Demihexeract (6-demicube) | ||
---|---|---|
![]() Petrie polygon projection | ||
Type | Uniform 6-polytope | |
Family | demihypercube | |
Schläfli symbol | {3,33,1} = h{4,34} s{21,1,1,1,1} | |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
| |
Coxeter symbol | 131 | |
5-faces | 44 | 12
{31,2,1}
![]() 32 {34} ![]() |
4-faces | 252 | 60
{31,1,1}
![]() 192 {33} ![]() |
Cells | 640 | 160
{31,0,1}
![]() 480 {3,3} ![]() |
Faces | 640 |
{3}
![]() |
Edges | 240 | |
Vertices | 32 | |
Vertex figure |
Rectified 5-simplex![]() | |
Symmetry group | D6, [33,1,1] = [1+,4,34 [25+ | |
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 HM6 for a 6-dimensional half measure polytope.
Coxeter named this polytope as 131 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 {3,33,1}.
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]
D6 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | k-figure | notes | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A4 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
( ) | f0 | 32 | 15 | 60 | 20 | 60 | 15 | 30 | 6 | 6 | r{3,3,3,3} | D6/A4 = 32*6!/5! = 32 |
A3A1A1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{ } | f1 | 2 | 240 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | {}x{3,3} | D6/A3A1A1 = 32*6!/4!/2/2 = 240 |
A3A2 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3} | f2 | 3 | 3 | 640 | 1 | 3 | 3 | 3 | 3 | 1 | {3}v( ) | D6/A3A2 = 32*6!/4!/3! = 640 |
A3A1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
h{4,3} | f3 | 4 | 6 | 4 | 160 | * | 3 | 0 | 3 | 0 | {3} | D6/A3A1 = 32*6!/4!/2 = 160 |
A3A2 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,3} | 4 | 6 | 4 | * | 480 | 1 | 2 | 2 | 1 | {}v( ) | D6/A3A2 = 32*6!/4!/3! = 480 | |
D4A1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
h{4,3,3} | f4 | 8 | 24 | 32 | 8 | 8 | 60 | * | 2 | 0 | { } | D6/D4A1 = 32*6!/8/4!/2 = 60 |
A4 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,3,3} | 5 | 10 | 10 | 0 | 5 | * | 192 | 1 | 1 | D6/A4 = 32*6!/5! = 192 | ||
D5 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
h{4,3,3,3} | f5 | 16 | 80 | 160 | 40 | 80 | 10 | 16 | 12 | * | ( ) | D6/D5 = 32*6!/16/5! = 12 |
A5 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,3,3,3} | 6 | 15 | 20 | 0 | 15 | 0 | 6 | * | 32 | D6/A5 = 32*6!/6! = 32 |
Coxeter plane | B6 | |
---|---|---|
Graph |
![]() | |
Dihedral symmetry | [12/2] | |
Coxeter plane | D6 | D5 |
Graph |
![]() |
![]() |
Dihedral symmetry | [10] | [8] |
Coxeter plane | D4 | D3 |
Graph |
![]() |
![]() |
Dihedral symmetry | [6] | [4] |
Coxeter plane | A5 | A3 |
Graph |
![]() |
![]() |
Dihedral symmetry | [6] | [4] |
There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:
D6 polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
![]() h{4,34} |
![]() h2{4,34} |
![]() h3{4,34} |
![]() h4{4,34} |
![]() h5{4,34} |
![]() h2,3{4,34} |
![]() h2,4{4,34} |
![]() h2,5{4,34} | ||||
![]() h3,4{4,34} |
![]() h3,5{4,34} |
![]() h4,5{4,34} |
![]() h2,3,4{4,34} |
![]() h2,3,5{4,34} |
![]() h2,4,5{4,34} |
![]() h3,4,5{4,34} |
![]() h2,3,4,5{4,34} |
The 6-demicube, 131 is third in a dimensional series of uniform polytopes, expressed by Coxeter as k31 series. The fifth figure is a Euclidean honeycomb, 331, and the final is a noncompact hyperbolic honeycomb, 431. Each progressive uniform polytope is constructed from the previous as its vertex figure.
n | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|
Coxeter group |
A3A1 | A5 | D6 | E7 | = E7+ | =E7++ |
Coxeter diagram |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry | [3−1,3,1] | [30,3,1] | [31,3,1] | [32,3,1] | [33,3,1] | [34,3,1] |
Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | |
Graph |
![]() |
![]() |
![]() |
![]() |
- | - |
Name | −131 | 031 | 131 | 231 | 331 | 431 |
It is also the second in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The fourth figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134.
Space | Finite | Euclidean | Hyperbolic | |||
---|---|---|---|---|---|---|
n | 4 | 5 | 6 | 7 | 8 | 9 |
Coxeter group |
A3A1 | A5 | D6 | E7 | =E7+ | =E7++ |
Coxeter diagram |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry | [3−1,3,1] | [30,3,1] | [31,3,1] | [32,3,1] | [[33,3,1]] | [34,3,1] |
Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | |
Graph |
![]() |
![]() |
![]() |
- | - | |
Name | 13,-1 | 130 | 131 | 132 | 133 | 134 |
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]
Demihexeract (6-demicube) | ||
---|---|---|
![]() Petrie polygon projection | ||
Type | Uniform 6-polytope | |
Family | demihypercube | |
Schläfli symbol | {3,33,1} = h{4,34} s{21,1,1,1,1} | |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
| |
Coxeter symbol | 131 | |
5-faces | 44 | 12
{31,2,1}
![]() 32 {34} ![]() |
4-faces | 252 | 60
{31,1,1}
![]() 192 {33} ![]() |
Cells | 640 | 160
{31,0,1}
![]() 480 {3,3} ![]() |
Faces | 640 |
{3}
![]() |
Edges | 240 | |
Vertices | 32 | |
Vertex figure |
Rectified 5-simplex![]() | |
Symmetry group | D6, [33,1,1] = [1+,4,34 [25+ | |
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 HM6 for a 6-dimensional half measure polytope.
Coxeter named this polytope as 131 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 {3,33,1}.
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]
D6 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | k-figure | notes | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A4 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
( ) | f0 | 32 | 15 | 60 | 20 | 60 | 15 | 30 | 6 | 6 | r{3,3,3,3} | D6/A4 = 32*6!/5! = 32 |
A3A1A1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{ } | f1 | 2 | 240 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | {}x{3,3} | D6/A3A1A1 = 32*6!/4!/2/2 = 240 |
A3A2 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3} | f2 | 3 | 3 | 640 | 1 | 3 | 3 | 3 | 3 | 1 | {3}v( ) | D6/A3A2 = 32*6!/4!/3! = 640 |
A3A1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
h{4,3} | f3 | 4 | 6 | 4 | 160 | * | 3 | 0 | 3 | 0 | {3} | D6/A3A1 = 32*6!/4!/2 = 160 |
A3A2 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,3} | 4 | 6 | 4 | * | 480 | 1 | 2 | 2 | 1 | {}v( ) | D6/A3A2 = 32*6!/4!/3! = 480 | |
D4A1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
h{4,3,3} | f4 | 8 | 24 | 32 | 8 | 8 | 60 | * | 2 | 0 | { } | D6/D4A1 = 32*6!/8/4!/2 = 60 |
A4 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,3,3} | 5 | 10 | 10 | 0 | 5 | * | 192 | 1 | 1 | D6/A4 = 32*6!/5! = 192 | ||
D5 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
h{4,3,3,3} | f5 | 16 | 80 | 160 | 40 | 80 | 10 | 16 | 12 | * | ( ) | D6/D5 = 32*6!/16/5! = 12 |
A5 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,3,3,3} | 6 | 15 | 20 | 0 | 15 | 0 | 6 | * | 32 | D6/A5 = 32*6!/6! = 32 |
Coxeter plane | B6 | |
---|---|---|
Graph |
![]() | |
Dihedral symmetry | [12/2] | |
Coxeter plane | D6 | D5 |
Graph |
![]() |
![]() |
Dihedral symmetry | [10] | [8] |
Coxeter plane | D4 | D3 |
Graph |
![]() |
![]() |
Dihedral symmetry | [6] | [4] |
Coxeter plane | A5 | A3 |
Graph |
![]() |
![]() |
Dihedral symmetry | [6] | [4] |
There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:
D6 polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
![]() h{4,34} |
![]() h2{4,34} |
![]() h3{4,34} |
![]() h4{4,34} |
![]() h5{4,34} |
![]() h2,3{4,34} |
![]() h2,4{4,34} |
![]() h2,5{4,34} | ||||
![]() h3,4{4,34} |
![]() h3,5{4,34} |
![]() h4,5{4,34} |
![]() h2,3,4{4,34} |
![]() h2,3,5{4,34} |
![]() h2,4,5{4,34} |
![]() h3,4,5{4,34} |
![]() h2,3,4,5{4,34} |
The 6-demicube, 131 is third in a dimensional series of uniform polytopes, expressed by Coxeter as k31 series. The fifth figure is a Euclidean honeycomb, 331, and the final is a noncompact hyperbolic honeycomb, 431. Each progressive uniform polytope is constructed from the previous as its vertex figure.
n | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|
Coxeter group |
A3A1 | A5 | D6 | E7 | = E7+ | =E7++ |
Coxeter diagram |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry | [3−1,3,1] | [30,3,1] | [31,3,1] | [32,3,1] | [33,3,1] | [34,3,1] |
Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | |
Graph |
![]() |
![]() |
![]() |
![]() |
- | - |
Name | −131 | 031 | 131 | 231 | 331 | 431 |
It is also the second in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The fourth figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134.
Space | Finite | Euclidean | Hyperbolic | |||
---|---|---|---|---|---|---|
n | 4 | 5 | 6 | 7 | 8 | 9 |
Coxeter group |
A3A1 | A5 | D6 | E7 | =E7+ | =E7++ |
Coxeter diagram |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry | [3−1,3,1] | [30,3,1] | [31,3,1] | [32,3,1] | [[33,3,1]] | [34,3,1] |
Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | |
Graph |
![]() |
![]() |
![]() |
- | - | |
Name | 13,-1 | 130 | 131 | 132 | 133 | 134 |
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]