10-demicube

Last updated
Demidekeract
(10-demicube)
Demidekeract ortho petrie.svg
Petrie polygon projection
TypeUniform 10-polytope
Family demihypercube
Coxeter symbol 171
Schläfli symbol {31,7,1}
h{4,38}
s{21,1,1,1,1,1,1,1,1}
Coxeter diagram CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.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 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.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.png
9-faces53220 {31,6,1} Demienneract ortho petrie.svg
512 {38} 9-simplex t0.svg
8-faces5300180 {31,5,1} Demiocteract ortho petrie.svg
5120 {37} 8-simplex t0.svg
7-faces24000960 {31,4,1} Demihepteract ortho petrie.svg
23040 {36} 7-simplex t0.svg
6-faces648003360 {31,3,1} Demihexeract ortho petrie.svg
61440 {35} 6-simplex t0.svg
5-faces1155848064 {31,2,1} Demipenteract graph ortho.svg
107520 {34} 5-simplex t0.svg
4-faces14246413440 {31,1,1} Cross graph 4.svg
129024 {33} 4-simplex t0.svg
Cells12288015360 {31,0,1} 3-simplex t0.svg
107520 {3,3} 3-simplex t0.svg
Faces61440 {3} 2-simplex t0.svg
Edges11520
Vertices512
Vertex figure Rectified 9-simplex
Rectified 9-simplex.png
Symmetry group D10, [37,1,1] = [1+,4,38]
[29]+
Dual?
Properties convex

In geometry, a 10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube 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.

10-cube ten-dimensional hypercube

In geometry, a 10-cube is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces.

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.

Contents

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM10 for a ten-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 171 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.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png and Schläfli symbol or {3,37,1}.

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 demidekeract centered at the origin are alternate halves of the dekeract:

(±1,±1,±1,±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

Images

10-demicube graph.png
B10 coxeter plane
10-demicube.svg
D10 coxeter plane
(Vertices are colored by multiplicity: red, orange, yellow, green = 1,2,4,8)

Related Research Articles

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

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.

5-orthoplex convex regular 5-polytope

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(1/5), or approximately 78.46°.

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.

7-demicube uniform 7-polytope

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.

8-demicube uniform 8-polytope

In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

9-demicube uniform 9-polytope

In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

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.

9-cube convex regular 9-polytope

In geometry, a 9-cube is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces.

Rectified 6-cubes

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

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.

Stericated 8-simplexes

In eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique sterications for the 8-simplex including permutations of truncation, cantellation, and runcination.

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.

Rectified 8-cubes

In eight-dimensional geometry, a rectified 8-cube is a convex uniform 8-polytope, being a rectification of the regular 8-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.

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 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

Harold Scott MacDonald Coxeter Canadian mathematician

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London, received his BA (1929) and PhD (1931) from Cambridge, but lived in Canada from age 29. He was always called Donald, from his third name MacDonald. He was most noted for his work on regular polytopes and higher-dimensional geometries. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra.

<i>Regular Polytopes</i> (book) book about mathematical geometry

Regular Polytopes is a mathematical geometry book written by Canadian mathematician Harold Scott MacDonald Coxeter. Originally published in 1947, the book was updated and republished in 1963 and 1973.

International Standard Book Number Unique numeric book identifier

The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

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