Demihypercube
In geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγ_{n} for being half of the hypercube family, γ_{n}. Half of the vertices are deleted and new facets are formed. The 2n facets become 2n (n-1)-demicubes, and 2^{n} (n-1)-simplex facets are formed in place of the deleted vertices.^{[1]}
They have been named with a demi- prefix to each hypercube name: demicube, demitesseract, etc. The demicube is identical to the regular tetrahedron, and the demitesseract is identical to the regular 16-cell. The demipenteract is considered semiregular for having only regular facets. Higher forms don't have all regular facets but are all uniform polytopes.
The vertices and edges of a demihypercube form two copies of the halved cube graph.
Discovery
Thorold Gosset described the demipenteract in his 1900 publication listing all of the regular and semiregular figures in n-dimensions above 3. He called it a 5-ic semi-regular. It also exists within the semiregular k_{21} polytope family.
The demihypercubes can be represented by extended Schläfli symbols of the form h{4,3,...,3} as half the vertices of {4,3,...,3}. The vertex figures of demihypercubes are rectified n-simplexes.
Constructions
They are represented by Coxeter-Dynkin diagrams of three constructive forms:
- ... (As an alternated orthotope) s{2^{1,1...,1}}
- ... (As an alternated hypercube) h{4,3^{n-1}}
- .... (As a demihypercube) {3^{1,n-3,1}}
H.S.M. Coxeter also labeled the third bifurcating diagrams as 1_{k1} representing the lengths of the 3 branches and lead by the ringed branch.
An n-demicube, n greater than 2, has n*(n-1)/2 edges meeting at each vertex. The graphs below show less edges at each vertex due to overlapping edges in the symmetry projection.
n | 1_{k1} | Petrie polygon |
Schläfli symbol | Coxeter diagrams A_{1}^{n} B_{n} D_{n} |
Elements | Facets: Demihypercubes & Simplexes |
Vertex figure | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Vertices | Edges | Faces | Cells | 4-faces | 5-faces | 6-faces | 7-faces | 8-faces | 9-faces | |||||||
2 | 1_{-1,1} | demisquare (digon) |
s{2} h{4} {3^{1,-1,1}} |
2 | 2 | 2 edges |
-- | |||||||||
3 | 1_{01} | demicube (tetrahedron) |
s{2^{1,1}} h{4,3} {3^{1,0,1}} |
4 | 6 | 4 | (6 digons) 4 triangles |
Triangle (Rectified triangle) | ||||||||
4 | 1_{11} | demitesseract (16-cell) |
s{2^{1,1,1}} h{4,3,3} {3^{1,1,1}} |
8 | 24 | 32 | 16 | 8 demicubes (tetrahedra) 8 tetrahedra |
Octahedron (Rectified tetrahedron) | |||||||
5 | 1_{21} | demipenteract |
s{2^{1,1,1,1}} h{4,3^{3}}{3^{1,2,1}} |
16 | 80 | 160 | 120 | 26 | 10 16-cells 16 5-cells |
Rectified 5-cell | ||||||
6 | 1_{31} | demihexeract |
s{2^{1,1,1,1,1}} h{4,3^{4}}{3^{1,3,1}} |
32 | 240 | 640 | 640 | 252 | 44 | 12 demipenteracts 32 5-simplices |
Rectified hexateron | |||||
7 | 1_{41} | demihepteract |
s{2^{1,1,1,1,1,1}} h{4,3^{5}}{3^{1,4,1}} |
64 | 672 | 2240 | 2800 | 1624 | 532 | 78 | 14 demihexeracts 64 6-simplices |
Rectified 6-simplex | ||||
8 | 1_{51} | demiocteract |
s{2^{1,1,1,1,1,1,1}} h{4,3^{6}}{3^{1,5,1}} |
128 | 1792 | 7168 | 10752 | 8288 | 4032 | 1136 | 144 | 16 demihepteracts 128 7-simplices |
Rectified 7-simplex | |||
9 | 1_{61} | demienneract |
s{2^{1,1,1,1,1,1,1,1}} h{4,3^{7}}{3^{1,6,1}} |
256 | 4608 | 21504 | 37632 | 36288 | 23520 | 9888 | 2448 | 274 | 18 demiocteracts 256 8-simplices |
Rectified 8-simplex | ||
10 | 1_{71} | demidekeract |
s{2^{1,1,1,1,1,1,1,1,1}} h{4,3^{8}}{3^{1,7,1}} |
512 | 11520 | 61440 | 122880 | 142464 | 115584 | 64800 | 24000 | 5300 | 532 | 20 demienneracts 512 9-simplices |
Rectified 9-simplex | |
... | ||||||||||||||||
n | 1_{n-3,1} | n-demicube | s{2^{1,1,...,1}} h{4,3^{n-2}}{3^{1,n-3,1}} |
... ... ... |
2^{n-1} | n (n-1)-demicubes 2^{n} (n-1)-simplices |
Rectified (n-1)-simplex |
In general, a demicube's elements can be determined from the original n-cube: (With C_{n,m} = m^{th}-face count in n-cube = 2^{n-m}*n!/(m!*(n-m)!))
- Vertices: D_{n,0} = 1/2 * C_{n,0} = 2^{n-1} (Half the n-cube vertices remain)
- Edges: D_{n,1} = C_{n,2} = 1/2 n(n-1)2^{n-2} (All original edges lost, each square faces create a new edge)
- Faces: D_{n,2} = 4 * C_{n,3} = n(n-1)(n-2)2^{n-3} (All original faces lost, each cube creates 4 new triangular faces)
- Cells: D_{n,3} = C_{n,3} + 2^{n-4}C_{n,4} (tetrahedra from original cells plus new ones)
- Hypercells: D_{n,4} = C_{n,4} + 2^{n-5}C_{n,5} (16-cells and 5-cells respectively)
- ...
- [For m=3...n-1]: D_{n,m} = C_{n,m} + 2^{n-1-m}C_{n,m+1} (m-demicubes and m-simplexes respectively)
- ...
- Facets: D_{n,n-1} = n + 2^{n} ((n-1)-demicubes and (n-1)-simplices respectively)
Symmetry group
The symmetry group of the demihypercube is the Coxeter group [3^{n-3,1,1}] has order and is an index 2 subgroup of the hyperoctahedral group (which is the Coxeter group [4,3^{n-1}]). It is generated by permutations of the coordinate axes and reflections along pairs of coordinate axes.^{[2]}
Orthotopic constructions
Constructions as alternated orthotopes have the same topology, but can be stretched with different lengths in n-axes of symmetry.
The rhombic disphenoid is the three-dimensional example as alternated cuboid. It has three sets of edge lengths, and scalene triangle faces.
See also
References
- ↑ Regular and semi-regular polytopes III, p. 315-316
- ↑ http://math.ucr.edu/home/baez/week187.html
- T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_{n1})
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
External links
- Olshevsky, George. "Half measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.