 6 | 120° |
arccos[-sqrt(2/5)] 129.231520° | 135° |
arccos[-sqrt(4/7)] 139.106605° | arccos[-sqrt((n-5)/(n-2))] |
Dihedral angles types 3 - 7 |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(3)] 125.264390° |
arccos[-sqrt(3/7)] 130.893395° | arccos[-sqrt((n-6)/(n-2))] | |
Dihedral angles types 3 - 8 |
arccos[-1/sqrt(6)] 114.094843° |
arccos[-sqrt(2/7)] 122.311533° | arccos[-sqrt((n-7)/(n-2))] | ||
Dihedral angles types 3 - 9 |
arccos[-1/sqrt(7)] 112.207654° | arccos[-sqrt((n-8)/(n-2))] | |||
Dihedral angles types 4 - 5 |
arccos[-sqrt(2/3)] 144.735610° | 150° |
arccos[-2/sqrt(5)] 153.434949° |
arccos[-sqrt(5/6)] 155.905157° | arccos[-sqrt((n-4)/(n-3))] |
Dihedral angles types 4 - 6 |
arccos[-1/sqrt(3)] 125.264390° | 135° |
arccos[-sqrt(3/5)] 140.768480° |
arccos[-sqrt(2/3)] 144.735610° | arccos[-sqrt((n-5)/(n-3))] |
Dihedral angles types 4 - 7 | 120° |
arccos[-sqrt(2/5)] 129.231520° | 135° | arccos[-sqrt((n-6)/(n-3))] | |
Dihedral angles types 4 - 8 |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(3)] 125.264390° | arccos[-sqrt((n-7)/(n-3))] | ||
Dihedral angles types 4 - 9 |
arccos[-1/sqrt(6)] 114.094843° | arccos[-sqrt((n-8)/(n-3))] | |||
Dihedral angles types 5 - 6 | 135° |
arccos[-sqrt(2/3)] 144.735610° | 150° |
arccos[-2/sqrt(5)] 153.434949° | arccos[-sqrt((n-5)/(n-4))] |
Dihedral angles types 5 - 7 |
arccos[-1/sqrt(3)] 125.264390° | 135° |
arccos[-sqrt(3/5)] 140.768480° | arccos[-sqrt((n-6)/(n-4))] | |
Dihedral angles types 5 - 8 | 120° |
arccos[-sqrt(2/5)] 129.231520° | arccos[-sqrt((n-7)/(n-4))] | ||
Dihedral angles types 5 - 9 |
arccos[-1/sqrt(5)] 116.565051° | arccos[-sqrt((n-8)/(n-4))] | |||
Dihedral angles types 6 - 7 | 135° |
arccos[-sqrt(2/3)] 144.735610° | 150° | arccos[-sqrt((n-6)/(n-5))] | |
Dihedral angles types 6 - 8 |
arccos[-1/sqrt(3)] 125.264390° | 135° | arccos[-sqrt((n-7)/(n-5))] | ||
Dihedral angles types 6 - 9 | 120° | arccos[-sqrt((n-8)/(n-5))] | |||
Dihedral angles types 7 - 8 | 135° |
arccos[-sqrt(2/3)] 144.735610° | arccos[-sqrt((n-7)/(n-6))] | ||
Dihedral angles types 7 - 9 |
arccos[-1/sqrt(3)] 125.264390° | arccos[-sqrt((n-8)/(n-6))] | |||
Dihedral angles types 8 - 9 | 135° | arccos[-sqrt((n-8)/(n-7))] |
As these polytopes Dn generally are nothing but the alternation of the regular hypercube Cn, and Cn in turn is the prism of Cn-1 atop Cn-1, so Dn likewise can be described as the segmentotope of the demihypercube Dn-1 atop the alternate demihypercube ~Dn-1. Thence, by means of the lace prism notation, Dn = x3o3o *b3o...o3o (n nodes) can be described as well as xo3oo3ox *b3oo...oo3oo&#x (n-1 node positions), which as such is nothing else than the demihypercubic alterprism.
A short consideration of general demihypercubes already occured here as well. Furthermore are demihypercubes special cases of the Coxeter-Elte-Gosset polytopes km,n, in fact those generally are clearly the ones of the form 1(n-2),1.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
x3o3o |
x3o3o *b3o |
x3o3o *b3o3o |
x3o3o *b3o3o3o |
x3o3o *b3o...o3o |
Acronym |
tet |
hex |
hin |
hax | n-demihypercube |
Vertex Count | 4 trig | 8 oct | 16 rap | 32 rix | 2n-1 |
Facet Count simplex | 4 trig | 8 tet | 16 pen | 32 hix | 2n-1 |
Facet Count demihyp.cube | 8 tet | 10 hex | 12 hin | 2n | |
Circumradius |
sqrt(3/8) 0.612372 |
1/sqrt(2) 0.707107 |
sqrt(5/8) 0.790569 |
sqrt(3)/2 0.866025 | sqrt(n/8) |
Inradius wrt. simplex |
1/sqrt(24) 0.204124 |
1/sqrt(8) 0.353553 |
3/sqrt(40) 0.474342 |
1/sqrt(3) 0.577350 | (n-2)/sqrt(8n) |
Inradius wrt. demihyp.cube |
1/sqrt(8) 0.353553 |
1/sqrt(8) 0.353553 |
1/sqrt(8) 0.353553 | ||
Volume |
sqrt(2)/12 0.117851 |
1/6 0.166667 |
13 sqrt(2)/120 0.153206 |
43/360 0.119444 | (1-2n-1/n!)/sqrt(2n) |
Surface |
sqrt(3) 1.732051 |
4 sqrt(2)/3 1.885618 |
(10+sqrt(5))/6 2.039345 |
[39 sqrt(2)+2 sqrt(3)]/30 1.953948 | [2 n!-2n-1(n-sqrt(n))]/[(n-1)! sqrt(2n-1)] |
Dihedral angles simp. - demi. |
arccos(1/3) 70.528779° (simp. - simp.) | 120° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(6)] 114.094843° | arccos[-1/sqrt(n)] |
Dihedral angles demi. - demi. | 90° | 90° | 90° | ||
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
x3o3o *b3o3o3o3o |
x3o3o *b3o3o3o3o3o |
x3o3o *b3o3o3o3o3o3o |
x3o3o *b3o3o3o3o3o3o3o |
x3o3o *b3o...o3o |
Acronym |
hesa |
hocto |
henne |
hede | n-demihypercube |
Vertex Count | 64 ril | 128 roc | 256 rene | 512 reday | 2n-1 |
Facet Count simplex | 64 hop | 128 oca | 256 ene | 512 day | 2n-1 |
Facet Count demihyp.cube | 14 hax | 16 hesa | 18 hocto | 20 henne | 2n |
Circumradius |
sqrt(7/8) 0.935414 | 1 |
3/sqrt(8) 1.060660 |
sqrt(5)/2 1.118034 | sqrt(n/8) |
Inradius wrt. simplex |
5/sqrt(56) 0.668153 |
3/4 0.75 |
7/sqrt(72) 0.824958 |
2/sqrt(5) 0.894427 | (n-2)/sqrt(8n) |
Inradius wrt. demihyp.cube |
1/sqrt(8) 0.353553 |
1/sqrt(8) 0.353553 |
1/sqrt(8) 0.353553 |
1/sqrt(8) 0.353553 |
1/sqrt(8) 0.353553 |
Volume |
311 sqrt(2)/5040 0.087266 |
157/2520 0.062302 |
2833 sqrt(2)/90720 0.044163 |
14173/453600 0.031246 | (1-2n-1/n!)/sqrt(2n) |
Surface |
[301+2 sqrt(7)]/180 1.701619 |
[2+311 sqrt(2)]/315 1.402605 |
943/840 1.122619 |
[14165 sqrt(2)+2 sqrt(5)]/22680 0.883457 | [2 n!-2n-1(n-sqrt(n))]/[(n-1)! sqrt(2n-1)] |
Dihedral angles simp. - demi. |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos(-1/3) 109.471221° |
arccos[-1/sqrt(10)] 108.434949° | arccos[-1/sqrt(n)] |
Dihedral angles demi. - demi. | 90° | 90° | 90° | 90° | 90° |
These non-convex polytopes hOn generally are facetings of the regular orthoplex On using the maximal count of it's hemifacets, thereby reducing the facet simplices to the half of the former. Moreover it happens that generally hOn-1 is the vertex figure of hOn. As On could be seen as the Sn-1-antiprism thence too hOn generally is the (non-convex) segmentotope of the regular simplex Sn-1 atop the dual (pseudo?) simplex -(Sn-1). In fact the even dimensional demicrosses have inversion symmetry, i.e. the pseudo part does not apply, while for the odd dimensional ones the inversion would just result in the complement of the original demicross wrt. its convex hull (the orthoplex), i.e. here the pseudo part does apply.
These polytopes never are orientable. Accordingly no volume can be calculated either.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
hemi ( x3/2o3x ) |
hemi ( x3o3/2o3o3*a ) |
hemi ( o3o3/2o3o3*a3x ) |
hemi ( o3o3/2o3o3*a3o3x ) |
hemi ( o3o3/2o3o3*a3o...o3x ) |
Acronym |
thah |
tho |
hehad |
thox | n-demicross |
Vertex Count | 6 | 8 | 10 | 12 | 2n |
Facet Count simplex | 4 trig | 8 tet | 16 pen | 32 hix | 2n-1 |
Facet Count hemi facets | 3 square | 4 oct | 5 hex | 6 tac | n |
Circumradius |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
Inradius wrt. simplex |
1/sqrt(6) 0.408248 |
1/sqrt(8) 0.353553 |
1/sqrt(10) 0.316228 |
1/sqrt(12) 0.288675 | 1/sqrt(2n) |
Inradius wrt. hemi facets | 0 | 0 | 0 | 0 | 0 |
Surface |
3+sqrt(3) 4.732051 |
sqrt(8) 2.828427 |
[5+sqrt(5)]/6 1.206011 |
[3 sqrt(2)+sqrt(3)]/15 0.398313 | (n+sqrt(n)) sqrt(2n-1)/(n-1)! |
Dihedral angles |
arccos[1/sqrt(3)] 54.735610° | 60° |
arccos[1/sqrt(5)] 63.434949° |
arccos[1/sqrt(6)] 65.905157° | arccos[1/sqrt(n)] |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
hemi ( o3o3/2o3o3*a3o3o3x ) |
hemi ( o3o3/2o3o3*a3o3o3o3x ) |
hemi ( o3o3/2o3o3*a3o3o3o3o3x ) |
hemi ( o3o3/2o3o3*a3o3o3o3o3o3x ) |
hemi ( o3o3/2o3o3*a3o...o3x ) |
Acronym |
guhsa |
zeho |
ekhen |
vehde | n-demicross |
Vertex Count | 14 | 16 | 18 | 20 | 2n |
Facet Count simplex | 64 hop | 128 oca | 256 ene | 512 day | 2n-1 |
Facet Count hemi facets | 7 gee | 8 zee | 9 ek | 10 vee | n |
Circumradius |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
1/sqrt(2) 0.707107 |
Inradius wrt. simplex |
1/sqrt(14) 0.267261 |
1/4 0.25 |
1/sqrt(18) 0.235702 |
1/sqrt(20) 0.223607 | 1/sqrt(2n) |
Inradius wrt. hemi facets | 0 | 0 | 0 | 0 | 0 |
Surface |
[7+sqrt(7)]/90 0.107175 |
[4+8 sqrt(2)]/630 0.0243075 |
1/210 0.00476190 |
[5 sqrt(2)+sqrt(5)]/11340 0.000820735 | (n+sqrt(n)) sqrt(2n-1)/(n-1)! |
Dihedral angles |
arccos[1/sqrt(7)] 67.792346° |
arccos[1/sqrt(8)] 69.295189° |
arccos(1/3) 70.528779° |
arccos[1/sqrt(10)] 71.565051° | arccos[1/sqrt(n)] |
As the non-truncated demihypercubes Dn generally could be described as the segmentotope of the demihypercube Dn-1 atop the alternate demihypercube ~Dn-1, their truncations tDn become tristratic lace towers with the truncated demihypercube tDn-1 at the top side and the alternate truncated demihypercube ~tDn-1 at the bottom side. Inbetween there will be 2 vertex layers which happen to be non-uniform variants of the rectified hypercube rCn-1. In fact, by means of the lace tower notation, tDn = x3o3o *b3o...o3o (n nodes) can be described as well as xuxo3xoox3oxux *b3oooo...oooo3oooo&#xt (n-1 node positions).
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
x3x3o |
x3x3o *b3o |
x3x3o *b3o3o |
x3x3o *b3o3o3o |
x3x3o *b3o...o3o |
Acronym |
tut |
thex |
thin |
thax | n-trunc. demihyp.cube |
Vertex Count | 12 | 48 | 160 | 480 | 2n-2 n(n-1) |
Facet Count trunc. simpl. | 4 hig | 8 tut | 16 tip | 32 tix | 2n-1 |
Facet Count rect. simpl. | 4 trig | 8 oct | 16 rap | 32 rix | 2n-1 |
Facet Count trunc. demi. | 8 tut | 10 thex | 12 thin | 2n | |
Circumradius |
sqrt(11/8) 1.172604 |
sqrt(5/2) 1.581139 |
sqrt(29/8) 1.903943 |
sqrt(19)/2 2.179449 | sqrt[(9n-16)/8] |
Inradius wrt. trunc. simpl. |
sqrt(3/8) 0.612372 |
3/sqrt(8) 1.060660 |
9/sqrt(40) 1.423025 |
sqrt(3) 1.732051 | 3(n-2)/sqrt(8n) |
Inradius wrt. rect. simpl. |
5/sqrt(24) 1.020621 |
sqrt(2) 1.414214 |
11/sqrt(40) 1.739253 |
7/sqrt(12) 2.020726 | (3n-4)/sqrt(8n) |
Inradius wrt. trunc. demi. |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 | ||
Volume |
23 sqrt(2)/12 2.710576 |
77/6 12.833333 |
623 sqrt(2)/24 36.710627 |
31243/360 86.786111 | ? |
Surface |
7 sqrt(3) 12.124356 |
100 sqrt(2)/3 47.140452 |
(770+87 sqrt(5))/6 160.756319 |
[9345 sqrt(2)+526 sqrt(3)]/30 470.896149 | ? |
Dihedral angles tr.simp. - re.simp. |
arccos(-1/3) 109.471221° | 120° |
arccos(-3/5) 126.869898° |
arccos(-2/3) 131.810315° | arccos[-(n-2)/n] |
Dihedral angles tr.simp. - tr.demi. |
arccos(1/3) 70.528779° (tr.simp. - tr.simp.) | 120° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(6)] 114.094843° | arccos[-1/sqrt(n)] |
Dihedral angles re.simp. -tr.demi. | 120° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(6)] 114.094843° | arccos[-1/sqrt(n)] | |
Dihedral angles tr.demi. - tr.demi. | 90° | 90° | 90° | ||
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
x3x3o *b3o3o3o3o |
x3x3o *b3o3o3o3o3o |
x3x3o *b3o3o3o3o3o3o |
x3x3o *b3o3o3o3o3o3o3o |
x3x3o *b3o...o3o |
Acronym |
thesa |
thocto |
thenne |
thede | n-trunc. demihyp.cube |
Vertex Count | 1344 | 3584 | 9216 | 23040 | 2n-2 n(n-1) |
Facet Count trunc. simpl. | 64 til | 128 toc | 256 tene | 512 teday | 2n-1 |
Facet Count rect. simpl. | 64 ril | 128 roc | 256 rene | 512 reday | 2n-1 |
Facet Count demihyp.cube | 14 thax | 16 thesa | 18 thocto | 20 thenne | 2n |
Circumradius |
sqrt(47/8) 2.423840 |
sqrt(7) 2.645751 |
sqrt(65/8) 2.850439 |
sqrt(37)/2 3.041381 | sqrt[(9n-16)/8] |
Inradius wrt. trunc. simpl. |
15/sqrt(56) 2.004459 |
9/4 2.25 |
7/sqrt(8) 2.474874 |
6/sqrt(5) 2.683282 | 3(n-2)/sqrt(8n) |
Inradius wrt. rect. simpl. |
17/sqrt(56) 2.271721 |
5/2 2.5 |
23/sqrt(72) 2.710576 |
13/sqrt(20) 2.906888 | (3n-4)/sqrt(8n) |
Inradius wrt. trunc. demi. |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 |
Volume |
34081 sqrt(2)/240 200.824218 | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles tr.simp. - re.simp. |
arccos(-5/7) 135.584691° |
arccos(-3/4) 138.590378° |
arccos(-7/9) 141.057559° |
arccos(-4/5) 143.130102° | arccos[-(n-2)/n] |
Dihedral angles tr.simp. - tr.demi. |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos(-1/3) 109.471221° |
arccos[-1/sqrt(10)] 108.434949° | arccos[-1/sqrt(n)] |
Dihedral angles re.simp. - tr.demi. |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos(-1/3) 109.471221° |
arccos[-1/sqrt(10)] 108.434949° | arccos[-1/sqrt(n)] |
Dihedral angles tr.demi. - tr.demi. | 90° | 90° | 90° | 90° | 90° |
Within these polytopes eDn generally can be described as the tristratic lace tower of the demihypercube Dn-1 atop the maximal expanded demihypercube eDn-1 atop the maximal expanded alternate demihypercube ~eDn-1 atop the alternate demihypercube ~Dn-1. Thence, by means of the lace tower notation, eDn = x3o3o *b3o...o3x (n nodes) can be described as well as xxoo3oooo3ooxx *b3oooo...oooo3oxxo&#xt (n-1 node positions).
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
x3x3o |
x3o3o *b3x |
x3o3o *b3o3x |
x3o3o *b3o3o3x |
x3o3o *b3o...o3x |
Acronym |
tut |
rit |
siphin |
sochax | max-exp. n-demihyp.cube |
Vertex Count | 12 | 32 | 80 | 192 | n 2n-1 |
Facet Count simplex | 4 trig | 8 tet | 16 pen | 32 hix | 2n-1 |
Facet Count exp. simpl. | 4 hig | 8 co | 16 spid | 32 scad | 2n-1 |
Facet Count duoprism I | 160 tratet | 4n(n-1)(n-2)/3 | |||
Facet Count prism | 40 tepe | 60 hexip | 2n(n-1) | ||
Facet Count demihyp.cube | 8 tet | 10 hex | 12 hin | 2n | |
Circumradius |
sqrt(11/8) 1.172604 |
sqrt(3/2) 1.224745 |
sqrt(13/8) 1.274755 |
sqrt(7)/2 1.322876 | sqrt[(n+8)/8] |
Inradius wrt. simplex facets |
5/sqrt(24) 1.020621 |
3/sqrt(8) 1.060660 |
7/sqrt(40) 1.106797 |
2/sqrt(3) 1.154701 | (n+2)/sqrt(8n) |
Inradius wrt. exp. simpl. fac. |
sqrt(3/8) 0.612372 |
1/sqrt(2) 0.707107 |
sqrt(5/8) 0.790569 |
sqrt(3)/2 0.866025 | sqrt(n/8) |
Inradius wrt. d.pr. I fac. |
5/sqrt(24) 1.020621 |
5/sqrt(24) 1.020621 | |||
Inradius wrt. prism facets | 1 | 1 | 1 | ||
Inradius wrt. demihyp.c. fac. |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 | |
Volume |
23 sqrt(2)/12 2.710576 |
23/6 3.833333 |
467 sqrt(2)/120 5.503648 |
2737/360 7.602778 | ? |
Surface |
7 sqrt(3) 12.124356 |
44 sqrt(2)/3 20.741799 |
[10+20 sqrt(2)+71 sqrt(5)]/6 32.840850 |
[300+39 sqrt(2)+506 sqrt(3)+100 sqrt(6)]/30 49.217367 | ? |
Dihedral angles simpl. - e.sim. |
arccos(-1/3) 109.471221° | 120° |
arccos(-3/5) 126.869898° |
arccos(-2/3) 131.810315° | arccos[-(n-2)/n] |
Dihedral angles e.sim. - e.sim. |
arccos(1/3) 70.528779° | 90° |
arccos(-1/5) 101.536959° |
arccos(-1/3) 109.471221° | arccos[-(n-4)/n] |
Dihedral angles e.sim. - d.pr. I | ? | ? | |||
Dihedral angles e.sim. - prism |
arccos[-sqrt(2/5)] 129.231520° | ? | ? | ||
Dihedral angles e.sim. - demi. | 120° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(6)] 114.094843° | arccos[-1/sqrt(n)] | |
Dihedral angles prism - d.pr. I | ? | ? | |||
Dihedral angles prism - demi. | 135° | 135° | 135° | ||
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
x3o3o *b3o3o3o3x |
x3o3o *b3o3o3o3o3x |
x3o3o *b3o3o3o3o3o3x |
x3o3o *b3o3o3o3o3o3o3x |
x3o3o *b3o...o3x |
Acronym |
suthesa |
spuho |
? |
? | max-exp. n-demihyp.cube |
Vertex Count | 448 | 1024 | 2304 | 5120 | n 2n-1 |
Facet Count simplex | 64 hop | 128 oca | 256 ene | 512 day | 2n-1 |
Facet Count exp. simpl. | 64 staf | 128 suph | 256 soxeb | 512 ? | 2n-1 |
Facet Count duoprism V | 15360 tethop | n! 27/[7!(n-7)!] | |||
Facet Count duoprism IV | 5376 tethix | 7680 hexhix | n! 26/[6!(n-6)!] | ||
Facet Count duoprism III | 1792 tetpen | 4032 penhex | 8064 penhin | n! 25/[5!(n-5)!] | |
Facet Count duoprism II | 560 tetdip | 1120 tethex | 2016 tethin | 3360 tethax | n! 24/[4!(n-4)!] |
Facet Count duoprism I | 280 trahex | 448 trahin | 672 trahax | 960 trahesa |
4n(n-1)(n-2)/3 n! 23/[3!(n-3)!] |
Facet Count prism | 84 hinnip | 112 haxip | 144 hesape | 180 hoctope |
2n(n-1) n! 22/[2!(n-2)!] |
Facet Count demihyp.cube | 14 hax | 16 hesa | 18 hocto | 20 henne |
2n n! 21/[1!(n-1)!] |
Circumradius |
sqrt(15/8) 1.369306 |
sqrt(2) 1.414214 |
sqrt(17/8) 1.457738 |
3/2 1.5 | sqrt[(n+8)/8] |
Inradius wrt. simplex |
9/sqrt(56) 1.202676 |
5/4 1.25 |
11/sqrt(72) 1.296362 |
3/sqrt(5) 1.341641 | (n+2)/sqrt(8n) |
Inradius wrt. exp. simpl. |
sqrt(7/8) 0.935414 | 1 |
3/sqrt(8) 1.060660 |
sqrt(5)/2 1.118034 | sqrt(n/8) |
Inradius wrt. duoprism V |
9/sqrt(56) 1.202676 |
9/sqrt(56) 1.202676 (3+6)/sqrt[(1+6)8] | |||
Inradius wrt. duoprism IV |
2/sqrt(3) 1.154701 |
2/sqrt(3) 1.154701 |
2/sqrt(3) 1.154701 (3+5)/sqrt[(1+5)8] | ||
Inradius wrt. duoprism III |
7/sqrt(40) 1.106797 |
7/sqrt(40) 1.106797 |
7/sqrt(40) 1.106797 |
7/sqrt(40) 1.106797 (3+4)/sqrt[(1+4)8] | |
Inradius wrt. duoprism II |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 (3+3)/sqrt[(1+3)8] |
Inradius wrt. duoprism I |
5/sqrt(24) 1.020621 |
5/sqrt(24) 1.020621 |
5/sqrt(24) 1.020621 |
5/sqrt(24) 1.020621 |
5/sqrt(24) 1.020621 (3+2)/sqrt[(1+2)8] |
Inradius wrt. prism | 1 | 1 | 1 | 1 |
1 (3+1)/sqrt[(1+1)8] |
Inradius wrt. demihyp.cube |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 |
3/sqrt(8) 1.060660 (3+0)/sqrt[(1+0)8] |
Volume | ? | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles simpl. - e.sim. |
arccos(-5/7) 135.584691° |
arccos[-3/4) 138.590378° |
arccos(-7/9) 141.057559° |
arccos(-4/5) 143.130102° | arccos[-(n-2)/n] |
Dihedral angles e.sim. - e.sim. |
arccos(-3/7) 115.376934° | 120° |
arccos(-5/9) 123.748989° |
arccos(-3/5) 126.869898° | arccos[-(n-4)/n] |
Dihedral angles e.sim. - d.pr. V | ? | ? | |||
Dihedral angles e.sim. - d.pr. IV | ? | ? | ? | ||
Dihedral angles e.sim. - d.pr. III | ? | ? | ? | ? | |
Dihedral angles e.sim. - d.pr. II | ? | ? | ? | ? | ? |
Dihedral angles e.sim. - d.pr. I | ? | ? | ? | ? | ? |
Dihedral angles e.sim. - prism | ? | ? | ? | ? | ? |
Dihedral angles e.sim. - demi. |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos(-1/3) 109.471221° |
arccos[-1/sqrt(10)] 108.434949° | arccos[-1/sqrt(n)] |
Dihedral angles prism - d.pr. V | ? | ? | |||
Dihedral angles prism - d.pr. IV | ? | ? | ? | ||
Dihedral angles prism - d.pr. III | ? | ? | ? | ? | |
Dihedral angles prism - d.pr. II | ? | ? | ? | ? | ? |
Dihedral angles prism - d.pr. I | ? | ? | ? | ? | ? |
Dihedral angles prism - demi. | 135° | 135° | 135° | 135° | 135° |
It is known that those series clearly terminate for n=8, i.e. that for n=9 they would result in a flat tesselations instead. This accordingly reflects itself in the provided dimension formulae: measures like circumradii and inradii all would become infinite for n=9 and dihedrals likewise would all become 180° then.
Dimension | 4D | 5D | 6D | 7D | 8D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
o3o3x3o |
o3o3o3x *c3o |
o3o3o3o3x *c3o |
o3o3o3o3o3x *c3o |
o3o3o3o3o3o3x *c3o |
o3o...o3x *c3o |
Acronym |
rap |
hin |
jak |
naq |
fy | (n-4)2,1 |
Vertex Count | 10 trip | 16 rap | 27 hin | 56 jak | 240 naq | ? |
Facet Count simplex | 5 tet | 16 pen | 72 hix | 576 hop | 17280 oca | ? |
Facet Count orthoplex | 5 oct | 10 hex | 27 tac | 126 gee | 2160 zee | ? |
Circumradius |
sqrt(3/5) 0.774597 |
sqrt(5/8) 0.790569 |
sqrt(2/3) 0.816497 |
sqrt(3)/2 0.866025 | 1 | sqrt[(10-n)/(18-2n)] |
Inradius wrt. simplex |
3/sqrt(40) 0.474342 |
3/sqrt(40) 0.474342 |
1/2 0.5 |
3/sqrt(28) 0.566947 |
3/4 0.75 | 3/sqrt[n(18-2n)] |
Inradius wrt. orthoplex |
1/sqrt(10) 0.316228 |
1/sqrt(8) 0.353553 |
1/sqrt(6) 0.408248 |
1/2 0.5 |
1/sqrt(2) 0.707107 | 1/sqrt(18-2n) |
Volume |
11 sqrt(5)/96 0.256216 |
13 sqrt(2)/120 0.153206 |
sqrt(3)/16 0.108253 |
17/140 0.121429 |
57/112 0.508929 | ? |
Surface |
25 sqrt(2)/12 2.946278 |
[10+sqrt(5)]/6 2.039345 |
[18 sqrt(2)+3 sqrt(3)]/20 1.532600 |
[14+sqrt(7)]/10 1.664575 |
[6+24 sqrt(2)]/7 5.705875 | ? |
Dihedral angles simpl. - ortho. |
arccos(-1/4) 104.477512° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-2/sqrt(7)] 139.106605° |
arccos[-5/sqrt(32)] 152.114433° | arccos[-(n-3)/sqrt(4n)] |
Dihedral angles ortho. - ortho. |
arccos(1/4) 75.522488° | 90° |
arccos(-1/4) 104.477512° | 120° |
arccos(-3/4) 138.590378° | arccos[-(n-5)/4] |
Dimension | 4D | 5D | 6D | 7D | 8D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
x3o3o3o |
x3o3o3o *c3o |
x3o3o3o3o *c3o |
x3o3o3o3o3o *c3o |
x3o3o3o3o3o3o *c3o |
x3o...o3o *c3o |
Acronym |
pen |
tac |
jak |
laq |
bay | 2n,1 |
Vertex Count | 5 tet | 10 hex | 27 hin | 126 hax | 2160 hesa | ? |
Facet Count simplex | 5 tet | 16 pen | 72 hix | 576 hop | 17280 oca | ? |
Facet Count Gossetic | 16 pen | 27 tac | 56 jak | 240 laq | ? | |
Circumradius |
sqrt(2/5) 0.632456 |
1/sqrt(2) 0.707107 |
sqrt(2/3) 0.816497 | 1 |
sqrt(2) 1.414214 | sqrt[2/(9-n)] |
Inradius wrt. simplex |
1/sqrt(40) 0.158114 |
1/sqrt(10) 0.316228 |
1/2 0.5 |
2/sqrt(7) 0.755929 |
5/4 1.25 | (n-3)/sqrt[2n(9-n)] |
Inradius wrt. Gossetic |
1/sqrt(10) 0.316228 |
1/sqrt(6) 0.408248 |
1/sqrt(3) 0.577350 | 1 | sqrt[2/((10-n)(9-n))] | |
Volume |
sqrt(5)/96 0.023292 |
sqrt(2)/30 0.047140 |
sqrt(3)/16 0.108253 |
37/70 0.528571 |
1791/112 15.991071 | ? |
Surface |
5 sqrt(2)/12 0.589256 |
sqrt(5)/3 0.745356 |
[18 sqrt(2)+3 sqrt(3)]/20 1.532600 |
[35 sqrt(3)+sqrt(7)]/10 6.326753 |
894/7 127.714286 | ? |
Dihedral angles simpl. - Goss. |
arccos(1/4) 75.522488° simpl. - simpl. |
arccos(-3/5) 126.869898° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-sqrt(3/7)] 130.893395° |
arccos(-3/4) 138.590378° | arccos[-3/sqrt(n(10-n))] |
Dihedral angles Goss. - Goss. |
arccos(-1/4) 104.477512° |
arccos(-1/3) 109.471221° | 120° | arccos[-1/(10-n)] |
Dimension | 4D | 5D | 6D | 7D | 8D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
o3o3o3x |
o3o3o3o *c3x |
o3o3o3o3o *c3x |
o3o3o3o3o3o *c3x |
o3o3o3o3o3o3o *c3x |
o3o...o3o *c3x |
Acronym |
pen |
hin |
mo |
lin |
bif | 1n,2 |
Vertex Count | 5 tet | 16 rap | 72 dot | 576 bril | 17280 broc | ? |
Facet Count demihypercube | 5 tet | 10 hex | 27 hin | 126 hax | 2160 hesa | ? |
Facet Count Gossetic | 16 pen | 27 hin | 56 mo | 240 lin | ? | |
Circumradius |
sqrt(2/5) 0.632456 |
sqrt(5/8) 0.790569 | 1 |
sqrt(7)/2 1.322876 | 2 | sqrt[n/(18-2n)] |
Inradius wrt. demihypercube |
1/sqrt(40) 0.158114 |
1/sqrt(8) 0.353553 |
sqrt(3/8) 0.612372 | 1 |
5/sqrt(8) 1.767767 | (n-3)/sqrt[8(9-n)] |
Inradius wrt. Gossetic |
3/sqrt(40) 0.474342 |
sqrt(3/8) 0.612372 |
sqrt(3)/2 0.866025 |
3/2 1.5 | 3/sqrt[2(10-n)(9-n)] | |
Volume |
sqrt(5)/96 0.023292 |
13 sqrt(2)/120 0.153206 |
39 sqrt(3)/80 0.844375 | 8 |
44985/112 401.651786 | ? |
Surface |
5 sqrt(2)/12 0.589256 |
[10+sqrt(5)]/6 2.039345 |
117 sqrt(2)/20 8.273149 |
[301+546 sqrt(3)]/20 62.334987 |
[13440+933 sqrt(2)]/7 2108.494465 | ? |
Dihedral angles demi. - demi. |
arccos(1/4) 75.522488° | 90° |
arccos(-1/4) 104.477512° | 120° |
arccos(-3/4) 138.590378° | arccos[-(n-5)/4] |
Dihedral angles demi. - Goss. |
arccos[-1/sqrt(5)] 116.565051° | 120° |
arccos[-1/sqrt(3)] 125.264390° | 135° | arccos[-1/sqrt(10-n)] | |
Dihedral angles Goss. - Goss. |
arccos(-1/4) 104.477512° |
arccos(-1/3) 109.471221° | 120° | arccos[-1/(10-n)] |
Dimension | 4D | 5D | 6D | 7D | 8D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
o3x3o3x |
o3o3x3o *c3o |
o3o3o3x3o *c3o |
o3o3o3o3x3o *c3o |
o3o3o3o3o3x3o *c3o |
o3o...o3x3o *c3o |
Acronym |
srip |
nit |
rojak |
ranq |
riffy | rect. (n-4)2,1 |
Vertex Count | 30 xx ox&#q | 80 tisdip | 216 rappip | 756 hinnip | 6720 jakip | ? |
Facet Count rect. simpl. | 5 oct | 16 rap | 72 rix | 576 ril | 17280 roc | ? |
Facet Count Gossetic | 10 trip | 16 rap | 27 hin | 56 jak | 240 naq | ? |
Facet Count rect. ortho. | 5 co | 10 ico | 27 rat | 126 rag | 2160 rez | ? |
Circumradius |
sqrt(7/5) 1.183216 |
sqrt(3/2) 1.224745 |
sqrt(5/3) 1.290994 |
sqrt(2) 1.414214 |
sqrt(3) 1.732051 | sqrt[(11-n)/(9-n)] |
Inradius wrt. rect. simpl. |
3/sqrt(10) 0.948683 |
3/sqrt(10) 0.948683 | 1 |
3/sqrt(7) 1.133893 |
3/2 1.5 | sqrt[18/(n(9-n))] |
Inradius wrt. Gossetic |
7/sqrt(60) 0.903696 |
3/sqrt(10) 0.948683 |
5/sqrt(24) 1.020621 |
2/sqrt(3) 1.154701 |
3/2 1.5 | (11-n)/sqrt[2(10-n)(9-n)] |
Inradius wrt. rect. ortho. |
sqrt(2/5) 0.632456 |
1/sqrt(2) 0.707107 |
sqrt(2/3) 0.816497 | 1 |
sqrt(2) 1.414214 | sqrt[2/(9-n)] |
Volume |
73 sqrt(5)/48 3.400687 |
31 sqrt(2)/10 4.384062 |
601 sqrt(3)/160 6.506016 |
1053/70 15.042857 |
3597/28 128.464286 | ? |
Surface |
[20 sqrt(2)+5 sqrt(3)]/2 18.472263 |
[60+11 sqrt(5)]/3 28.198916 |
[1089 sqrt(2)+156 sqrt(3)]/40 45.256962 |
[812+35 sqrt(3)+57 sqrt(7)]/10 102.342960 |
[924+2904 sqrt(2)]/7 718.696598 | ? |
Dihedral angles r.sim. - Goss. |
arccos[-sqrt(3/8)] 127.761244° |
arccos(-3/5) 126.869898° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-sqrt(3/7)] 130.893395° |
arccos(-3/4) 138.590378° | arccos[-3/sqrt(n(10-n))] |
Dihedral angles r.sim. - r.orth. |
arccos(-1/4) 104.477512° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-2/sqrt(7)] 139.106605° |
arccos[-5/sqrt(32)] 152.114433° | arccos[-(n-3)/sqrt(4n)] |
Dihedral angles Goss. - r.orth. |
arccos[-1/sqrt(6)] 114.094843° |
arccos[-1/sqrt(5)] 116.565051° | 120° |
arccos[-1/sqrt(3)] 125.264390° | 135° | arccos[-1/sqrt(10-n)] |
Dihedral angles r.orth. - r.orth. |
arccos(1/4) 75.522488° | 90° |
arccos(-1/4) 104.477512° | 120° |
arccos(-3/4) 138.590378° | arccos[-(n-5)/4] |
Dimension | 4D | 5D | 6D | 7D | 8D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
o3x3o3o |
o3x3o3o *c3o |
o3x3o3o3o *c3o |
o3x3o3o3o3o *c3o |
o3x3o3o3o3o3o *c3o |
o3x3o...o3o *c3o |
Acronym |
rap |
rat |
rojak |
rolaq |
robay | rectified 2n,1 |
Vertex Count | 10 trip | 40 ope | 216 rappip | 2016 rixip | 69120 rillip | ? |
Facet Count rect. simplex | 5 oct | 16 rap | 72 rix | 576 ril | 17280 roc | ? |
Facet Count rect. Gossetic | 16 rap | 27 rat | 56 rojak | 240 rolaq | ? | |
Facet Count demihypercube | 5 tet | 10 hex | 27 hin | 126 hax | 2160 hesa | ? |
Circumradius |
sqrt(3/5) 0.774597 | 1 |
sqrt(5/3) 1.290994 |
sqrt(3) 1.732051 |
sqrt(7) 2.645751 | sqrt[(n-1)/(9-n)] |
Inradius wrt. rect. simplex |
1/sqrt(10) 0.316228 |
sqrt(2/5) 0.632456 | 1 |
4/sqrt(7) 1.511858 |
5/2 2.5 | (n-3) sqrt[2/(n(9-n))] |
Inradius wrt. rect. Gossetic |
sqrt(2/5) 0.632456 |
sqrt(2/3) 0.816497 |
2/sqrt(3) 1.154701 | 2 | sqrt[8/((10-n)(9-n))] | |
Inradius wrt. demihypercube |
3/sqrt(40) 0.474342 |
1/sqrt(2) 0.707107 |
5/sqrt(24) 1.020621 |
3/2 1.5 |
7/sqrt(8) 2.474874 | (n-1)/sqrt[8(9-n)] |
Volume |
11 sqrt(5)/96 0.256216 |
9 sqrt(2)/10 1.272792 |
601 sqrt(3)/160 6.506016 |
18643/280 66.582143 |
457563/112 4085.383929 | ? |
Surface |
25 sqrt(2)/12 2.946278 |
(5+11 sqrt(5))/3 9.865583 |
[1089 sqrt(2)+156 sqrt(3)]/40 45.256962 |
[301+4207 sqrt(3)+114 sqrt(7)]/20 394.467670 | ? | ? |
Dihedral angles r.simp. - r.Goss. |
arccos(1/4) 75.522488° r.simp. - r.simp. |
arccos(-3/5) 126.869898° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-sqrt(3/7)] 130.893395° |
arccos(-3/4) 138.590378° | arccos[-3/sqrt(n(10-n))] |
Dihedral angles r.Goss. - r.Goss. |
arccos(-1/4) 104.477512° |
arccos(-1/3) 109.471221° | 120° | arccos[-1/(10-n)] | ||
Dihedral angles r.simp. - demi. |
arccos(-1/4) 104.477512° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-2/sqrt(7)] 139.106605° |
arccos[-5/sqrt(32)] 152.114433° | arccos[-(n-3)/sqrt(4n)] |
Dihedral angles r.Goss. - demi. |
arccos[-1/sqrt(5)] 116.565051° | 120° |
arccos[-1/sqrt(3)] 125.264390° | 135° | arccos[-1/sqrt(10-n)] |
The rectified Gossetic r(1n,2) surely can be described likewise as the birectified Gossetic br(2n,1). In fact it is that polytope, where in its Coxeter-Dynkin diagram exactly the bifurcation node is marked.
Dimension | 4D | 5D | 6D | 7D | 8D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
o3o3x3o |
o3o3x3o *c3o |
o3o3x3o3o *c3o |
o3o3x3o3o3o *c3o |
o3o3x3o3o3o3o *c3o |
o3o3x3o...o3o *c3o |
Acronym |
rap |
nit |
ram |
lanq |
buffy | rectified 1n,2 |
Vertex Count | 10 trip | 80 tisdip | 720 tratrip | 10080 tratepe | 483840 trippen | ? |
Facet Count birect. simp. | 5 tet | 16 rap | 72 dot | 576 bril | 17280 broc | ? |
Facet Count rect. Goss. | 16 rap | 27 nit | 56 ram | 240 lanq | ? | |
Facet Count birect. hyp.c. | 5 oct | 10 ico | 27 nit | 126 brox | 2160 bersa | ? |
Circumradius |
sqrt(3/5) 0.774597 |
sqrt(3/2) 1.224745 |
sqrt(3) 1.732051 |
sqrt(6) 2.449490 |
sqrt(15) 3.872983 | sqrt[3(n-3)/(9-n)] |
Inradius wrt. birect. simp. |
3/sqrt(40) 0.474342 |
3/sqrt(10) 0.948683 |
3/2 1.5 |
6/sqrt(7) 2.267787 |
15/4 3.75 | 3(n-3)/sqrt[2n(9-n)] |
Inradius wrt. rect. Gossetic |
3/sqrt(10) 0.948683 |
sqrt(3/2) 1.224745 |
sqrt(3) 1.732051 | 3 | sqrt[18/((10-n)(9-n))] | |
Inradius wrt. birect. hyp.c. |
1/sqrt(10) 0.316228 |
1/sqrt(2) 0.707107 |
sqrt(3/2) 1.224745 | 2 |
5/sqrt(2) 3.535534 | (n-3)/sqrt(18-2n) |
Volume |
11 sqrt(5)/96 0.256216 |
31 sqrt(2)/10 4.384062 |
243 sqrt(3)/8 52.611043 | ? | ? | ? |
Surface |
25 sqrt(2)/12 2.946278 |
[60+11 sqrt(5)]/3 28.198916 |
[1674 sqrt(2)+99 sqrt(3)]/10 253.886653 | ? | ? | ? |
Dihedral angles bir.s. - r.Goss. |
arccos(-3/5) 126.869898° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-sqrt(3/7)] 130.893395° |
arccos(-3/4) 138.590378° | arccos[-3/sqrt(n(10-n))] | |
Dihedral angles bir.s. - bir.h.c. |
arccos(-1/4) 104.477512° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-2/sqrt(7)] 139.106605° |
arccos[-5/sqrt(32)] 152.114433° | arccos[-(n-3)/sqrt(4n)] |
Dihedral angles r.Goss. - r.Goss. |
arccos(-1/4) 104.477512° |
arccos(-1/3) 109.471221° | 120° | arccos[-1/(10-n)] | ||
Dihedral angles r.Goss. - bir.h.c. |
arccos[-1/sqrt(5)] 116.565051° | 120° |
arccos[-1/sqrt(3)] 125.264390° | 135° | arccos[-1/sqrt(10-n)] | |
Dihedral angles bir.h.c. - bir.h.c. |
arccos(1/4) 75.522488° | 90° |
arccos(-1/4) 104.477512° | 120° |
arccos(-3/4) 138.590378° | arccos[-(n-5)/4] |
Dimension | 4D | 5D | 6D | 7D | 8D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
o3x3x3x |
o3o3x3x *c3o |
o3o3o3x3x *c3o |
o3o3o3o3x3x *c3o |
o3o3o3o3o3x3x *c3o |
o3o...o3x3x *c3o |
Acronym |
grip |
thin |
tojak |
tanq |
tiffy | truncated (n-4)2,1 |
Vertex Count | 60 | 160 | 432 | 1512 | 13440 | ? |
Facet Count trunc. simplex | 5 tut | 16 tip | 72 tix | 576 til | 17280 toc | ? |
Facet Count Gossetic | 10 trip | 16 rap | 27 hin | 56 jak | 240 naq | ? |
Facet Count trunc. ortho. | 5 toe | 10 thex | 27 tot | 126 tag | 2160 taz | ? |
Circumradius |
sqrt(17/5) 1.843909 |
sqrt(29/8) 1.903943 | 2 |
sqrt(19)/2 2.179449 |
sqrt(7) 2.645751 | sqrt[(54-5n)/(18-2n)] |
Inradius wrt. trunc. simplex |
9/sqrt(40) 1.423025 |
9/sqrt(40) 1.423025 |
3/2 1.5 |
9/sqrt(28) 1.700840 |
9/4 2.25 | 9/sqrt[n(18-2n)] |
Inradius wrt. Gossetic |
13/sqrt(60) 1.678293 |
11/sqrt(40) 1.739253 |
sqrt(27/8) 1.837117 |
7/sqrt(12) 2.020726 |
5/2 2.5 | (21-2n)/sqrt[(18-2n)(10-n)] |
Inradius wrt. trunc. ortho. |
3/sqrt(10) 0.948683 |
3/sqrt(8) 1.060660 |
sqrt(3/2) 1.224745 |
3/2 1.5 |
3/sqrt(2) 2.121320 | 3/sqrt(18-2n) |
Volume |
287 sqrt(5)/32 20.054735 |
623 sqrt(2)/24 36.710627 |
7251 sqrt(3)/160 78.494378 |
37109/140 265.064286 | ? | ? |
Surface |
[595 sqrt(2)+30 sqrt(3)]/12 74.451549 |
[770+87 sqrt(5)]/6 160.756319 |
[8685 sqrt(2)+1422 sqrt(3)]/40 368.635526 |
[10122+35 sqrt(3)+722 sqrt(7)]/10 1209.285422 | ? | ? |
Dihedral angles tr.simp. - Goss. |
arccos[-sqrt(3/8)] 127.761244° |
arccos(-3/5) 126.869898° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-sqrt(3/7)] 130.893395° |
arccos(-3/4) 138.590378° | arccos[-3/sqrt(n(10-n))] |
Dihedral angles tr.sim. - tr.orth. |
arccos(-1/4) 104.477512° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-2/sqrt(7)] 139.106605° |
arccos[-5/sqrt(32)] 152.114433° | arccos[-(n-3)/sqrt(4n)] |
Dihedral angles Goss. - tr.orth. |
arccos[-1/sqrt(6)] 114.094843° |
arccos[-1/sqrt(5)] 116.565051° | 120° |
arccos[-1/sqrt(3)] 125.264390° | 135° | arccos[-1/sqrt(10-n)] |
Dihedral angles tr.orth. - tr.orth. |
arccos(1/4) 75.522488° | 90° |
arccos(-1/4) 104.477512° | 120° |
arccos(-3/4) 138.590378° | arccos[-(n-5)/4] |
Dimension | 4D | 5D | 6D | 7D | 8D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
x3x3o3o |
x3x3o3o *c3o |
x3x3o3o3o *c3o |
x3x3o3o3o3o *c3o |
x3x3o3o3o3o3o *c3o |
x3x3o...o3o *c3o |
Acronym |
tip |
tot |
tojak |
talq |
toby | truncated 2n,1 |
Vertex Count | 20 | 80 | 432 | 4032 | 138240 | ? |
Facet Count trunc. simplex | 5 tut | 16 tip | 72 tix | 576 til | 17280 toc | ? |
Facet Count trunc. Gossetic | 16 tip | 27 tot | 56 tojak | 240 talq | ? | |
Facet Count demihypercube | 5 tet | 10 hex | 27 hin | 126 hax | 2160 hesa | ? |
Circumradius |
sqrt(8/5) 1.264911 |
sqrt(5/2) 1.581139 | 2 |
sqrt(7) 2.645751 | 4 | sqrt[2n/(9-n)] |
Inradius wrt. trunc. simplex |
3/sqrt(40) 0.474342 |
3/sqrt(10) 0.948683 |
3/2 1.5 |
6/sqrt(7) 2.267787 |
15/4 3.75 | 3(n-3)/sqrt[2n(9-n)] |
Inradius wrt. trunc. Gossetic |
3/sqrt(10) 0.948683 |
sqrt(3/2) 1.224745 |
sqrt(3) 1.732051 | 3 | sqrt[18/((9-n)(10-n))] | |
Inradius wrt. demihypercube |
7/sqrt(40) 1.106797 |
sqrt(2) 1.414214 |
sqrt(27/8) 1.837117 |
5/2 2.5 |
11/sqrt(8) 3.889087 | (n+3)/sqrt[8(9-n)] |
Volume |
19 sqrt(5)/24 1.770220 |
119 sqrt(2)/15 11.219428 |
7251 sqrt(3)/160 78.494378 | ? | ? | ? |
Surface |
10 sqrt(2) 14.142136 |
(5+76 sqrt(5))/3 58.313722 |
[8685 sqrt(2)+1422 sqrt(3)]/40 368.635526 | ? | ? | ? |
Dihedral angles tr.sim. - tr.Goss. |
arccos(1/4) 75.522488° tr.sim. - tr.sim. |
arccos(-3/5) 126.869898° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-sqrt(3/7)] 130.893395° |
arccos(-3/4) 138.590378° | arccos[-3/sqrt(n(10-n))] |
Dihedral angles tr.Goss. - tr.Goss. |
arccos(-1/4) 104.477512° |
arccos(-1/3) 109.471221° | 120° | arccos[-1/(10-n)] | ||
Dihedral angles tr.sim. - demi. |
arccos(-1/4) 104.477512° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-2/sqrt(7)] 139.106605° |
arccos[-5/sqrt(32)] 152.114433° | arccos[-(n-3)/sqrt(4n)] |
Dihedral angles tr.Goss. - demi. |
arccos[-1/sqrt(5)] 116.565051° | 120° |
arccos[-1/sqrt(3)] 125.264390° | 135° | arccos[-1/sqrt(10-n)] |
Dimension | 4D | 5D | 6D | 7D | 8D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
o3o3x3x |
o3o3x3o *c3x |
o3o3x3o3o *c3x |
o3o3x3o3o3o *c3x |
o3o3x3o3o3o3o *c3x |
o3o3x3o...o3o *c3x |
Acronym |
tip |
thin |
tim |
tolin |
tabif | truncated 1n,2 |
Vertex Count | 20 | 160 | 1440 | 20160 | 967680 | ? |
Facet Count birect. simp. | 5 tet | 16 rap | 72 dot | 576 bril | 17280 broc | ? |
Facet Count trunc. Goss. | 16 tip | 27 thin | 56 tim | 240 tolin | ? | |
Facet Count trunc. demih.c. | 5 tut | 10 thex | 27 thin | 126 thax | 2160 thesa | ? |
Circumradius |
sqrt(8/5) 1.264911 |
sqrt(29/8) 1.903943 |
sqrt(7) 2.645751 |
sqrt(55)/2 3.708099 |
sqrt(34) 5.830952 | sqrt[(13n-36)/(18-2n)] |
Inradius wrt. birect. simp. |
7/sqrt(40) 1.106797 |
11/sqrt(40) 1.739253 |
5/2 2.5 |
19/sqrt(28) 3.590662 |
23/4 5.75 | (4n-9)/sqrt[2n(9-n)] |
Inradius wrt. trunc. Goss. |
9/sqrt(40) 1.423025 |
sqrt(27/8) 1.837117 |
sqrt(27)/2 2.598076 |
9/2 4.5 | 9/sqrt[2(9-n)(10-n)] | |
Inradius wrt. trunc. demih.c. |
3/sqrt(40) 0.474342 |
3/sqrt(8) 1.060660 |
sqrt(27/8) 1.837117 | 3 |
15/sqrt(8) 5.303301 | 3(n-3)/sqrt[8(9-n)] |
Volume |
19 sqrt(5)/24 1.770220 |
623 sqrt(2)/24 36.710627 |
5673 sqrt(3)/16 614.120264 | ? | ? | ? |
Surface |
10 sqrt(2) 14.142136 |
[770+87 sqrt(5)]/6 160.756319 |
[28035 sqrt(2)+198 sqrt(3)]/20 1999.521164 | ? | ? | ? |
Dihedral angles bir.s. - tr.Goss. |
arccos(-3/5) 126.869898° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-sqrt(3/7)] 130.893395° |
arccos(-3/4) 138.590378° | arccos[-3/sqrt(n(10-n))] | |
Dihedral angles bir.s. - tr.demi. |
arccos(-1/4) 104.477512° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-2/sqrt(7)] 139.106605° |
arccos[-5/sqrt(32)] 152.114433° | arccos[-(n-3)/sqrt(4n)] |
Dihedral angles tr.Goss. - tr.Goss. |
arccos(-1/4) 104.477512° |
arccos(-1/3) 109.471221° | 120° | arccos[-1/(10-n)] | ||
Dihedral angles tr.Goss. - tr.demi. |
arccos[-1/sqrt(5)] 116.565051° | 120° |
arccos[-1/sqrt(3)] 125.264390° | 135° | arccos[-1/sqrt(10-n)] | |
Dihedral angles tr.demi. - tr.demi. |
arccos(1/4) 75.522488° | 90° |
arccos(-1/4) 104.477512° | 120° |
arccos(-3/4) 138.590378° | arccos[-(n-5)/4] |
Dimension | 4D | 5D | 6D | 7D | 8D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
x3o3x3x |
x3o3o3x *c3x |
x3o3o3o3x *c3x |
x3o3o3o3o3x *c3x |
x3o3o3o3o3o3x *c3x |
x3o...o3x *c3x |
Acronym |
prip |
spat |
spam |
sethalq |
spuffy | all-ends exp. En |
Vertex Count | 60 | 320 | 2160 | 24192 | ? | ? |
Facet Count exp. simpl. | 5 co | 16 spid | 72 scad | 576 staf | ? suph | ? |
Facet Count exp. Goss. | 10 trip | 16 spid | 27 siphin | 56 hejak | ? shilq | ? |
Facet Count exp. Goss. pr. | 216 spiddip | 756 siphinnip | ? hejakip | ? | ||
Facet Count duoprism I | 4032 traspid | ? trasiphin | ? | |||
Facet Count duoprism II | ? tetspid | ? | ||||
Facet Count dippip | 80 tisdip | 720 tratrip | 10080 tratepe | ? trippen | ? | |
Facet Count exp. simpl. pr. | 10 hip | 40 cope | 216 spiddip | 2016 scadip | ? staffip | ? |
Facet Count exp. demicube | 5 tut | 10 rit | 27 siphin | 126 sochax | ? suthesa | ? |
Circumradius |
sqrt(13/5) 1.612452 |
sqrt(7/2) 1.870829 |
sqrt(5) 2.236068 |
sqrt(8) 2.828427 |
sqrt(17) 4.123106 | sqrt[(9+n)/(9-n)] |
Inradius wrt. exp. simpl. |
sqrt(8/5) 1.264911 |
sqrt(5/2) 1.581139 | 2 |
sqrt(7) 2.645751 | 4 | sqrt[2n/(9-n)] |
Inradius wrt. exp. Goss. |
11/sqrt(60) 1.420094 |
sqrt(5/2) 1.581139 |
sqrt(27/8) 1.837117 |
4/sqrt(3) 2.309401 |
7/2 3.5 | (15-n)/sqrt[2(10-n)(9-n)] |
Inradius wrt. exp. Goss. pr. |
sqrt(15)/2 1.936492 |
7/sqrt(8) 2.474874 |
13/sqrt(12) 3.752777 | (21-n)/sqrt[4(11-n)(9-n)] | ||
Inradius wrt. duoprism I |
sqrt(20/3) 2.581989 | ? | ? | |||
Inradius wrt. duoprism II | ? | ? | ||||
Inradius wrt. dippip |
4/sqrt(6) 1.632993 |
7/sqrt(12) 2.020726 |
13/sqrt(24) 2.653614 |
31/sqrt(60) 4.002083 | (5n-9)/sqrt[12(n-3)(9-n)] | |
Inradius wrt. exp. simpl. pr. |
sqrt(27/20) 1.161895 |
3/2 1.5 |
sqrt(15)/2 1.936492 |
sqrt(27)/2 2.598076 |
sqrt(63)/2 3.968627 | sqrt[(9n-9)/(36-4n)] |
Inradius wrt. exp. demicube |
7/sqrt(40) 1.106797 |
sqrt(2) 1.414214 |
sqrt(27/8) 1.837117 |
5/2 2.5 |
11/sqrt(8) 3.889087 | (n+3)/sqrt[8(9-n)] |
Volume |
237 sqrt(5)/32 16.560878 |
142 sqrt(2)/3 66.939442 |
17811 sqrt(3)/80 385.619462 |
34715/8 4339.375 | ? | ? |
Surface |
[215 sqrt(2)+210 sqrt(3)]/12 55.648882 |
5 [23+40 sqrt(2)+ +12 sqrt(3)+14 sqrt(5)]/3 219.430173 |
[2700+4203 sqrt(2)+ +756 sqrt(3)+6300 sqrt(5)]/20 1202.029914 |
[19159+58842 sqrt(2)+ +38962 sqrt(3)+4200 sqrt(6)+ +1848 sqrt(7)+14700 sqrt(15)]/20 12098.418927 | ? | ? |
Dihedral angles e.sim. - e.Goss. |
arccos[-sqrt(3/8)] 127.761244° |
arccos(-3/5) 126.869898° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-sqrt(3/7)] 130.893395° |
arccos(-3/4) 138.590378° | arccos[-3/sqrt(n(10-n))] |
Dihedral angles e.sim. - e.Goss.p. | ? | ? | ? | ? | ||
Dihedral angles e.Goss. - e.Goss.p. | ? | ? | ? | ? | ||
Dihedral angles e.sim. - duop.I | ? | ? | ? | |||
Dihedral angles e.Goss.p. - duop.I | ? | ? | ? | |||
Dihedral angles e.sim. - duop.II | ? | ? | ||||
Dihedral angles duop.I - duop.II | ? | ? | ||||
Dihedral angles e.sim. - dippip | ? | ? | ? | ? | ? | |
Dihedral angles duop.II - dippip | ? | ? | ||||
Dihedral angles e.sim. - e.sim.p. |
arccos[-sqrt(1/6)] 114.094843° |
arccos[-sqrt(2/5)] 129.231520° | ? | ? | ? | ? |
Dihedral angles e.Goss. - e.sim.p. |
arccos(-2/3) 131.810315° |
arccos[-sqrt(2/5)] 129.231520° | ? | ? | ? | ? |
Dihedral angles e.Goss.p. - e.sim.p. | ? | ? | ? | ? | ||
Dihedral angles duop.I - e.sim.p. | ? | ? | ? | |||
Dihedral angles duop.II - e.sim.p. | ? | ? | ||||
Dihedral angles dippip - e.sim.p. | ? | ? | ? | ? | ? | |
Dihedral angles e.sim. - e.demic. |
arccos(-1/4) 104.477512° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-2/sqrt(7)] 139.106605° |
arccos[-5/sqrt(32)] 152.114433° | arccos[-(n-3)/sqrt(4n)] |
Dihedral angles e.Goss. - e.demic. |
arccos[-1/sqrt(5)] 116.565051° | 120° |
arccos[-1/sqrt(3)] 125.264390° | 135° | arccos[-1/sqrt(10-n)] | |
Dihedral angles e.Goss.p. - e.demic. | ? | ? | ? | ? | ||
Dihedral angles duop.I - e.demic. | ? | ? | ? | |||
Dihedral angles duop.II - e.demic. | ? | ? | ? | |||
Dihedral angles e.sim.p. - e.demic. |
arccos[-sqrt(3/8)] 127.761244° | 135° | ? | ? | ? | ? |
Dimension | 4D | 5D | 6D | 7D | 8D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
x3x3x3x |
x3x3x3x *c3x |
x3x3x3x3x *c3x |
x3x3x3x3x3x *c3x |
x3x3x3x3x3x3x *c3x |
x3x...x3x3x *c3x |
Acronym |
gippid |
gippit |
gopam |
gotanq |
gupofy | omnitr. (n-4)2,1 |
Vertex Count | 120 | 1920 | 51840 | 2903040 | 696729600 | ? |
Facet Count wrt. type 1 | 5 toe | 16 gippid | 72 gocad | 576 gotaf | 17280 guph | ? |
Facet Count wrt. type 2 | 10 hip | 16 gippid | 27 gippit | 56 gopam | 240 gotanq | ? |
Facet Count wrt. type 3 | 10 hip | 80 shiddip | 216 gippiddip | 756 gippitip | 6720 gopamp | ? |
Facet Count wrt. type 4 | 5 toe | 40 tope | 720 hahip | 4032 hagippid | 60480 hagippit | ? |
Facet Count wrt. type 5 | 10 tico | 216 gippiddip | 2016 gocadip | 241920 toegippid | ? | |
Facet Count wrt. type 6 | 27 gippit | 10080 hatope | 483840 hagippiddip | ? | ||
Facet Count wrt. type 7 | 126 gocog | 69120 gotafip | ? | |||
Facet Count wrt. type 8 | 2160 gotaz | ? | ||||
Circumradius |
sqrt(5) 2.236068 |
sqrt(15) 3.872983 |
sqrt(39) 6.244998 |
sqrt(399)/2 9.987492 |
sqrt(310) 17.606817 | ? |
Inradius wrt. facet type 1 |
sqrt(5/2) 1.581139 |
sqrt(10) 3.162278 |
11/2 5.5 |
sqrt(343)/2 9.260130 | 17 | ? |
Inradius wrt. facet type 2 |
sqrt(15)/2 1.936492 |
sqrt(10) 3.162278 |
sqrt(24) 4.898980 |
sqrt(243)/2 7.794229 |
29/2 14.5 | ? |
Inradius wrt. facet type 3 |
sqrt(15)/2 1.936492 |
sqrt(27/2) 3.674235 |
sqrt(135)/2 5.809475 |
13/sqrt(2) 9.192388 |
sqrt(1083)/2 16.454483 | ? |
Inradius wrt. facet type 4 |
sqrt(5/2) 1.581139 |
7/2 3.5 |
sqrt(147)/2 6.062178 |
sqrt(375)/2 9.682458 |
sqrt(294) 17.146428 | ? |
Inradius wrt. facet type 5 |
sqrt(8) 2.828427 |
sqrt(135)/2 5.809475 |
sqrt(363)/2 9.526279 |
sqrt(605/2) 17.392527 | ? | |
Inradius wrt. facet type 6 |
sqrt(24) 4.898980 |
sqrt(96) 9.797959 |
sqrt(1215)/2 17.428425 | ? | ||
Inradius wrt. facet type 7 |
17/2 8.5 |
sqrt(1183)/2 17.197384 | ? | |||
Inradius wrt. facet type 8 |
sqrt(529/2) 16.263456 | ? | ||||
Volume |
125 sqrt(5)/4 69.877124 | ? | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? | ? |
Dihedral angles types 1 - 2 |
arccos[-sqrt(3/8)] 127.761244° |
arccos(-3/5) 126.869898° |
arccos[-sqrt(3/8)] 127.761244° |
arccos[-sqrt(3/7)] 130.893395° |
arccos(-3/4) 138.590378° | arccos[-3/sqrt(n(10-n))] |
Dihedral angles types 1 - 3 |
arccos[-sqrt(1/6)] 114.094843° | ? | ? | ? | ? | ? |
Dihedral angles types 1 - 4 |
arccos(-1/4) 104.477512° |
arccos[-sqrt(2/5)] 129.231520° | ? | ? | ? | ? |
Dihedral angles types 1 - 5 |
arccos[-1/sqrt(5)] 116.565051° | ? | ? | ? | ? | |
Dihedral angles types 1 - 6 |
arccos[-sqrt(3/8)] 127.761244° | ? | ? | ? | ||
Dihedral angles types 1 - 7 |
arccos[-2/sqrt(7)] 139.106605° | ? | ? | |||
Dihedral angles types 1 - 8 |
arccos[-5/sqrt(32)] 152.114433° | ? | ||||
Dihedral angles types 2 - 3 |
arccos(-2/3) 131.810315° | ? | ? | ? | ? | ? |
Dihedral angles types 2 - 4 |
arccos(-1/4) 104.477512° |
arccos[-sqrt(2/5)] 129.231520° | ? | ? | ? | ? |
Dihedral angles types 2 - 5 |
arccos[-1/sqrt(5)] 116.565051° | ? | ? | ? | ? | |
Dihedral angles types 2 - 6 | 120° | ? | ? | ? | ||
Dihedral angles types 2 - 7 |
arccos[-1/sqrt(3)] 125.264390° | ? | ? | |||
Dihedral angles types 2 - 8 | 135° | ? | ||||
Dihedral angles types 3 - 4 |
arccos[-sqrt(3/8)] 127.761244° | ? | ? | ? | ? | ? |
Dihedral angles types 3 - 5 | ? | ? | ? | ? | ? | |
Dihedral angles types 3 - 6 | ? | ? | ? | ? | ||
Dihedral angles types 3 - 7 | ? | ? | ? | |||
Dihedral angles types 3 - 8 | ? | ? | ||||
Dihedral angles types 4 - 5 | 135° | ? | ? | ? | ? | |
Dihedral angles types 4 - 6 | ? | ? | ? | ? | ||
Dihedral angles types 4 - 7 | ? | ? | ? | |||
Dihedral angles types 4 - 8 | ? | ? | ||||
Dihedral angles types 5 - 6 | ? | ? | ? | ? | ||
Dihedral angles types 5 - 7 | ? | ? | ? | |||
Dihedral angles types 5 - 8 | ? | ? | ||||
Dihedral angles types 6 - 7 | ? | ? | ? | |||
Dihedral angles types 6 - 8 | ? | ? | ||||
Dihedral angles types 7 - 8 | ? | ? |
The name of the Ursatopes derives from the acronym of the 3D sequence member, teddi (J63), being homonym to the toy-bear, or Latinized "urs". The simplexial ones are defined generally as the bistratic lace towers ofx3xoo3ooo...ooo3ooo&#xt, i.e. the n-dimensional simplexial ursatope Un can be described as the rectified simplex rSn-1 atop the f-scaled regular simplex f·Sn-1 atop the (unit) regular simplex Sn-1. All those ursatopes happen to be orbiform CRFs, i.e. are circumscribable, convex, and regular faced.
It could be mentioned here additionally that the simplexial ursatope Un generally is nothing but the vertex figure of s3s4o3o...o3o, which for low dimensions is spherical, at rank 5 (i.e. 5 nodes) becomes an euclidean tetracomb, and thereafter will belong to hyperbolic geometry. This then gets reflected too in the table below by the values of the circumradii of Un, which traverse unity at n=4.
Further the vertex figures of these polytopes could be described uniformely. At the lower 2 of its vertex types one has ox3oo...oo3oo&#f spike-like tall simplex pyramides, the top vertices however are xf xo...oo3oo&#x, i.e. simplex prism wedges, where the additional wedge-edge has size f and runs axis parallel to the base (simplex prism).
Dimension | 2D | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|---|
Dynkin diagram |
ofx&#xt |
ofx3xoo&#xt |
ofx3xoo3ooo&#xt |
ofx3xoo3ooo3ooo&#xt |
ofx3xoo3ooo3ooo3ooo&#xt |
ofx3xoo3ooo...ooo3ooo&#xt |
Acronym |
peg |
teddi |
tetu |
penu |
hixu | simpl. n-ursatope |
Vertex Count top layer | 1 | 3 | 6 | 10 | 15 | n(n-1)/2 |
Vertex Count medial layer | 2 | 3 | 4 | 5 | 6 | n |
Vertex Count bottom layer | 2 | 3 | 4 | 5 | 6 | n |
Facet Count top | 1 trig | 1 oct | 1 rap | 1 rix | 1 | |
Facet Count upper lacing | 2 line | 3 trig | 4 tet | 5 pen | 6 hix | n |
Facet Count lower lacing | 2 line | 3 peg | 4 teddi | 5 tetu | 6 penu | n |
Facet Count bottom | 1 line | 1 trig | 1 tet | 1 pen | 1 hix | 1 |
Circumradius |
sqrt[(5+sqrt(5))/10] 0.850651 |
sqrt[(5+sqrt(5))/8] 0.951057 | 1 |
sqrt[2+sqrt(5)]/2 1.029086 |
sqrt[(13+5 sqrt(5))/22] 1.048383 |
sqrt[((29n2+36n+7)+(13n2+16n+3) sqrt(5)) / ((22n2+50n+28)+(10n2+22n+12) sqrt(5))] |
Inradius wrt. top facet |
sqrt[(7+3 sqrt(5))/24] 0.755761 |
1/sqrt(2) 0.707107 |
sqrt[(5 sqrt(5)-2)/20] 0.677508 |
sqrt[(15 sqrt(5)-5)/66] 0.657601 | ? | |
Inradius wrt. upper lacing |
sqrt[(5+2 sqrt(5))/20] 0.688191 |
sqrt[(7+3 sqrt(5))/24] 0.755761 |
sqrt(5/8) 0.790569 |
sqrt[(1+3 sqrt(5))/12] 0.801468 |
sqrt[(23+30 sqrt(5))/132] 0.826099 | ? |
Inradius wrt. lower lacing |
sqrt[(5+2 sqrt(5))/20] 0.688191 |
sqrt[(5+sqrt(5))/40] 0.425325 |
[sqrt(5)-1]/4 0.309017 |
sqrt[sqrt(5)-2]/2 0.242934 |
sqrt[(4-sqrt(5))/44] 0.200223 | ? |
Inradius wrt. bottom facet |
sqrt[(5+2 sqrt(5))/20] 0.688191 |
sqrt[(7+3 sqrt(5))/24] 0.755761 |
sqrt(5/8) 0.790569 |
sqrt[(1+3 sqrt(5))/12] 0.801468 |
sqrt[(23+30 sqrt(5))/132] 0.826099 | ? |
Volume |
sqrt[25+10 sqrt(5)]/4 1.720477 |
[15+7 sqrt(5)]/24 1.277186 |
[28+13 sqrt(5)]/96 0.594468 |
[11 sqrt(5 sqrt(5)-2)+2 sqrt(15+45 sqrt(5))+ +(140+65 sqrt(5)) sqrt(sqrt(5)-2)]/960 0.201536 | ? | ? |
Surface | 5 |
[5 sqrt(3)+3 sqrt(25+10 sqrt(5))]/4 7.326496 |
[30+9 sqrt(2)+14 sqrt(5)]/12 6.169406 |
[70+41 sqrt(5)]/48 3.368308 | ? | ? |
Dihedral angles top - upper |
108° upper - upper |
arccos(-sqrt(5)/3) 138.189685° | ? | ? | ? | ? |
Dihedral angles top - lower | ? | ? | ? | ? | ||
Dihedral angles lower - upper | 108° |
arccos(-sqrt[(5-2 sqrt(5))/15]) 100.812317° | ? | ? | ? | ? |
Dihedral angles lower - lower |
arccos(1/sqrt(5)) 63.434949° | ? | ? | ? | ? | |
Dihedral angles lower - bottom | 108° |
arccos(-sqrt[(5-2 sqrt(5))/15]) 100.812317° | ? | ? | ? | ? |
The orthoplexial Ursatopes are defined generally as the bistratic lace towers ofx3xoo3ooo...ooo3ooo4ooo&#xt, i.e. the n-dimensional orthoplexial ursatope oUn can be described as the rectified orthoplex rOn-1 atop the f-scaled regular orthoplex f·On-1 atop the (unit) regular orthoplex On-1. Nearly all those ursatopes happen to be orbiform CRFs, i.e. are circumscribable, convex, and regular faced. Only the 3D representant shows up external q-edges; none the less it still remains circumscribable.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
ofx4qoo&#xt(non-orbiform) |
ofx3xoo4ooo&#xt |
ofx3xoo3ooo4ooo&#xt |
ofx3xoo3ooo3ooo4ooo&#xt |
ofx3xoo3ooo...ooo4ooo&#xt |
Acronym |
-- |
octu |
hexu |
tacu | orthopl. n-ursatope |
Vertex Count top layer | 4 | 12 | 24 | 40 | 2(n-1)(n-2) |
Vertex Count medial layer | 4 | 6 | 8 | 10 | 2(n-1) |
Vertex Count bottom layer | 4 | 6 | 8 | 10 | 2(n-1) |
Facet Count top | 1 q-square | 1 co | 1 ico | 1 rit | 1 |
Facet Count upper lacing | 4 oq&#x | 6 squippy | 8 octpy | 10 hexpy | 2(n-1) |
Facet Count lower lacing | 4 peg | 8 teddi | 16 tetu | 32 penu | 2n-1 |
Facet Count bottom | 1 square | 1 oct | 1 hex | 1 tac | 1 |
Circumradius |
sqrt[3+sqrt(5)]/2 1.144123 |
sqrt[3+sqrt(5)]/2 1.144123 |
sqrt[3+sqrt(5)]/2 1.144123 |
sqrt[3+sqrt(5)]/2 1.144123 |
sqrt[3+sqrt(5)]/2 1.144123 |
Inradius wrt. top facet |
sqrt[sqrt(5)-1]/2 0.555893 |
sqrt[sqrt(5)-1]/2 0.555893 |
sqrt[sqrt(5)-1]/2 0.555893 |
sqrt[sqrt(5)-1]/2 0.555893 |
sqrt[sqrt(5)-1]/2 0.555893 |
Inradius wrt. upper lacing |
sqrt[1+sqrt(5)]/2 0.899454 |
sqrt[1+sqrt(5)]/2 0.899454 |
sqrt[1+sqrt(5)]/2 0.899454 |
sqrt[1+sqrt(5)]/2 0.899454 |
sqrt[1+sqrt(5)]/2 0.899454 |
Inradius wrt. lower lacing |
sqrt[(5+3 sqrt(5))/20] 0.765121 |
sqrt[(1+sqrt(5))/8] 0.636010 |
sqrt[sqrt(5)-1]/2 0.555893 |
1/2 0.5 |
sqrt[(29(n-1)+13(n-1)sqrt(5)) / ((22n2-38 n+16)+(10n2-18n+8)sqrt(5))] |
Inradius wrt. bottom facet |
sqrt[1+sqrt(5)]/2 0.899454 |
sqrt[1+sqrt(5)]/2 0.899454 |
sqrt[1+sqrt(5)]/2 0.899454 |
sqrt[1+sqrt(5)]/2 0.899454 |
sqrt[1+sqrt(5)]/2 0.899454 |
Volume | ? | ? | ? | ? | ? |
Surface | ? | ? | ? | ? | ? |
Dihedral angles top - upper | ? | ? | ? | ? | ? |
Dihedral angles top - lower | ? | ? | ? | ? | |
Dihedral angles lower - upper | ? | ? | ? | ? | ? |
Dihedral angles lower - lower | ? | ? | ? | ? | ? |
Dihedral angles lower - bottom | ? | ? | ? | ? | ? |
As such these polytopes oo3ox3oo...oo3oo&#x look just to be a mere similar concept to the pyramids on simplex base, which, for sure, as such are nothing but simplices of the next dimension themselves. However it happens that the demihypercube Dn, when seen as lace tower with vertex first orientation, becomes generally ooo..-3-oxo..-3-ooo..-3-oox..-...-ooo..&#xt (n-1 node positions, n/2 or (n+1)/2 layers). Thence this very pyramid of consideration is nothing but the vertex pyramid thereof.
Below it is shown that the dihedral angle at the base decreases to zero with increasing dimension. This is what makes the possibilities to augment other polytopes with this component ever more likely, esp. the possibilities for higher dimensional CRF would explode.
Dimension | 3D | 4D | 5D | 6D | nD |
---|---|---|---|---|---|
Dynkin diagram |
oo3ox&#x |
oo3ox3oo&#x |
oo3ox3oo3oo&#x |
oo3ox3oo3oo&#x |
oo3ox3oo...oo3oo&#x |
Acronym |
tet |
octpy |
rappy |
rixpy | rect. n-simplex pyr. |
Vertex Count | 1+3 | 1+6 | 1+10 | 1+15 | 1+n(n-1)/2 |
Facet Count simpl. lacing | 3 trig | 4 tet | 5 pen | 6 hix | n |
Facet Count other lacing | 4 tet | 5 octpy | 6 rappy | n | |
Facet Count base | 1 trig | 1 oct | 1 rap | 1 rix | 1 |
Circumradius |
sqrt(3/8) 0.612372 |
1/sqrt(2) 0.707107 |
sqrt(5/8) 0.790569 |
sqrt(3)/2 0.866025 | sqrt(n/8) |
Inradius wrt. simpl. lacing |
1/sqrt(24) 0.204124 |
1/sqrt(8) 0.353553 |
3/sqrt(40) 0.474342 |
1/sqrt(3) 0.577350 | (n-2)/sqrt(8n) |
Inradius wrt. other lacing |
1/sqrt(8) 0.353553 |
1/sqrt(8) 0.353553 |
1/sqrt(8) 0.353553 | ||
Inradius wrt. base |
1/sqrt(24) 0.204124 | 0 |
-1/sqrt(40) -0.158114 |
-1/sqrt(12) -0.288675 | -(n-4)/sqrt(8n) |
Volume |
sqrt(2)/12 0.117851 |
1/12 0.833333 |
11 sqrt(2)/480 0.032409 |
13/1440 0.0090278 | (2n-1-n)/(n! sqrt(2n-2)) |
Surface |
sqrt(3) 1.732051 |
sqrt(2) 1.414214 | ? | ? | ? |
Dihedral angles simp. - other |
arccos(1/3) 70.528779° (simp. - simp.) | 120° |
arccos[-1/sqrt(5)] 116.565051° |
arccos[-1/sqrt(6)] 114.094843° | arccos[-1/sqrt(n)] |
Dihedral angles other - other | 90° | 90° | 90° | ||
Dihedral angles simp. - base |
arccos(1/3) 70.528779° | 60° |
arccos(3/5) 53.130102° |
arccos(2/3) 48.189685° | arccos[(n-2)/n] |
Dihedral angles other - base | 60° |
arccos[1/sqrt(5)] 63.434949° |
arccos[1/sqrt(6)] 65.905157° | arccos[1/sqrt(n)] | |
Height |
sqrt(2/3) 0.816497 |
1/sqrt(2) 0.707107 |
sqrt(2/5) 0.632456 |
1/sqrt(3) 0.577350 | sqrt(2/n) |
Dimension | 7D | 8D | 9D | 10D | nD |
Dynkin diagram |
oo3ox3oo3oo3oo3oo |
oo3ox3oo3oo3oo3oo3oo |
oo3ox3oo3oo3oo3oo3oo3oo |
oo3ox3oo3oo3oo3oo3oo3oo3oo |
oo3ox3oo...oo3oo |
Acronym |
rilpy |
rocpy |
renepy |
? | rect. n-simplex pyr. |
Vertex Count | 1+21 | 1+28 | 1+36 | 1+45 | 1+n(n-1)/2 |
Facet Count simpl. lacing | 7 hop | 8 oca | 9 ene | 10 day | n |
Facet Count other lacing | 7 rixpy | 8 rilpy | 9 rocpy | 10 renepy | n |
Facet Count base | 1 ril | 1 roc | 1 rene | 1 reday | 1 |
Circumradius |
sqrt(7/8) 0.935414 | 1 |
3/sqrt(8) 1.060660 |
sqrt(5)/2 1.118034 | sqrt(n/8) |
Inradius wrt. simpl. lacing |
5/sqrt(56) 0.668153 |
3/4 0.75 |
7/sqrt(72) 0.824958 |
2/sqrt(5) 0.894427 | (n-2)/sqrt(8n) |
Inradius wrt. other lacing |
1/sqrt(8) 0.353553 |
1/sqrt(8) 0.353553 |
1/sqrt(8) 0.353553 |
1/sqrt(8) 0.353553 |
1/sqrt(8) 0.353553 |
Inradius wrt. base |
-3/sqrt(56) -0.400892 |
-1/2 -0.5 |
-5/sqrt(72) -0.589256 |
-6/sqrt(80) -0.670820 | -(n-4)/sqrt(8n) |
Volume |
19 sqrt(2)/13440 0.0019993 |
1/2688 0.00037202 |
247 sqrt(2)/5806080 0.000060163 |
251/29030400 0.0000086461 | (2n-1-n)/(n! sqrt(2n-2)) |
Surface | ? | ? | ? | ? | ? |
Dihedral angles simp. - other |
arccos[-1/sqrt(7)] 112.207654° |
arccos[-1/sqrt(8)] 110.704811° |
arccos(-1/3) 109.471221° |
arccos[-1/sqrt(10)] 108.434949° | arccos[-1/sqrt(n)] |
Dihedral angles other - other | 90° | 90° | 90° | 90° | 90° |
Dihedral angles simp. - base |
arccos(5/7) 44.415309° |
arccos(3/4) 41.409622° |
arccos(7/9) 38.942441° |
arccos(4/5) 36.869898° | arccos[(n-2)/n] |
Dihedral angles other - base |
arccos[1/sqrt(7)] 67.792346° |
arccos[1/sqrt(8)] 69.295189° |
arccos(1/3) 70.528779° |
arccos[1/sqrt(10)] 71.565051° | arccos[1/sqrt(n)] |
Height |
sqrt(2/7) 0.534522 |
1/2 0.5 |
sqrt(2)/3 0.471405 |
1/sqrt(5) 0.447214 | sqrt(2/n) |
Volumes of duoprisms A×B are easily calculated as the product of the subdimensional volumes of A resp. of B. Thus plain prisms (of unit height, for sure) have the same numeric volume value, as the subdimensional volume of its base.
Vertex counts of duoprisms A×B likewise are given as the product of the vertex counts of A resp. of B.
This case results in even dimensions only.
From the axial representation of one of the factors, i.e. of Sn,
it becomes clear that Sn×Sn can well be represented
as the segmentotope of the regular simplex Sn
atop the simplex duoprism Sn×Sn-1.
Thence, by means of the lace prism notation,
Sn×Sn
x3o3o...o3o x3o3o...o3o (2n nodes) can be described as well as xx3oo3oo...oo3oo ox3oo...oo3oo&#x (2n-1 nodes).
It could be mentioned here additionally that the simplex duoprism Sn×Sn generally is nothing but the vertex figure of the mid-rectified simplex mrS2n+1.
Dimension | 2D | 4D | 6D | 8D | 10D | (2n)D |
---|---|---|---|---|---|---|
Dynkin diagram |
x x |
x3o x3o |
x3o3o x3o3o |
x3o3o3o x3o3o3o |
x3o3o3o3o x3o3o3o3o |
x3o...o3o x3o...o3o |
Acronym |
square |
triddip |
tetdip |
pendip |
hixdip | n-simplex duoprism |
Vertex Count | 4 | 9 | 16 | 25 | 36 | (n+1)2 |
Facet Count | 4 line | 6 trip | 8 tratet | 10 tetpen | 12 penhix | 2(n+1) |
Circumradius |
1/sqrt(2) 0.707107 |
sqrt(2/3) 0.816497 |
sqrt(3)/2 0.866025 |
2/sqrt(5) 0.894427 |
sqrt(5/6) 0.912871 | sqrt[n/(n+1)] |
Inradius |
1/2 0.5 |
1/sqrt(12) 0.288675 |
1/sqrt(24) 0.204124 |
1/sqrt(40) 0.158114 |
1/sqrt(60) 0.129099 | 1/sqrt[2n(n+1)] |
Volume | 1 |
3/16 0.1875 |
1/72 0.013889 |
5/9216 0.00054253 |
1/76800 0.000013021 | (n+1)/[2n (n!)2] |
Surface | 4 |
sqrt(27)/2 2.598076 |
1/sqrt(6) 0.408248 |
5 sqrt(10)/576 0.027450 |
1/[256 sqrt(15)] 0.0010086 | sqrt[(n+1)3/(n 22n-3 ((n-1)!)4)] |
Dihedral angles at Sn-1×Sn-1 | 90° | 90° | 90° | 90° | 90° | 90° |
Dihedral angles at Sn×Sn-2 | 60° |
arccos(1/3) 70.528779 |
arccos(1/4) 75.522488 |
arccos(1/5) 78.463041 | arccos(1/n) |
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