Demihepteract (7-demicube) | ||
---|---|---|
Petrie polygon projection | ||
Type | Uniform 7-polytope | |
Family | demihypercube | |
Coxeter symbol | 1_{41} | |
Schläfli symbol | {3,3^{4,1}} = h{4,3^{5}} s{2^{1,1,1,1,1,1}} | |
Coxeter diagrams | =
| |
6-faces | 78 | 14 {3^{1,3,1}} 64 {3^{5}} |
5-faces | 532 | 84 {3^{1,2,1}} 448 {3^{4}} |
4-faces | 1624 | 280 {3^{1,1,1}} 1344 {3^{3}} |
Cells | 2800 | 560 {3^{1,0,1}} 2240 {3,3} |
Faces | 2240 | {3} |
Edges | 672 | |
Vertices | 64 | |
Vertex figure | Rectified 6-simplex | |
Symmetry group | D_{7}, [3^{4,1,1}] = [1^{+},4,3^{5}] [2^{6}]^{+} | |
Dual | ? | |
Properties | convex |
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM_{7} for a 7-dimensional half measure polytope.
Coxeter named this polytope as 1_{41} from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol or {3,3^{4,1}}.
Cartesian coordinates for the vertices of a demihepteract centered at the origin are alternate halves of the hepteract:
with an odd number of plus signs.
Coxeter plane |
B_{7} | D_{7} | D_{6} |
---|---|---|---|
Graph | |||
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D_{5} | D_{4} | D_{3} |
Graph | |||
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A_{5} | A_{3} | |
Graph | |||
Dihedral symmetry |
[6] | [4] |
The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element.^{[1]}^{[2]}
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.^{[3]}
D_{7} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | k-figures | notes | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A_{6} | ( ) | f_{0} | 64 | 21 | 105 | 35 | 140 | 35 | 105 | 21 | 42 | 7 | 7 | 0_{41} | D_{7}/A_{6} = 64*7!/7! = 64 | |
A_{4}A_{1}A_{1} | { } | f_{1} | 2 | 672 | 10 | 5 | 20 | 10 | 20 | 10 | 10 | 5 | 2 | { }×{3,3,3} | D_{7}/A_{4}A_{1}A_{1} = 64*7!/5!/2/2 = 672 | |
A_{3}A_{2} | 1_{00} | f_{2} | 3 | 3 | 2240 | 1 | 4 | 4 | 6 | 6 | 4 | 4 | 1 | {3,3}v( ) | D_{7}/A_{3}A_{2} = 64*7!/4!/3! = 2240 | |
A_{3}A_{3} | 1_{01} | f_{3} | 4 | 6 | 4 | 560 | * | 4 | 0 | 6 | 0 | 4 | 0 | {3,3} | D_{7}/A_{3}A_{3} = 64*7!/4!/4! = 560 | |
A_{3}A_{2} | 1_{10} | 4 | 6 | 4 | * | 2240 | 1 | 3 | 3 | 3 | 3 | 1 | {3}v( ) | D_{7}/A_{3}A_{2} = 64*7!/4!/3! = 2240 | ||
D_{4}A_{2} | 1_{11} | f_{4} | 8 | 24 | 32 | 8 | 8 | 280 | * | 3 | 0 | 3 | 0 | {3} | D_{7}/D_{4}A_{2} = 64*7!/8/4!/2 = 280 | |
A_{4}A_{1} | 1_{20} | 5 | 10 | 10 | 0 | 5 | * | 1344 | 1 | 2 | 2 | 1 | { }v( ) | D_{7}/A_{4}A_{1} = 64*7!/5!/2 = 1344 | ||
D_{5}A_{1} | 1_{21} | f_{5} | 16 | 80 | 160 | 40 | 80 | 10 | 16 | 84 | * | 2 | 0 | { } | D_{7}/D_{5}A_{1} = 64*7!/16/5!/2 = 84 | |
A_{5} | 1_{30} | 6 | 15 | 20 | 0 | 15 | 0 | 6 | * | 448 | 1 | 1 | D_{7}/A_{5} = 64*7!/6! = 448 | |||
D_{6} | 1_{31} | f_{6} | 32 | 240 | 640 | 160 | 480 | 60 | 192 | 12 | 32 | 14 | * | ( ) | D_{7}/D_{6} = 64*7!/32/6! = 14 | |
A_{6} | 1_{40} | 7 | 21 | 35 | 0 | 35 | 0 | 21 | 0 | 7 | * | 64 | D_{7}/A_{6} = 64*7!/7! = 64 |
There are 95 uniform polytopes with D_{6} symmetry, 63 are shared by the B_{6} symmetry, and 32 are unique: