Demipenteract (5-demicube) | ||
---|---|---|
Petrie polygon projection | ||
Type | Uniform 5-polytope | |
Family (D_{n}) | 5-demicube | |
Families (E_{n}) | k_{21} polytope 1_{k2} polytope | |
Coxeter symbol |
1_{21} | |
Schläfli symbols |
{3,3^{2,1}} = h{4,3^{3}} s{2,4,3,3} or h{2}h{4,3,3} sr{2,2,4,3} or h{2}h{2}h{4,3} h{2}h{2}h{2}h{4} s{2^{1,1,1,1}} or h{2}h{2}h{2}s{2} | |
Coxeter diagrams |
= | |
4-faces | 26 | 10 {3^{1,1,1}} 16 {3,3,3} |
Cells | 120 | 40 {3^{1,0,1}} 80 {3,3} |
Faces | 160 | {3} |
Edges | 80 | |
Vertices | 16 | |
Vertex figure |
rectified 5-cell | |
Petrie polygon |
Octagon | |
Symmetry | D_{5}, [3^{2,1,1}] = [1^{+},4,3^{3}] [2^{4}]^{+} | |
Properties | convex |
In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.
It was discovered by Thorold Gosset. Since it was the only semiregular 5-polytope (made of more than one type of regular facets), he called it a 5-ic semi-regular. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM_{5} for a 5-dimensional half measure polytope.
Coxeter named this polytope as 1_{21} from its Coxeter diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches, and Schläfli symbol or {3,3^{2,1}}.
It exists in the k_{21} polytope family as 1_{21} with the Gosset polytopes: 2_{21}, 3_{21}, and 4_{21}.
The graph formed by the vertices and edges of the demipenteract is sometimes called the Clebsch graph, though that name sometimes refers to the folded cube graph of order five instead.
Cartesian coordinates for the vertices of a demipenteract centered at the origin and edge length 2√2 are alternate halves of the penteract:
with an odd number of plus signs.
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_{5} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | k-figure | notes | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|
A_{4} | ( ) | f_{0} | 16 | 10 | 30 | 10 | 20 | 5 | 5 | rectified 5-cell | D_{5}/A_{4} = 16*5!/5! = 16 | |
A_{2}A_{1}A_{1} | { } | f_{1} | 2 | 80 | 6 | 3 | 6 | 3 | 2 | triangular prism | D_{5}/A_{2}A_{1}A_{1} = 16*5!/3!/2/2 = 80 | |
A_{2}A_{1} | {3} | f_{2} | 3 | 3 | 160 | 1 | 2 | 2 | 1 | Isosceles triangle | D_{5}/A_{2}A_{1} = 16*5!/3!/2 = 160 | |
A_{3}A_{1} | h{4,3} | f_{3} | 4 | 6 | 4 | 40 | * | 2 | 0 | { } | D_{5}/A_{3}A_{1} = 16*5!/4!/2 = 40 | |
A_{3} | {3,3} | 4 | 6 | 4 | * | 80 | 1 | 1 | { } | D_{5}/A_{3} = 16*5!/4! = 80 | ||
D_{4} | h{4,3,3} | f_{4} | 8 | 24 | 32 | 8 | 8 | 10 | * | ( ) | D_{5}/D_{4} = 16*5!/8/4! = 10 | |
A_{4} | {3,3,3} | 5 | 10 | 10 | 0 | 5 | * | 16 | ( ) | D_{5}/A_{4} = 16*5!/5! = 16 |
Perspective projection. |
Coxeter plane | B_{5} | |
---|---|---|
Graph | ||
Dihedral symmetry | [10/2] | |
Coxeter plane | D_{5} | D_{4} |
Graph | ||
Dihedral symmetry | [8] | [6] |
Coxeter plane | D_{3} | A_{3} |
Graph | ||
Dihedral symmetry | [4] | [4] |
It is a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 23 Uniform 5-polytopes (uniform 5-polytopes) that can be constructed from the D_{5} symmetry of the demipenteract, 8 of which are unique to this family, and 15 are shared within the penteractic family.
D5 polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
h{4,3,3,3} |
h_{2}{4,3,3,3} |
h_{3}{4,3,3,3} |
h_{4}{4,3,3,3} |
h_{2,3}{4,3,3,3} |
h_{2,4}{4,3,3,3} |
h_{3,4}{4,3,3,3} |
h_{2,3,4}{4,3,3,3} |
The 5-demicube is third in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (5-cells and 16-cells in the case of the rectified 5-cell). In Coxeter's notation the 5-demicube is given the symbol 1_{21}.
k_{21} figures in n dimensional | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | Finite | Euclidean | Hyperbolic | ||||||||
E_{n} | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||
Coxeter group |
E_{3}=A_{2}A_{1} | E_{4}=A_{4} | E_{5}=D_{5} | E_{6} | E_{7} | E_{8} | E_{9} = = E_{8}^{+} | E_{10} = = E_{8}^{++} | |||
Coxeter diagram |
|||||||||||
Symmetry | [3^{−1,2,1}] | [3^{0,2,1}] | [3^{1,2,1}] | [3^{2,2,1}] | [3^{3,2,1}] | [3^{4,2,1}] | [3^{5,2,1}] | [3^{6,2,1}] | |||
Order | 12 | 120 | 192 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ||||
Graph | - | - | |||||||||
Name | −1_{21} | 0_{21} | 1_{21} | 2_{21} | 3_{21} | 4_{21} | 5_{21} | 6_{21} |
1_{k2} figures in n dimensions | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | Finite | Euclidean | Hyperbolic | ||||||||
n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||
Coxeter group |
E_{3}=A_{2}A_{1} | E_{4}=A_{4} | E_{5}=D_{5} | E_{6} | E_{7} | E_{8} | E_{9} = = E_{8}^{+} | E_{10} = = E_{8}^{++} | |||
Coxeter diagram |
|||||||||||
Symmetry (order) |
[3^{−1,2,1}] | [3^{0,2,1}] | [3^{1,2,1}] | [[3^{2,2,1}]] | [3^{3,2,1}] | [3^{4,2,1}] | [3^{5,2,1}] | [3^{6,2,1}] | |||
Order | 12 | 120 | 192 | 103,680 | 2,903,040 | 696,729,600 | ∞ | ||||
Graph | - | - | |||||||||
Name | 1_{−1,2} | 1_{02} | 1_{12} | 1_{22} | 1_{32} | 1_{42} | 1_{52} | 1_{62} |