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 alogogy 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 4 5 6 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 8 9 10 11 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)

For 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 0n,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 6 10 15 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 28 36 45 55 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
sqrt(4/5)
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)

Birectified Simplex brSn   (up)

For 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 0n,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 10 20 35 (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 84 120 165 (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)

For 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 tegum 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)

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.

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 20 70 252 (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.

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 these polytopes can be decomposed accordingly.

For 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 tegum notation, eSn = x3o...o3x (n nodes) can be described as well as xxo3ooo...ooo3oxx&#xt (n-1 node positions).

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)
? ?



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).

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 6 8 10 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 14 16 18 20 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)

Regular Hypercube Cn   (up)

These polytopes are closely related to the prism product. In fact Cn generally can be described as 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).

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 8 16 32 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 64 128 256 512 1024 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. Furthermore it forces that the facet-to-bodycenter pyramids all are CRF, i.e. that these polytopes can be decomposed accordingly.

For these polytopes rOn generally can be described as the bistratic lace tower of the orthoplex On-1 atop the rectified orthoplex rOn-1 atop the orthoplex On-1. Thence, by means of the lace prism 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 prism notation, rOn = o3x3o...o3o4o (n nodes) can be described as well as xxo3ooo...ooo3oxx&#xt (n-1 node positions).

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 24 40 60 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 84 112 144 180 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/sqrt(9)]
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)

For 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 prism 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 32 80 192 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
?
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 ? ? ? ? ?
Dihedral angles
rect. - orthopl.
arccos[-1/sqrt(7)]
112.207654°
arccos[-1/sqrt(8)]
110.704811°
arccos[-1/sqrt(9)]
109.471221°
arccos[-1/sqrt(10)]
108.434949°
arccos[-1/sqrt(n)]
Dihedral angles
rect. - rect.
90° 90° 90° 90° 90°

Truncated Orthoplex tOn   (up)

For these polytopes tOn generally can be described as the tetrastratic lace tower of the orthoplex On-1 atop the u-scaled orthoplex On-1 atop the truncated orthoplex tOn-1 atop the u-scaled orthoplex On-1 atop the orthoplex On-1. Thence, by means of the lace prism 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)

For these polytopes tCn generally can be described as the tristratic lace tower of the truncated hypercube tCn-1 atop the w-scaled hypercube Cn-1 atop the w-scaled hypercube Cn-1 atop the truncated hypercube tCn-1. Thence, by means of the lace prism 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
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° 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
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°

Maximal Expanded Hypercube eCn   (up)

For 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 the maximal expanded hypercube eCn-1 atop the regular hypercube Cn-1. Thence, by means of the lace tegum 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 trates 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 ? ? ? ? ?
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°


Symmetry Dn

Demihypercube Dn   (up)

As these polytopes Dn generally are nothing but the alternation of the 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).

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 1n,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 8 16 32 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 64 128 256 512 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°

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 prism notation, tDn = x3o3o *b3o...o3o (n nodes) can be described as well as xuxo3xoox3oxux *b3oooo...oooo3oooo&#x (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)

For 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 tegum 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 16 27 56 240 ?
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 10 27 126 2160 ?
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 16 72 576 17280 ?
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 80 216 756 6720 ?
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 40 216 2016 69120 ?
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
? ?
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
rolin
buffy
rectified 1n,2
Vertex Count 10 80 720 10080 483840 ?
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 rolin ?
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]


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 simplex Sn-1 atop the (unit) 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 cross unity at n=4.

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°
? ? ? ?


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
? ?
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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