Demihexeract (6-demicube) | ||
---|---|---|
Petrie polygon projection | ||
Type | Uniform 6-polytope | |
Family | demihypercube | |
Schläfli symbol | {3,3^{3,1}} = h{4,3^{4}} s{2^{1,1,1,1,1}} | |
Coxeter diagrams | = =
| |
Coxeter symbol | 1_{31} | |
5-faces | 44 | 12 {3^{1,2,1}} 32 {3^{4}} |
4-faces | 252 | 60 {3^{1,1,1}} 192 {3^{3}} |
Cells | 640 | 160 {3^{1,0,1}} 480 {3,3} |
Faces | 640 | {3} |
Edges | 240 | |
Vertices | 32 | |
Vertex figure | Rectified 5-simplex | |
Symmetry group | D_{6}, [3^{3,1,1}] = [1^{+},4,3^{4}] [2^{5}]^{+} | |
Petrie polygon | decagon | |
Properties | convex |
In geometry, a 6-demicube or demihexeract is a uniform 6-polytope, constructed from a 6-cube (hexeract) 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_{6} for a 6-dimensional half measure polytope.
Coxeter named this polytope as 1_{31} from its Coxeter diagram, with a ring on one of the 1-length branches, . It can named similarly by a 3-dimensional exponential Schläfli symbol or {3,3^{3,1}}.
Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract:
with an odd number of plus signs.
This configuration matrix represents the 6-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's 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_{6} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | k-figure | notes | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A_{4} | ( ) | f_{0} | 32 | 15 | 60 | 20 | 60 | 15 | 30 | 6 | 6 | r{3,3,3,3} | D_{6}/A_{4} = 32*6!/5! = 32 | |
A_{3}A_{1}A_{1} | { } | f_{1} | 2 | 240 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | {}x{3,3} | D_{6}/A_{3}A_{1}A_{1} = 32*6!/4!/2/2 = 240 | |
A_{3}A_{2} | {3} | f_{2} | 3 | 3 | 640 | 1 | 3 | 3 | 3 | 3 | 1 | {3}v( ) | D_{6}/A_{3}A_{2} = 32*6!/4!/3! = 640 | |
A_{3}A_{1} | h{4,3} | f_{3} | 4 | 6 | 4 | 160 | * | 3 | 0 | 3 | 0 | {3} | D_{6}/A_{3}A_{1} = 32*6!/4!/2 = 160 | |
A_{3}A_{2} | {3,3} | 4 | 6 | 4 | * | 480 | 1 | 2 | 2 | 1 | {}v( ) | D_{6}/A_{3}A_{2} = 32*6!/4!/3! = 480 | ||
D_{4}A_{1} | h{4,3,3} | f_{4} | 8 | 24 | 32 | 8 | 8 | 60 | * | 2 | 0 | { } | D_{6}/D_{4}A_{1} = 32*6!/8/4!/2 = 60 | |
A_{4} | {3,3,3} | 5 | 10 | 10 | 0 | 5 | * | 192 | 1 | 1 | D_{6}/A_{4} = 32*6!/5! = 192 | |||
D_{5} | h{4,3,3,3} | f_{5} | 16 | 80 | 160 | 40 | 80 | 10 | 16 | 12 | * | ( ) | D_{6}/D_{5} = 32*6!/16/5! = 12 | |
A_{5} | {3,3,3,3} | 6 | 15 | 20 | 0 | 15 | 0 | 6 | * | 32 | D_{6}/A_{5} = 32*6!/6! = 32 |
Coxeter plane | B_{6} | |
---|---|---|
Graph | ||
Dihedral symmetry | [12/2] | |
Coxeter plane | D_{6} | D_{5} |
Graph | ||
Dihedral symmetry | [10] | [8] |
Coxeter plane | D_{4} | D_{3} |
Graph | ||
Dihedral symmetry | [6] | [4] |
Coxeter plane | A_{5} | A_{3} |
Graph | ||
Dihedral symmetry | [6] | [4] |
There are 47 uniform polytopes with D_{6} symmetry, 31 are shared by the B_{6} symmetry, and 16 are unique:
The 6-demicube, 1_{31} is third in a dimensional series of uniform polytopes, expressed by Coxeter as k_{31} series. The fifth figure is a Euclidean honeycomb, 3_{31}, and the final is a noncompact hyperbolic honeycomb, 4_{31}. Each progressive uniform polytope is constructed from the previous as its vertex figure.
n | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|
Coxeter group |
A_{3}A_{1} | A_{5} | D_{6} | E_{7} | = E_{7}^{+} | =E_{7}^{++} |
Coxeter diagram |
||||||
Symmetry | [3^{−1,3,1}] | [3^{0,3,1}] | [3^{1,3,1}] | [3^{2,3,1}] | [3^{3,3,1}] | [3^{4,3,1}] |
Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | |
Graph | - | - | ||||
Name | −1_{31} | 0_{31} | 1_{31} | 2_{31} | 3_{31} | 4_{31} |
It is also the second in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_{3k} series. The fourth figure is the Euclidean honeycomb 1_{33} and the final is a noncompact hyperbolic honeycomb, 1_{34}.
Space | Finite | Euclidean | Hyperbolic | |||
---|---|---|---|---|---|---|
n | 4 | 5 | 6 | 7 | 8 | 9 |
Coxeter group |
A_{3}A_{1} | A_{5} | D_{6} | E_{7} | =E_{7}^{+} | =E_{7}^{++} |
Coxeter diagram |
||||||
Symmetry | [3^{−1,3,1}] | [3^{0,3,1}] | [3^{1,3,1}] | [3^{2,3,1}] | [[3^{3,3,1}]] | [3^{4,3,1}] |
Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | |
Graph | - | - | ||||
Name | 1_{3,-1} | 1_{30} | 1_{31} | 1_{32} | 1_{33} | 1_{34} |
Coxeter identified a subset of 12 vertices that form a regular skew icosahedron {3, 5} with the same symmetries as the icosahedron itself, but at different angles. He dubbed this the regular skew icosahedron.^{[4]}^{[5]}