 Demihypercube

Not to be confused with Hemicube (geometry).
In geometry, demihypercubes (also called ndemicubes, nhemicubes, and half measure polytopes) are a class of npolytopes constructed from alternation of an nhypercube, 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 (n1)demicubes and 2^{n} (n1)simplex facets are formed in place of the deleted vertices.
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 16cell. The demipenteract is considered semiregular for having only regular facets. Higher forms don't have all regular facets but are all uniform polytopes.
Contents
Discovery
Thorold Gosset described the demipenteract in his 1900 publication listing all of the regular and semiregular figures in ndimensions above 3. He called it a 5ic semiregular. 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 nsimplexes.
Constructions
They are represented by CoxeterDynkin diagrams of three constructive forms:
 ... (As an alternated orthotope) s{2^{n1}}
 ... (As an alternated hypercube) h{4,3^{n1}}
 .... (As a demihypercube) {3^{1,n3,1}}
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 ndemicube, n greater than 2, has n*(n1)/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
polygonSchläfli symbol CoxeterDynkin diagrams
C_{n} family
D_{n} familyElements Facets:
Demihypercubes &
SimplexesVertex figure Vertices Edges Faces Cells 4faces 5faces 6faces 7faces 8faces 9faces 2 1_{1,1} Demisquare
(digon)
s{2^{1}}
h{4}
{3^{1,1,1}}
2 2
2 edges 3 1_{01} demicube
(tetrahedron)
s{2^{2}}
h{4,3}
{3^{1,0,1}}
4 6 4 (6 digons)
4 trianglesTriangle
(Rectified triangle)4 1_{11} demitesseract
(16cell)
s{2^{3}}
h{4,3,3}
{3^{1,1,1}}
8 24 32 16 8 demicubes
(tetrahedra)
8 tetrahedraOctahedron
(Rectified tetrahedron)5 1_{21} demipenteract
s{2^{4}}
h{4,3^{3}}{3^{1,2,1}}
16 80 160 120 26 10 16cells
16 5cellsRectified 5cell 6 1_{31} demihexeract
s{2^{5}}
h{4,3^{4}}{3^{1,3,1}}
32 240 640 640 252 44 12 demipenteracts
32 5simplicesRectified hexateron 7 1_{41} demihepteract
s{2^{6}}
h{4,3^{5}}{3^{1,4,1}}
64 672 2240 2800 1624 532 78 14 demihexeracts
64 6simplicesRectified 6simplex 8 1_{51} demiocteract
s{2^{7}}
h{4,3^{6}}{3^{1,5,1}}
128 1792 7168 10752 8288 4032 1136 144 16 demihepteracts
128 7simplicesRectified 7simplex 9 1_{61} demienneract
s{2^{8}}
h{4,3^{7}}{3^{1,6,1}}
256 4608 21504 37632 36288 23520 9888 2448 274 18 demiocteracts
256 8simplicesRectified 8simplex 10 1_{71} demidekeract
s{2^{9}}
h{4,3^{8}}{3^{1,7,1}}
512 11520 61440 122880 142464 115584 64800 24000 5300 532 20 demienneracts
512 9simplicesRectified 9simplex ... n 1_{n3,1} ndemicube s{2^{n1}}
h{4,3^{n2}}{3^{1,n3,1}}...
...
...2^{n1} n (n1)demicubes
2^{n} (n1)simplicesRectified (n1)simplex In general, a demicube's elements can be determined from the original ncube: (With C_{n,m} = m^{th}face count in ncube = 2^{nm}*n!/(m!*(nm)!))
 Vertices: D_{n,0} = 1/2 * C_{n,0} = 2^{n1} (Half the ncube vertices remain)
 Edges: D_{n,1} = C_{n,2} = 1/2 n(n1)2^{n2} (All original edges lost, each square faces create a new edge)
 Faces: D_{n,2} = 4 * C_{n,3} = n(n1)(n2)2^{n3} (All original faces lost, each cube creates 4 new triangular faces)
 Cells: D_{n,3} = C_{n,3} + 2^{n4}C_{n,4} (tetrahedra from original cells plus new ones)
 Hypercells: D_{n,4} = C_{n,4} + 2^{n5}C_{n,5} (16cells and 5cells respectively)
 ...
 [For m=3...n1]: D_{n,m} = C_{n,m} + 2^{n1m}C_{n,m+1} (mdemicubes and msimplexes respectively)
 ...
 Facets: D_{n,n1} = n + 2^{n} ((n1)demicubes and (n1)simplices respectively)
Symmetry group
The symmetry group of the demihypercube is the Coxeter group D_{n}, [3^{n3,1,1}] has order 2^{n − 1}n!, and is an index 2 subgroup of the hyperoctahedral group (which is the Coxeter group BC_{n} [4,3^{n1}]).
Orthotopic constructions
Constructions as alternated orthotopes have the same topology, but can be stretched with different lengths in naxes of symmetry.
The rhombic disphenoid is the threedimensional example as alternated cuboid. It has three sets of edge lengths, and scalene triangle faces.
See also
 Hypercube honeycomb
 Semiregular Epolytope
References
 T. Gosset: On the Regular and SemiRegular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205 (Chapter 26. pp. 409: Hemicubes: 1_{n1})
External links
 Olshevsky, George, Half measure polytope at Glossary for Hyperspace.
Dimension Dimensional spaces One · Two · Three · Four · Five · Six · Seven · Eight · ndimensions · Spacetime · Projective space · HyperplanePolytopes and Shapes Concepts and mathematics Cartesian coordinates · Linear algebra · Geometric algebra · Conformal geometry · Reflection · Rotation · Plane of rotation · Space · Fractal dimension · MultiverseCategory Fundamental convex regular and uniform polytopes in dimensions 2–10 Family A_{n} BC_{n} D_{n} E_{6} / E_{7} / E_{8} / F_{4} / G_{2} H_{n} Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5cell 16cell • Tesseract Demitesseract 24cell 120cell • 600cell Uniform 5polytope 5simplex 5orthoplex • 5cube 5demicube Uniform 6polytope 6simplex 6orthoplex • 6cube 6demicube 1_{22} • 2_{21} Uniform 7polytope 7simplex 7orthoplex • 7cube 7demicube 1_{32} • 2_{31} • 3_{21} Uniform 8polytope 8simplex 8orthoplex • 8cube 8demicube 1_{42} • 2_{41} • 4_{21} Uniform 9polytope 9simplex 9orthoplex • 9cube 9demicube Uniform 10polytope 10simplex 10orthoplex • 10cube 10demicube npolytopes nsimplex northoplex • ncube ndemicube 1_{k2} • 2_{k1} • k_{21} pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes Categories: Polytopes
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