6-demicube

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6-demicube
Demihexeract
(6-demicube) Petrie polygon projection
Type Uniform 6-polytope
Family demihypercube
Schläfli symbol {3,33,1}
h{4,3,3,3,3}
s{2,2,2,2,2}
Coxeter-Dynkin diagram                               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, [35,1,1] = [1+,4,34]
+
Petrie polygon decagon
Properties convex

In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternate vertices deleted. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

Coxeter named this polytope as 131 from its Coxeter-Dynkin diagram, with a ring on one of the 1-length Coxeter-Dynkin diagram branches. It can named similarly by a 3-dimensional exponential Schläfli symbol, {3,33,1}.

Cartesian coordinates

Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract:

(±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph  Dihedral symmetry  
Coxeter plane D4 D3
Graph  Dihedral symmetry  
Coxeter plane A5 A3
Graph  Dihedral symmetry  

Related polytopes

There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:

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