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Uniform 6-polytope
In geometry, a 6-demicube, demihexeract or hemihexeract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed
6-demicube
6-dimensional hypercube
of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has 12 5-demicube and 32
6-cube
6-cube, , has half of the vertices of a pentellated 6-cube, . Stericated 6-demicube Stericated demihexeract Small cellated hemihexeract (Acronym: sochax)
Pentic_6-cubes
Pentellated 6-simplex 6-demicube, Truncated 6-demicube, Cantellated 6-demicube, Runcinated 6-demicube, Stericated 6-demicube 6-cube, Rectified 6-cube, 6-cube
List_of_mathematical_shapes
Uniform Polytope
pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 221). Its vertex figure is a 6-demicube. Its 126 vertices represent the root vectors
2_31_polytope
Pentellated 6-simplex 6-demicube, Truncated 6-demicube, Cantellated 6-demicube, Runcinated 6-demicube, Stericated 6-demicube 6-cube, Rectified 6-cube, Truncated
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube. Runcinated demihexeract Runcinated 6-demicube Small prismated
Steric_6-cubes
alternation of the regular 6-cube honeycomb. It is composed of two different types of facets. The 6-cubes become alternated into 6-demicubes h{4,3,3,3,3} and the
6-demicubic_honeycomb
Four-dimensional analogue of the cube
configuration matrix [ 16 4 6 4 2 32 3 3 4 4 24 2 8 12 6 8 ] . {\displaystyle {\begin{bmatrix}{\begin{matrix}16&4&6&4\\2&32&3&3\\4&4&24&2\\8&12&6&8\end{matrix}}\end{bmatrix}}
Tesseract
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
9-demicube
6-dimensional geometric object
The expanded 6-simplex is the vertex figure of the uniform 6-simplex honeycomb, . The 6-demicube honeycomb, , vertex figure is a rectified 6-orthoplex and
6-polytope
Polyhedron with four faces
{3}{2}}}} , 1 2 {\displaystyle {\sqrt {\tfrac {1}{2}}}} , 1 6 {\displaystyle {\sqrt {\tfrac {1}{6}}}} (edges that are the characteristic radii of the regular
Tetrahedron
six-dimensional geometry, a runcic 6-cube is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube. Cantellated 6-demicube Cantellated demihexeract
Runcic_6-cubes
Shape with six sides
circle, equal to 120°. The Schläfli symbol denotes this polygon as { 6 } {\displaystyle \{6\}} . However, the regular hexagon can also be considered as cutting
Hexagon
7-cube is a convex uniform 7-polytope, constructed from the uniform 7-demicube. There are 16 unique forms. Small terated demihepteract (acronym: suthesa)
Hexic_7-cubes
(Acronym: gocog) (Jonathan Bowers) The snub 6-demicube defined as an alternation of the omnitruncated 6-demicube is not uniform, but it can be given Coxeter
Stericated_6-orthoplexes
Regular 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
5-demicube
Shape in six-dimensional geometry
six-dimensional geometry, a cantic 6-cube (or a truncated 6-demicube) is a uniform 6-polytope. Truncated 6-demicube Truncaced demihexeract Truncated hemihexeract
Cantic_6-cube
Uniform 8 dimensional 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
8-demicube
Polyhedron with 12 faces
angle of approximately 121.6° in between two angles of approximately 106.6° and opposite two angles of approximately 102.6°. The following formulas show
Dodecahedron
Uniform 6-dimensional polytope
[33,1,1] - 47 uniform 6-polytopes (16 unique) as permutations of rings in the group diagram, including: {3,32,1}, 121 6-demicube (demihexeract) - ; also
Uniform_6-polytope
7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube. A uniform 7-polytope is vertex-transitive and constructed from uniform 6-polytope
Cantic_7-cube
Geometrical Shape
(acronym: brox) (Jonathan Bowers) Rectified 6-demicube The birectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints
Rectified_6-cubes
In 6-dimensional geometry, there are 39 uniform polytopes with E6 symmetry. The two simplest forms are the 221 and 122 polytopes, composed of 27 and 72
E6_polytope
Uniform 10-polytope
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
10-demicube
Polytope constructed from alternation of a hypercube
In geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation
Demihypercube
Plane figure bounded by line segments
section, the coordinates of the centroid of a solid simple polygon are C x = 1 6 A ∑ i = 0 n − 1 ( x i + x i + 1 ) ( x i y i + 1 − x i + 1 y i ) , {\displaystyle
Polygon
Shape with three equal sides
three sides of an equilateral at their midpoints r = 3 t 6 . {\displaystyle r={\frac {{\sqrt {3}}t}{6}}.} The circumscribed circle of an equilateral triangle
Equilateral_triangle
5-dimensional hypercube
vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the demihypercubes. The
5-cube
Multi-dimensional generalization of triangle
cos ( 6 π / 5 ) sin ( 6 π / 5 ) cos ( 2 π / 5 ) sin ( 2 π / 5 ) ) , ( cos ( 8 π / 5 ) sin ( 8 π / 5 ) cos ( 6 π / 5 ) sin ( 6 π / 5 )
Simplex
10-dimensional hypercube
vertices of the dekeract, creates another uniform polytope, called a 10-demicube, (part of an infinite family called demihypercubes), which has 20 demienneractic
10-cube
Geometric space with six dimensions
Each uniform polytope is defined by a ringed Coxeter–Dynkin diagram. The 6-demicube is a unique polytope from the D6 family, and 221 and 122 polytopes from
Six-dimensional_space
Uniform 8-polytope
Cartesian coordinates for the vertices of a truncated 8-demicube centered at the origin and edge length 6√2 are coordinate permutations: (±1,±1,±3,±3,±3,±3
Cantic_8-cube
honeycomb, and a quarter of the vertices of a 6-cube honeycomb. Its facets are 6-demicubes, stericated 6-demicubes, and {3,3}×{3,3} duoprisms. This honeycomb
Quarter_6-cubic_honeycomb
Shape with five sides
i = 1 5 d i 6 = 5 ( ( R 2 + L 2 ) 3 + 6 R 2 L 2 ( R 2 + L 2 ) ) , ∑ i = 1 5 d i 8 = 5 ( ( R 2 + L 2 ) 4 + 12 R 2 L 2 ( R 2 + L 2 ) 2 + 6 R 4 L 4 ) . {\displaystyle
Pentagon
Solid with twenty equal triangular faces
8.660 a 2 , V = 5 φ 2 6 a 3 ≈ 2.182 a 3 . {\displaystyle A=5{\sqrt {3}}a^{2}\approx 8.660a^{2},\qquad V={\frac {5\varphi ^{2}}{6}}a^{3}\approx 2.182a^{3}
Regular_icosahedron
geometry, a stericated 6-cube is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-cube. There are 8 unique
Stericated_6-cubes
Equiangular and equilateral polygon
Hexagons – {6/2}, {6/3} Octagons – {8/2}, {8/4} Enneagon – {9/3} Decagons – {10/2}, {10/4}, and {10/5} Dodecagons – {12/2}, {12/3}, {12/4}, and {12/6} Depending
Regular_polygon
pentic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 8 unique forms. Small cellated demihepteract (acronym: sochesa)
Pentic_7-cubes
Geometric object with flat sides
polytopes (2nd ed.), New York & London: Springer-Verlag, ISBN 0-387-00424-6. Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics
Polytope
9-dimensional hypercube
another uniform polytope, called a 9-demicube, (part of an infinite family called demihypercubes), which has 18 8-demicube and 256 8-simplex facets. Klitzing
9-cube
{\displaystyle \left(0,1,2,3,4\right)} The snub 5-demicube defined as an alternation of the omnitruncated 5-demicube is not uniform, but it can be given Coxeter
Runcinated_5-orthoplexes
Uniform 6-polytope
7 6 15 20 15 6 2 21 5 10 10 5 3 3 35 4 6 4 4 6 4 35 3 3 5 10 10 5 21 2 6 15 20 15 6 7 ] {\displaystyle {\begin{bmatrix}{\begin{matrix}7&6&15&20&15&6
6-simplex
6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex. There are 8 unique sterications for the 6-simplex
Stericated_6-simplexes
runcinated 7-demicube) is a convex uniform 7-polytope, being a runcination of the uniform 7-demicube. There are 4 unique runcinations for the 7-demicube including
Steric_7-cubes
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
7-demicube
Polytope in 8-dimensional geometry
ringed node and adding rings to the neighboring nodes. This makes a 5-demicube-triangular duoprism. These graphs represent orthographic projections in
4_21_polytope
8-dimensional hypercube
vertices of the octeract, creates another uniform polytope, called an 8-demicube, (part of an infinite family called demihypercubes), which has 16 demihepteractic
8-cube
Shape with four equal sides and angles
square. The answer to the puzzle is n ( n + 1 ) ( 2 n + 1 ) / 6 {\displaystyle n(n+1)(2n+1)/6} , a square pyramidal number. For n = 1 , 2 , 3 , … {\displaystyle
Square
runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-cube. There are 12 unique runcinations of the 6-cube
Runcinated_6-cubes
the 6-orthoplex, and 6-cube with 12 and 64 vertices respectively. The 6-demicube is added with half the symmetry. They can be visualized as symmetric orthographic
B6_polytope
Convex polytope, the n-dimensional analogue of a square and a cube
{\displaystyle 4} sides or edges; a 3 {\displaystyle 3} -dimensional cube has 6 {\displaystyle 6} square faces; a ( 4 {\displaystyle 4} -dimensional) tesseract has
Hypercube
Four-dimensional analog of the octahedron
There is a lower symmetry form of the 16-cell, called a demitesseract or 4-demicube, a member of the demihypercube family, and represented by h{4,3,3}, and
16-cell
of the birectified 7-orthoplex. The rectified 6-simplex polytope is the vertex figure of the 7-demicube, and the edge figure of the uniform 241 polytope
Rectified_6-simplexes
Uniform 6-polytope
5-orthoplexes and 72 5-simplices. Its vertex figure is a 5-demicube. For visualization this 6-dimensional polytope is often displayed in a special skewed
2_21_polytope
Concept in geometry
In five-dimensional geometry, a runcic 5-cube, runcic 5-demicube or runcihalf 5-cube, is a convex uniform 5-polytope. There are 2 runcic forms for the
Runcic_5-cubes
Uniform polytope
diagram, Removing the node on the end of the 2-length branch leaves the 6-demicube, 131, Removing the node on the end of the 3-length branch leaves the 122
1_32_polytope
geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination (3rd order truncations) of the regular 6-simplex. There are 8
Runcinated_6-simplexes
Group of geometric symmetries with at least one fixed point
crystallographic restriction theorem restricts n to values 1, 2, 3, 4, and 6 for both families, yielding 10 groups. The subset of pure reflectional point
Point_group
Convex regular 8-polytope
56 70 56 28 8 2 36 7 21 35 35 21 7 3 3 84 6 15 20 15 6 4 6 4 126 5 10 10 5 5 10 10 5 126 4 6 4 6 15 20 15 6 84 3 3 7 21 35 35 21 7 36 2 8 28 56 70 56
8-simplex
Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6 (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit
Runcinated_7-cubes
six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex. There are unique 3 degrees
Truncated_6-simplexes
of the cubical cells are connected to the other 24 cubical cells via all 6 square faces. The other 24 cubical cells are connected to the former 8 cells
Runcinated_tesseracts
Four-dimensional geometric object with flat sides
infinite families) 47 non-prismatic convex uniform 4-polytope including: 6 Convex regular 4-polytope Prismatic uniform 4-polytopes: {} × {p,q} : 18 polyhedral
4-polytope
Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6 (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit
Stericated_7-orthoplexes
5-dimensional geometric object
5-simplex honeycomb, . The 5-demicube honeycomb, , vertex figure is a rectified 5-orthoplex and facets are the 5-orthoplex and 5-demicube. Pyramidal 5-polytopes
5-polytope
Regular 5-polytope
[ 6 5 10 10 5 2 15 4 6 4 3 3 20 3 3 4 6 4 15 2 5 10 10 5 6 ] {\displaystyle {\begin{bmatrix}{\begin{matrix}6&5&10&10&5\\2&15&4&6&4\\3&3&20&3&3\\4&6
5-simplex
Regular object in four dimensional geometry
row's element. [ 24 8 12 6 2 96 3 3 3 3 96 2 6 12 8 24 ] . {\displaystyle {\begin{bmatrix}24&8&12&6\\2&96&3&3\\3&3&96&2\\6&12&8&24\end{bmatrix}}.} The
24-cell
Penteractitriacontaditeron (acronym: nit) (Jonathan Bowers) Rectified 5-demicube/demipenteract The birectified 5-cube may be constructed by birectifying
Rectified_5-cubes
Geometric object
9-orthoplex facets with all vertices at infinity.) The family starts uniquely as 6-polytopes. The triangular prism and rectified 5-cell are included at the beginning
Uniform_k_21_polytope
Seven-dimensional geometric object
{\tilde {C}}_{6}} , 32 new Uniform 6-demicube honeycomb, represented by symbols h{4,34,4} = {31,1,33,4}, = D ~ 6 {\displaystyle {\tilde {D}}_{6}} , [31,1
Uniform_7-polytope
Regular 6 dimensional polytope
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell
6-orthoplex
runcic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 2 unique forms. A runcic 7-cube, h3{4,35}, has half the vertices
Runcic_7-cubes
7-polytope
uniform 7-polytope, being a truncation of the regular 7-orthoplex. There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are
Truncated_7-orthoplexes
Uniform polytope
{3,3k,2}. The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-demicube (demipenteract) in 5 dimensions, and the
Uniform_1_k2_polytope
Four-dimensional analogue of the tetrahedron
1 6 , 1 3 , ± 1 ) {\displaystyle \left({\frac {1}{\sqrt {10}}},\ {\frac {1}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)} ( 1 10 , 1 6 ,
5-cell
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 •
Rectified_24-cell
Type of geometrical object
10-demicube - Demihypercube D10 family: [37,1,1] - 767 uniform 10-polytopes as permutations of rings in the group diagram, including: 17,1 - 10-demicube
Uniform_10-polytope
Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6 (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit
E8_polytope
six-dimensional geometry, a runcinated 6-orthplex is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-orthoplex. There are 12 unique
Runcinated_6-orthoplexes
Type of geometric object
as permutations of rings in the group diagram, including: {31,6,1} - 9-demicube or demienneract, 161 - ; also as h{4,38} - {36,1,1} - 9-orthoplex, 611
Uniform_9-polytope
7-dimensional hypercube
or at the row's element. [ 128 7 21 35 35 21 7 2 448 6 15 20 15 6 4 4 672 5 10 10 5 8 12 6 560 4 6 4 16 32 24 8 280 3 3 32 80 80 40 10 84 2 64 192 240
7-cube
Four-dimensional analog of the icosahedron
dual polytope is the 120-cell. The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes in order of complexity(as measured by comparing
600-cell
shared with the B6 symmetry. There are two regular forms, the 6-orthoplex, and 6-demicube with 12 and 32 vertices respectively. They can be visualized
D6_polytope
the 6-cubes with 6-demicubes, and the alternated gaps are filled by 6-orthoplex facets. A trirectified 6-cubic honeycomb, , contains all birectified 6-orthoplex
6-cubic_honeycomb
Uniform polychoron
penteract respectively. The rectified 5-cell is the vertex figure of the 5-demicube, and the edge figure of the uniform 221 polytope. The convex hull the rectified
Rectified_5-cell
Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6 (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit
Stericated_7-simplexes
previous as its vertex figure. This polytope is the vertex figure of the 6-demicube, and the edge figure of the uniform 231 polytope. It is also one of 19
Rectified_5-simplexes
Type of 7-polytope
degree rotation. [ 8 7 21 35 35 21 7 2 28 6 15 20 15 6 3 3 56 5 10 10 5 4 6 4 70 4 6 4 5 10 10 5 56 3 3 6 15 20 15 6 28 2 7 21 35 35 21 7 8 ] {\displaystyle
7-simplex
octeract Acronym: recto (Jonathan Bowers) Birectified octeract Rectified 8-demicube Acronym: bro (Jonathan Bowers) Trirectified octeract Acronym: tro (Jonathan
Rectified_8-cubes
In five-dimensional geometry, a steric 5-cube, steric 5-demicube or sterihalf 5-cube, is a convex uniform 5-polytope. There are unique 4 steric forms of
Steric_5-cubes
cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex. There are unique 4 degrees of cantellation for the 6-simplex
Cantellated_6-simplexes
Four-dimensional geometrical object
The 6 square faces of the cuboctahedron are joined to the edges of the central tetrahedron via distorted triangular prisms. These are the images of 6 of
Runcinated_5-cell
truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube. There are 5 truncations for the 6-cube.
Truncated_6-cubes
Isogonal polyhedron with regular faces
Convex and Computational, Springer, pp. 43–70, doi:10.1007/978-94-011-0924-6_3, ISBN 978-94-010-4398-4 McMullen, Peter; Schulte, Egon (2002), Abstract
Uniform_polyhedron
Solid with eight equal triangular faces
, r i = 6 6 a ≈ 0.408 a , r m = 1 2 a = 0.5 a . {\displaystyle r_{u}={\frac {\sqrt {2}}{2}}a\approx 0.707a,\qquad r_{i}={\frac {\sqrt {6}}{6}}a\approx
Regular_octahedron
Convex regular 9 dimensional polytope
tetrahedron cells, 4032 5-cell 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces. It has two
9-orthoplex
octahedra, 15 cuboctahedra and 60 triangular prisms), and 27 4-faces (6 cantellated 5-cell, 6 rectified 5-cells, and 15 tetrahedral prisms). Cantellated hexateron
Cantellated_5-simplexes
Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6 (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit
Truncated_8-simplexes
10|3}, {10,12|3}, {12,4|3}, {4,12|3}, {6,4|6}, {4,6|6}, {8,4|4}, {4,8|4}, {12,6|3}, {6,12|3}, {12,12|3}, {6,6|6}, {8,6|4}, {6,8|4}, {12,8|3}, {8,12|3}, and
List_of_regular_polytopes
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