7-demicube

Last updated
Demihepteract
(7-demicube)
Demihepteract ortho petrie.svg
Petrie polygon projection
TypeUniform 7-polytope
Family demihypercube
Coxeter symbol 141
Schläfli symbol {3,34,1} = h{4,35}
s{21,1,1,1,1,1}
Coxeter diagrams CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.png

6-faces7814 {31,3,1} Demihexeract ortho petrie.svg
64 {35} 6-simplex t0.svg
5-faces53284 {31,2,1} Demipenteract graph ortho.svg
448 {34} 5-simplex t0.svg
4-faces1624280 {31,1,1} 4-orthoplex.svg
1344 {33} 4-simplex t0.svg
Cells2800560 {31,0,1} 3-simplex t0.svg
2240 {3,3} 3-simplex t0.svg
Faces2240 {3} 2-simplex t0.svg
Edges672
Vertices64
Vertex figure Rectified 6-simplex
6-simplex t1.svg
Symmetry group D7, [34,1,1] = [1+,4,35]
[26]+
Dual?
Properties convex

In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

Geometry branch of mathematics that measures the shape, size and position of objects

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

Alternation (geometry) operation on a polyhedron or tiling that removes alternate vertices

In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

Uniform polytope vertex-transitive polytope bounded by uniform facets

A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

Contents

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM7 for a 7-dimensional half measure polytope.

Emanuel Lodewijk Elte was a Dutch mathematician. He is noted for discovering and classifying semiregular polytopes in dimensions four and higher.

Coxeter named this polytope as 141 from its Coxeter diagram, with a ring on one of the 1-length branches, CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png and Schläfli symbol or {3,34,1}.

Schläfli symbol notation that defines regular polytopes and tessellations

In geometry, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations.

Cartesian coordinates

Cartesian coordinates for the vertices of a demihepteract centered at the origin are alternate halves of the hepteract:

(±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

Images

orthographic projections
Coxeter
plane
B7D7D6
Graph 7-demicube t0 B7.svg 7-demicube t0 D7.svg 7-demicube t0 D6.svg
Dihedral
symmetry
[14/2][12][10]
Coxeter planeD5D4D3
Graph 7-demicube t0 D5.svg 7-demicube t0 D4.svg 7-demicube t0 D3.svg
Dihedral
symmetry
[8][6][4]
Coxeter
plane
A5A3
Graph 7-demicube t0 A5.svg 7-demicube t0 A3.svg
Dihedral
symmetry
[6][4]

Construction

The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. [1] [2]

In geometry, H. S. M. Coxeter called a regular polytope a special kind of configuration.

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time. [3]

Wythoff construction method for constructing a uniform polyhedron or plane tiling

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.

D7CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png k-face fk f0 f1f2 f3 f4f5f6 k-figures notes
A6CDel nodea x.pngCDel 2.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png ( ) f064211053514035105214277 041 D7/A6 = 64*7!/7! = 64
A4A1A1CDel nodea 1.pngCDel 2.pngCDel nodes x0.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png { } f12672105201020101052 { }×{3,3,3} D7/A4A1A1 = 64*7!/5!/2/2 = 672
A3A2CDel nodea 1.pngCDel 3a.pngCDel nodes 0x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 100 f233224014466441 {3,3}v( ) D7/A3A2 = 64*7!/4!/3! = 2240
A3A3CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 101 f3464560*406040 {3,3} D7/A3A3 = 64*7!/4!/4! = 560
A3A2CDel nodea 1.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 110 464*2240133331 {3}v( ) D7/A3A2 = 64*7!/4!/3! = 2240
D4A2CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 111 f48243288280*3030 {3} D7/D4A2 = 64*7!/8/4!/2 = 280
A4A1CDel nodea 1.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 120 5101005*13441221 { }v( ) D7/A4A1 = 64*7!/5!/2 = 1344
D5A1CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 121 f516801604080101684*20{ } D7/D5A1 = 64*7!/16/5!/2 = 84
A5CDel nodea 1.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 130 6152001506*44811 D7/A5 = 64*7!/6! = 448
D6CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 131 f63224064016048060192123214*( ) D7/D6 = 64*7!/32/6! = 14
A6CDel nodea 1.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 140 7213503502107*64 D7/A6 = 64*7!/7! = 64

There are 95 uniform polytopes with D6 symmetry, 63 are shared by the B6 symmetry, and 32 are unique:

Related Research Articles

5-demicube semiregular 5-polytope

In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.

6-cube 6-dimensional hypercube

In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

6-demicube uniform 6-polytope

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

7-cube convex regular 7-polytope

In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.

8-cube convex regular 8-polytope

In geometry, an 8-cube is an eight-dimensional hypercube (8-cube). It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.

7-orthoplex convex regular 7-polytope

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.

8-orthoplex convex regular 8-polytope

In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.

Rectified 6-simplexes

In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

Cantic 5-cube

In geometry of five dimensions or higher, a cantic 5-cube, cantihalf 5-cube, truncated 5-demicube is a uniform 5-polytope, being a truncation of the 5-demicube. It has half the vertices of a cantellated 5-cube.

Runcinated 5-orthoplexes

In five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex.

Runcic 5-cubes

In six-dimensional geometry, a runcic 5-cube or is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.

Cantic 7-cube

In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube.

In seven-dimensional geometry, a rectified 7-cube is a convex uniform 7-polytope, being a rectification of the regular 7-cube.

Runcic 6-cubes

In six-dimensional geometry, a runcic 6-cube is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube.

Steric 6-cubes

In six-dimensional geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube.

Runcic 7-cubes

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.

Pentic 7-cubes

In seven-dimensional geometry, a pentic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 8 unique forms.

Hexic 7-cubes

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.

Steric 5-cubes

In five-dimensional geometry, a steric 5-cube or is a convex uniform 5-polytope. There are unique 4 steric forms of the 5-cube. Steric 5-cubes have half the vertices of stericated 5-cubes.

Steric 7-cubes

In seven-dimensional geometry, a stericated 7-cube is a convex uniform 7-polytope, being a runcination of the uniform 7-demicube. There are 4 unique runcinations for the 7-demicube including truncation and cantellation.

References

  1. Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. Coxeter, Complex Regular Polytopes, p.117
  3. Klitzing, Richard. "x3o3o *b3o3o3o - hax".
Fundamental 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 OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds