Hexic 7-cubes

Orthogonal projections in D7 Coxeter plane
7-demicube	Hexic 7-cube	Hexicantic 7-cube	Hexiruncic 7-cube	Hexiruncicantic 7-cube
Hexisteric 7-cube	Hexistericantic 7-cube	Hexisteriruncic 7-cube	Hexisteriruncicantic 7-cube	Hexipentic 7-cube
Hexipenticantic 7-cube	Hexipentiruncic 7-cube	Hexipentiruncicantic 7-cube	Hexipentisteric 7-cube	Hexipentistericantic 7-cube
Hexipentisteriruncic 7-cube	Hexipentisteriruncicantic 7-cube

In seven-dimensional geometry, a hexic 7-cube is a convex uniform 7-polytope, constructed from the uniform 7-demicube. There are 16 unique forms.

Hexic 7-cube

Hexic 7-cube
Type	uniform 7-polytope
Schläfli symbol	t0,5{3,34,1} h6{4,35}
Coxeter-Dynkin diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	4704
Vertices	448
Vertex figure
Coxeter groups	D7, [34,1,1]
Properties	convex

Alternate names

Small terated demihepteract (acronym: suthesa)

Cartesian coordinates

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

(±1,±1,±1,±1,±1,±1,±3)

with an odd number of plus signs.

Images

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]

Hexicantic 7-cube

Alternate names

Teritruncated demihepteract (acronym: tuthesa)

Images

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]

Hexiruncic 7-cube

Alternate names

Terirhombated demihepteract (acronym: turhesa)

Images

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]

Hexisteric 7-cube

Alternate names

Teriprismated demihepteract (acronym: tuphesa)

Images

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]

Hexipentic 7-cube

Alternate names

Tericellated demihepteract (acronym: tuchesa)

Images

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]

Hexiruncicantic 7-cube

Alternate names

Terigreatorhombated demihepteract (acronym: tugrohesa)

Images

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]

Hexistericantic 7-cube

Alternate names

Teriprismatotruncated demihepteract (acronym: tupthesa)

Images

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]

Hexipenticantic 7-cube

Alternate names

Tericellitruncated demihepteract (acronym: tucothesa)

Images

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]

Hexisteriruncic 7-cube

Alternate names

Teriprismatorhombated demihepteract (acronym: tuprohesa)

Images

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]

Hexipentiruncic 7-cube

Alternate names

Tericellirhombated demihepteract (acronym: tucrohesa)

Images

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]

Hexipentisteric 7-cube

Alternate names

Tericelliprismated demihepteract (acronym: tucophesa)

Images

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]

Hexisteriruncicantic 7-cube

Alternate names

Terigreatoprismated demihepteract (acronym: tugphesa)

Images

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]

Hexipentiruncicantic 7-cube

Alternate names

Tericelligreatorhombated demihepteract (acronym: tucagrohesa)

Images

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]

Hexipentisteriruncic 7-cube

Alternate names

Tericelliprismatorhombated demihepteract (acronym: tucprohesa)

Images

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]

Hexipentistericantic 7-cube

Alternate names

Tericelliprismatotruncated demihepteract (acronym: tucpathesa)

Images

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]

Hexipentisteriruncicantic 7-cube

Alternate names

Great terated demihepteract (acronym: guthesa)

Images

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]

Related polytopes

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 BC7 symmetry, and 32 are unique:

D7 polytopes
t0(141)	t0,1(141)	t0,2(141)	t0,3(141)	t0,4(141)	t0,5(141)	t0,1,2(141)	t0,1,3(141)
t0,1,4(141)	t0,1,5(141)	t0,2,3(141)	t0,2,4(141)	t0,2,5(141)	t0,3,4(141)	t0,3,5(141)	t0,4,5(141)
t0,1,2,3(141)	t0,1,2,4(141)	t0,1,2,5(141)	t0,1,3,4(141)	t0,1,3,5(141)	t0,1,4,5(141)	t0,2,3,4(141)	t0,2,3,5(141)
t0,2,4,5(141)	t0,3,4,5(141)	t0,1,2,3,4(141)	t0,1,2,3,5(141)	t0,1,2,4,5(141)	t0,1,3,4,5(141)	t0,2,3,4,5(141)	t0,1,2,3,4,5(141)

Notes

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. .

External links

Weisstein, Eric W. . MathWorld.

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