Runcic 7-cubes

Orthogonal projections in D7 Coxeter plane
7-demicube	Runcic 7-cube	Runcicantic 7-cube

In seven-dimensional geometry, a runcic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 2 unique forms.

Runcic 7-cube

A runcic 7-cube, h3{4,35}, has half the vertices of a runcinated 7-cube, t0,3{4,35}.

The Cartesian coordinates for the vertices of a cantellated demihepteract centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3,±3,±3)

with an odd number of plus signs.

Orthographic projections
Coxeter plane	B7	D7	D6
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D5	D4	D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

A runcicantic 7-cube, h2,3{4,35}, has half the vertices of a runcicantellated 7-cube, t0,1,3{4,35}.

The Cartesian coordinates for the vertices of a runcicantic 7-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±5,±5)

with an odd number of plus signs.

Orthographic projections
Coxeter plane	B7	D7	D6
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D5	D4	D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

These polytopes are based on the 7-demicube, a member of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 95 uniform polytopes with D7 symmetry, 63 are shared by the BC6 symmetry, and 32 are unique:

H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd edition, Dover, New York, 1973 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, , ISBN 978-0-471-01003-6 (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
Klitzing, Richard. . x3o3o *b3x3o3o3o - sirhesa, x3x3o *b3x3o3o3o - girhesa

vteFundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations