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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
8-demicube, Cantellated 8-demicube, Runcinated 8-demicube, Stericated 8-demicube, Pentellated 8-demicube, Hexicated 8-demicube[citation needed] 142 polytope
List_of_mathematical_shapes
Hexicated 8-simplex, Heptellated 8-simplex 8-demicube, Truncated 8-demicube, Cantellated 8-demicube, Runcinated 8-demicube, Stericated 8-demicube, Pentellated
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
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
Group of polytopes
In 8-dimensional geometry, there are 255 uniform polytopes with B8 symmetry (to which this article adds for illustration the 8-demicube as an alternation
B8_polytope
Uniform 8-polytope
eight-dimensional geometry, a cantic 8-cube or truncated 8-demicube is a uniform 8-polytope, being a truncation of the 8-demicube. Truncated demiocteract Truncated
Cantic_8-cube
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
9-dimensional hypercube
uniform polytope, called a 9-demicube, (part of an infinite family called demihypercubes), which has 18 8-demicube and 256 8-simplex facets. Klitzing, Richard
9-cube
Geometric space with eight dimensions
Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram. The 8-demicube is a unique polytope from the D8 family, and 421, 241, and 142 polytopes
Eight-dimensional_space
Polyhedron with four faces
thereby reduced from 12 to 8. The four relations given by this sine law further reduce the number of degrees of freedom, from 8 down to not 4 but 5, since
Tetrahedron
8-dimensional hypercube
polytope, called an 8-demicube, (part of an infinite family called demihypercubes), which has 16 demihepteractic and 128 8-simplex facets. The 8-cube is 8th in
8-cube
6-polytope. There are 8 pentic forms of the 6-cube. The pentic 6-cube, , has half of the vertices of a pentellated 6-cube, . Stericated 6-demicube Stericated demihexeract
Pentic_6-cubes
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
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 contained by 7-polytope facets
- 8-demicube or demiocteract, 151 - ; also as h{4,36} - {3,3,3,3,3,31,1} - 8-orthoplex, 511 - E-polytope family E8 family: [34,1,1] - 255 uniform 8-polytopes
Uniform_8-polytope
Four-dimensional analogue of the cube
one of the six convex regular 4-polytopes. The tesseract is also called an 8-cell, C8, (regular) octachoron, or cubic prism. It is the four-dimensional
Tesseract
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
honeycomb, and a quarter of the vertices of a 8-cube honeycomb. Its facets are 8-demicubes h{4,36}, pentic 8-cubes h6{4,36}, {3,3}×{32,1,1} and {31,1,1}×{31
Quarter_8-cubic_honeycomb
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
octeract Acronym: recto (Jonathan Bowers) Birectified octeract Rectified 8-demicube Acronym: bro (Jonathan Bowers) Trirectified octeract Acronym: tro (Jonathan
Rectified_8-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
6-dimensional hypercube
another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has 12 5-demicube and 32 5-simplex facets. This configuration
6-cube
Shape with six sides
regular hexagon A = 3 3 2 R 2 = 3 R r = 2 3 r 2 ≈ 2.598 R 2 ≈ 3.464 r 2 = 3 3 8 D 2 = 3 4 D d = 3 2 d 2 ≈ 0.6495 D 2 ≈ 0.866 d 2 . {\displaystyle {\begin{aligned}A&={\frac
Hexagon
Plane figure bounded by line segments
density 2. The two triangular regions of a cross-quadrilateral (like a figure 8) have opposite-signed densities, and adding their areas together can give
Polygon
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
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
Polyhedron with 12 faces
convex dodecahedra, excluding mirror images—the number of vertices ranges from 8 to 20. Two polyhedra are topologically distinct if they have intrinsically
Dodecahedron
Multi-dimensional generalization of triangle
( 8 π / 5 ) sin ( 8 π / 5 ) ) , ( cos ( 6 π / 5 ) sin ( 6 π / 5 ) cos ( 2 π / 5 ) sin ( 2 π / 5 ) ) , ( cos ( 8 π / 5 ) sin ( 8 π / 5
Simplex
alternation of the regular 8-cubic honeycomb. It is composed of two different types of facets. The 8-cubes become alternated into 8-demicubes h{4,3,3,3,3,3,3} and
8-demicubic_honeycomb
Convex polytope, the n-dimensional analogue of a square and a cube
square faces; a ( 4 {\displaystyle 4} -dimensional) tesseract has 8 {\displaystyle 8} three-dimensional cubes as its facets. The number of vertices of
Hypercube
In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry. The three simplest forms are the 421, 241, and 142 polytopes, composed of
E8_polytope
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
E6_polytope
cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex. There are six unique cantellations for the 8-simplex
Cantellated_8-simplexes
Shape with four equal sides and angles
Problems". The Arithmetic Teacher. 25 (8). National Council of Teachers of Mathematics: 39–43. doi:10.5951/at.25.8.0039. JSTOR 41190469. Sloane, N. J. A
Square
Shape with five sides
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
Type of geometric object
with C ~ 8 {\displaystyle {\tilde {C}}_{8}} , 128 new 8-demicube honeycomb: h{4,36,4} or {31,1,35,4}, or D ~ 8 {\displaystyle {\tilde {D}}_{8}} , [31,1
Uniform_9-polytope
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-cell
8-orthoplex
Class of eight-dimensional polytopes
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
Stericated_8-simplexes
Solid with twenty equal triangular faces
expressions are: A = 5 3 a 2 ≈ 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
Regular_icosahedron
Shape with three equal sides
egg, and the cathedral Sant'Ivo alla Sapienza floor plan. The cue sports of 8-ball pool arranged fifteen object balls with a tool of an equilateral triangular
Equilateral_triangle
figure of the 9-demicube, and the edge figure of the uniform 261 honeycomb. The Cartesian coordinates of the vertices of the rectified 8-simplex can be
Rectified_8-simplexes
eight-dimensional geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex. There are four unique degrees
Truncated_8-simplexes
7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 8 unique forms. Small cellated demihepteract (acronym: sochesa) The
Pentic_7-cubes
Equiangular and equilateral polygon
to 12 sides are: Tetragon – {4/2} Hexagons – {6/2}, {6/3} Octagons – {8/2}, {8/4} Enneagon – {9/3} Decagons – {10/2}, {10/4}, and {10/5} Dodecagons –
Regular_polygon
Convex regular 8-polytope
In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces
8-simplex
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 Schläfli-Hess 4-polytope with 600 vertices
Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things
Great grand stellated 120-cell
Great_grand_stellated_120-cell
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
eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex. There are eleven
Runcinated_8-simplexes
Geometric object with flat sides
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
Polytope
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
Cantellated_5-simplexes
Eight-dimensional geometric tessellation
8-demicube vertex figure. It is the final figure in the 2k1 family. It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional
2_51_honeycomb
eight-dimensional geometry, a hexicated 8-simplex is a uniform 8-polytope, being a hexication (6th order truncation) of the regular 8-simplex. Acronym: supane (Jonathan
Hexicated_8-simplexes
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
Uniform 6-polytope
Richard. "6D uniform polytopes (polypeta) x3o3o3o3o3o — hop". Coxeter 1973, §1.8 Configurations Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.)
6-simplex
In 8-dimensional geometry, there are 135 uniform polytopes with A8 symmetry. There is one self-dual regular form, the 8-simplex with 9 vertices. Each can
A8_polytope
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
honeycomb, replacing the 8-cubes with 8-demicubes, and the alternated gaps are filled by 8-orthoplex facets. A quadrirectified 8-cubic honeycomb, , contains
8-cubic_honeycomb
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
Truncated_5-cubes
Regular 5-polytope
vertices connected between. The form { }∨{ }∨{ } has symmetry [2,2,1,1], order 8, extended by permuting 3 segments as [3[2,2],1] or [4,3,1,1], order 48. These
5-simplex
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
Truncated_5-orthoplexes
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
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
List_of_F4_polytopes
Solid with eight equal triangular faces
by placing the vertices of a regular octahedron inscribed in a sphere. An 8-sided die is usually a regular octahedron. In roleplaying games, this is one
Regular_octahedron
Skew polygon derived from a polytope
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
Petrie_polygon
Type of 7-polytope
7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its
7-simplex
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
Stericated_7-orthoplexes
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
Rectified_9-cubes
geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube. A uniform 7-polytope is vertex-transitive
Cantic_7-cube
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
Truncated_6-simplexes
geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex. There are unique 8 degrees of rectifications
Rectified_8-orthoplexes
runcinations of the 7-cube with permutations of truncations, and cantellations. 8 are more simply constructed from the 7-orthoplex. These polytopes are among
Runcinated_7-cubes
Uniform polytopes with D8 symmetry
In 8-dimensional geometry, there are 191 uniform polytopes with D8 symmetry, of which 64 are unique and 127 are shared with the B8 symmetry. There is
D8_polytope
Type of tesseract
creates the truncated 16-cell. The truncated tesseract is bounded by 24 cells: 8 truncated cubes, and 16 tetrahedra. Truncated tesseract (Norman W. Johnson)
Truncated_tesseract
Four-dimensional geometric object with flat sides
Artificial Neural Networks and Arts. Springer. p. 598. doi:10.1007/978-90-481-8581-8. ISBN 978-90-481-8580-1. Coxeter 1973, p. 141, §7-x. Historical remarks. Coxeter
4-polytope
Uniform Polytope
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 of the simple Lie group E7
2_31_polytope
Group of geometric symmetries with at least one fixed point
) 7-cube, 7-orthoplex D7 [3,3,3,3,31,1] 322560 (26×7!) 7-demicube E7 [3,3,3,32,1] 2903040 (8×9!) 321, 231, 132 A6×A1 [3,3,3,3,3,2] 10080 (2×7!) BC6×A1
Point_group
There are many enumerations that fit in the plane (1/p + 1/q = 1/2), like {8/3,8}, {10/3,5}, {5/2,10}, {12/5,12}, etc., but none repeat periodically. Tessellations
List_of_regular_polytopes
a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex. There are 7 truncation for the 8-orthoplex. Vertices
Truncated_8-orthoplexes
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
7-dimensional hypercube
[ 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 64 192 240 160 60 12 14 ] {\displaystyle
7-cube
Four-dimensional analogue of the tetrahedron
the characteristic orthoscheme of the 4-cube (also called the tesseract or 8-cell), the 4-dimensional analogue of the 3-dimensional cube. If the three
5-cell
the rectified tesseract, rectified 8-cell is a uniform 4-polytope (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. It
Rectified_tesseract
Uniform 6-polytope
facets: 27 5-orthoplexes and 72 5-simplices. Its vertex figure is a 5-demicube. For visualization this 6-dimensional polytope is often displayed in a
2_21_polytope
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
Stericated_7-cubes
symmetry. There are two regular forms, the tesseract and 16-cell, with 16 and 8 vertices respectively. They can be visualized as symmetric orthographic projections
B4_polytope
Uniform 4-polytope
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
Truncated_120-cells
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
Runcinated_120-cells
Four-dimensional geometric objects
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
H4_polytope
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
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 (Acronym:
Cantic_6-cube
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
Runcinated_5-simplexes
Four-dimensional analog of the dodecahedron
{\displaystyle (0,0,0,-1)} , comprise all 8 vertices of the 4-orthoplex. Uniquely, the 4-orthoplex is also the 4-demicube, half the vertices of the 4-cube. This
120-cell
uniform 6-polytope. There are 2 unique runcic for the 6-cube. Cantellated 6-demicube Cantellated demihexeract Small rhombated hemihexeract (Acronym: sirhax)
Runcic_6-cubes
Regular polytope dual to the hypercube in any number of dimensions
compound polytopes: In two dimensions, we obtain the octagrammic star figure {8/2}, In three dimensions we obtain the compound of cube and octahedron, In
Cross-polytope
Convex uniform 8-polytope in 8-dimensional geometry
truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube. There are unique 7 degrees of truncation for the 8-cube. Vertices
Truncated_8-cubes
Type of uniform space-filling tessellation
The 5-demicube honeycomb (or demipenteractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 5-space. It is constructed
5-demicubic_honeycomb
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
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
all 6 square faces. The other 24 cubical cells are connected to the former 8 cells via only two opposite square faces; the remaining 4 faces are connected
Runcinated_tesseracts
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