Site Map Polytopes Dynkin Diagrams Vertex Figures, etc. Incidence Matrices Index

Analogs

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.



Symmetry An

Regular Simplex Sn   (up)

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 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)

Rectified Simplex rSn   (up)

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)

Facetorectified Simplex frSn   (up)

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)
?

Birectified Simplex brSn   (up)

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)

Truncated Simplex tSn   (up)

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)

Bitruncated Simplex btSn   (up)

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)

Mid-rectified Simplex mrSn   (up)

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
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)

Mid-truncated Simplex mtSn   (up)

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)

Maximal Expanded Simplex eSn   (up)

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
...

Retroexpanded Simplex reSn   (up)

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)

Omnitruncated Simplex otSn   (up)

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)])]



Symmetry BCn

Regular Orthoplex On   (up)

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 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
...

Regular Hypercube Cn   (up)

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 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°

Rectified Orthoplex rOn   (up)

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)

Rectified Hypercube rCn   (up)

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°

Facetorectified Hypercube frCn   (up)

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)

Birectified Orthoplex brOn   (up)

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°

Birectified Hypercube brCn   (up)

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°

Truncated Orthoplex tOn   (up)

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)

Truncated Hypercube tCn   (up)

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°

Quasitruncated Hypercube qtCn   (up)

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°

Bitruncated Hypercube btCn   (up)

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°

Rhombated Hypercube rbCn   (up)
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°

Quasihombateded Hypercube qrbCn   (up)
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°

Maximal Expanded Hypercube eCn   (up)

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°


Quasiexpanded Hypercube qeCn   (up)

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°

Retroexpanded Hypercube reCn   (up)

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°

Quasiretroexpanded Hypercube qreCn   (up)

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°

Omnitruncated Hypercube otCn   (up)
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))]


Symmetry Dn

Demihypercube Dn   (up)

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°

Demicross hOn   (up)

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)]

Truncated Demihypercube tDn   (up)

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°

Maximal Expanded Demihypercube eDn   (up)

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°


Symmetry En

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.

Gossetic n2,1   (up)
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]

Gossetic 2n,1   (up)
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)]

Gossetic 1n,2   (up)
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)]

Rectified Gossetic r(n2,1)   (up)
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]

Rectified Gossetic r(2n,1)   (up)
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)]

Rectified Gossetic r(1n,2)   (up)

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]

Truncated Gossetic t(n2,1)   (up)
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]

Truncated Gossetic t(2n,1)   (up)
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)]

Truncated Gossetic t(1n,2)   (up)
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]

All-Ends Expanded Gossetic eEn   (up)
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° ? ? ? ?

Omnitruncated Gossetic otEn   (up)
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
? ?


Some Axial Cases

Simplexial Ursatope Un  (up)

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 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°
? ? ? ?

Orthoplexial Ursatope oUn  (up)

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 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
? ? ? ? ?

Rectified Simplex Pyramid rSn-py   (up)

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&#x
oo3ox3oo3oo3oo3oo3oo&#x
oo3ox3oo3oo3oo3oo3oo3oo&#x
oo3ox3oo3oo3oo3oo3oo3oo3oo&#x
oo3ox3oo...oo3oo&#x
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)


Some Duoprismatic Cases

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.


Simplex Duoprism Sn×Sn   (up)

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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