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Regular 5-polytope
five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices
5-demicube
steric 5-cube, steric 5-demicube or sterihalf 5-cube, is a convex uniform 5-polytope. There are unique 4 steric forms of the 5-cube. Steric 5-cubes have
Steric_5-cubes
Geometric space with five dimensions
(each a 5-cell). An important uniform 5-polytope is the 5-demicube, h{4,3,3,3} has half the vertices of the 5-cube (16), bounded by alternating 5-cell and
Five-dimensional_space
5-dimensional hypercube
alternating 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
5-cube
Uniform 5-polytope
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
Cantic_5-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
Concept in 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 5-cube. Runcic 5-cubes have
Runcic_5-cubes
6-dimensional hypercube
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
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
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
5-dimensional geometric object
, vertex figure is a rectified 5-orthoplex and facets are the 5-orthoplex and 5-demicube. Pyramidal 5-polytopes, or 5-pyramids, can be generated by a
5-polytope
(Jonathan Bowers) Rectified 5-demicube/demipenteract The birectified 5-cube may be constructed by birectifying the vertices of the 5-cube at 2 {\displaystyle
Rectified_5-cubes
Uniform 6-polytope
The 221 has 27 vertices, and 99 facets: 27 5-orthoplexes and 72 5-simplices. Its vertex figure is a 5-demicube. For visualization this 6-dimensional polytope
2_21_polytope
8-demicube, Truncated 8-demicube, Cantellated 8-demicube, Runcinated 8-demicube, Stericated 8-demicube, Pentellated 8-demicube, Hexicated 8-demicube[citation
List_of_mathematical_shapes
8-demicube, Truncated 8-demicube, Cantellated 8-demicube, Runcinated 8-demicube, Stericated 8-demicube, Pentellated 8-demicube, Hexicated 8-demicube 8-cube
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Five-dimensional geometric shape
the D5 family contains the 5-orthoplex, as well as a 5-demicube which is an alternated 5-cube. Each reflective uniform 5-polytope can be constructed
Uniform_5-polytope
Four-dimensional analogue of the cube
(16-cells). It can also be triangulated into 4-dimensional simplices (irregular 5-cells) that share their vertices with the tesseract. It is known that there
Tesseract
Polyhedron with four faces
down to not 4 but 5, since the fourth constraint is not independent of the first three. Thus, the space of all shapes of tetrahedra is 5-dimensional. Let
Tetrahedron
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
Uniform polytope
5-demicube (demipenteract) in 5 dimensions, and the 4-simplex (5-cell) in 4 dimensions. Each polytope is constructed from 1k−1,2 and (n−1)-demicube facets
Uniform_1_k2_polytope
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
Uniform 7-dimensional polytope
removing the ringed node and ringing the neighboring node. This makes 5-demicube prism, . Birectified hecatonicosihexa-pentacosiheptacontahexa-exon as
3_21_polytope
Four-dimensional analogue of the tetrahedron
{5}},-{\sqrt {5}},-{\sqrt {5}},-1\right)/4} ( − 5 , 5 , − 5 , − 1 ) / 4 {\displaystyle \left(-{\sqrt {5}},{\sqrt {5}},-{\sqrt {5}},-1\right)/4} ( − 5
5-cell
Shape with five sides
by: H = 5 + 2 5 2 t ≈ 1.539 t , W = D = 1 + 5 2 t ≈ 1.618 t , W = 2 − 2 5 ⋅ H ≈ 1.051 H , R = 5 + 5 10 t ≈ 0.8507 t , D = R 5 + 5 2 = 2 R
Pentagon
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
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 ^{2}}{6}}a^{3}\approx
Regular_icosahedron
Multi-dimensional generalization of triangle
5 ) − sin ( 2 π / 5 ) 0 0 sin ( 2 π / 5 ) cos ( 2 π / 5 ) 0 0 0 0 cos ( 4 π / 5 ) − sin ( 4 π / 5 ) 0 0 sin ( 4 π / 5 ) cos ( 4 π / 5 )
Simplex
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
Polyhedron with 12 faces
y|{\Big \}}} and α = 3 − 5 2 , β = 5 − 1 2 = 1 φ . {\displaystyle \alpha ={\frac {3-{\sqrt {5}}}{2}},\qquad \beta ={\frac {{\sqrt {5}}-1}{2}}={\frac {1}{\varphi
Dodecahedron
6-cube, , has half of the vertices of a pentellated 6-cube, . Stericated 6-demicube Stericated demihexeract Small cellated hemihexeract (Acronym: sochax) (Jonathan
Pentic_6-cubes
cantellated 5-simplex is a convex uniform 5-polytope, being a cantellation of the regular 5-simplex. There are unique 4 degrees of cantellation for the 5-simplex
Cantellated_5-simplexes
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
Regular 5-polytope
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.
5-simplex
Plane figure bounded by line segments
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
Polygon
Uniform polychoron
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 5-cell and its
Rectified_5-cell
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
Shape with six sides
have d 1 2 + d 4 2 = d 2 2 + d 5 2 = d 3 2 + d 6 2 = 2 ( R 2 + L 2 ) , {\displaystyle d_{1}^{2}+d_{4}^{2}=d_{2}^{2}+d_{5}^{2}=d_{3}^{2}+d_{6}^{2}=2\le
Hexagon
Uniform 8-polytope
geometry, a cantic 8-cube or truncated 8-demicube is a uniform 8-polytope, being a truncation of the 8-demicube. Truncated demiocteract Truncated hemiocteract;
Cantic_8-cube
them are more easily constructed from the dual 5-cube. Cantellated 5-orthoplex Bicantellated 5-demicube Small rhombated triacontaditeron (Acronym: sart)
Cantellated_5-orthoplexes
Polytope in 8-dimensional geometry
the 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
five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube. There are eight degrees
Stericated_5-cubes
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
Shape with three equal sides
brings about the area of Koch snowflake with infinite iterations 8 / 5 {\displaystyle 8/5} . The Reuleaux triangle is a curved triangle. The performance begins
Equilateral_triangle
Demicube Dodecahedron • Icosahedron Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex
E6_polytope
and 5-cube with 10 and 32 vertices respectively. The 5-demicube is added as an alternation of the 5-cube. They can be visualized as symmetric orthographic
B5_polytope
Convex regular 5-polytope in geometry
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
5-orthoplex
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
shared with the B5 symmetry. There are two special forms, the 5-orthoplex, and 5-demicube with 10 and 16 vertices respectively. They can be visualized
D5_polytope
Regular and uniform honeycombs in 5-space: 5-cubic honeycomb 5-demicube honeycomb Truncated 5-simplex honeycomb Omnitruncated 5-simplex honeycomb "The Lattice
5-simplex_honeycomb
Polygon shape with eight sides
constructed with meccano bars. Twelve bars of size 4, three bars of size 5 and two bars of size 6 are required. Each side of a regular octagon subtends
Octagon
Uniform Polytope
(4-simplexes), 4788 5-faces (756 pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 221). Its vertex figure is a 6-demicube. Its 126 vertices
2_31_polytope
a 5-cube honeycomb. Its facets are 5-demicubes and runcinated 5-demicubes. This honeycomb is one of 20 uniform honeycombs constructed by the D ~ 5 {\displaystyle
Quarter_5-cubic_honeycomb
Equiangular and equilateral polygon
diagonals is 1 2 n ( n − 3 ) {\displaystyle {\tfrac {1}{2}}n(n-3)} ; i.e., 0, 2, 5, 9, ..., for a triangle, square, pentagon, hexagon, ... . The diagonals divide
Regular_polygon
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
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
coordinate permutations: (±1,±1,±3,±5,±5,±5) with an odd number of plus signs. This polytope is based on the 6-demicube, a part of a dimensional family of
Runcic_6-cubes
Geometric object with flat sides
infinitely many regular polygons of n-fold symmetry, both convex and (for n ≥ 5) star. But in higher dimensions there are no other regular polytopes. In three
Polytope
Graph of the vertices and edges of a demihypercube
commonly called the Clebsch graph. It exists in the 5-dimensional uniform 5-polytope, the 5-demicube. Because it is the bipartite half of a distance-regular
Halved_cube_graph
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
Four-dimensional analog of the icosahedron
given as: ({<10}𝜋/5, {≤5}𝜋/10, {<10}𝜋/5) where {<10} is the permutation of the ten digits (0 1 2 3 4 5 6 7 8 9) and {≤5} is the permutation of
600-cell
Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1) Norman Johnson Uniform Polytopes, Manuscript
Rectified_24-cell
Group of geometric symmetries with at least one fixed point
also be given by their Coxeter group and related polygons. These include 5 crystallographic groups. The symmetry of the reflectional groups can be doubled
Point_group
geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex. There are six unique
Stericated_5-simplexes
permutations: (±1,±1,±1,±3,±3,±5) with an odd number of plus signs. Runcicantellated demihexeract Runcicantellated 6-demicube Prismatorhombated hemihexeract
Steric_6-cubes
Shape with four equal sides and angles
Instagram to Polaroid". Afterimage. 45 (5). University of California Press: 10–15. doi:10.1525/aft.2018.45.5.10. Adams, Ansel (1980). "Medium-Format Cameras"
Square
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
Geometrical Shape
1)} Birectified hexeract (acronym: brox) (Jonathan Bowers) Rectified 6-demicube The birectified 6-cube may be constructed from the 6-cube by truncating
Rectified_6-cubes
geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube. There are 8 unique
Runcinated_5-cubes
Four-dimensional geometrical object
a runcinated 5-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation, up to face-planing) of the regular 5-cell. There are
Runcinated_5-cell
5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex. There are 4 unique runcinations of the 5-simplex
Runcinated_5-simplexes
rectified 5-orthoplex is the vertex figure for the 5-demicube honeycomb: or This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube
Rectified_5-orthoplexes
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
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
Convex polytope, the n-dimensional analogue of a square and a cube
MR 0370328. Bowen, J. P. (April 1982). "Hypercube". Practical Computing. 5 (4): 97–99. Archived from the original on 2008-06-30. Retrieved June 30, 2008
Hypercube
Type of tesseract
I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n ≥ 5) Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk
Truncated_tesseract
Regular Schläfli-Hess 4-polytope with 600 vertices
stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol {5/2,3,3}, one of 10 regular Schläfli-Hess 4-polytopes. It is unique among the
Great grand stellated 120-cell
Great_grand_stellated_120-cell
truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex. There are 4 unique truncations of the 5-orthoplex
Truncated_5-orthoplexes
7-dimensional hypercube
84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol {4,35}, being composed of 3 6-cubes around each 5-face. It can be
7-cube
Uniform 6-polytope
the 5-demicube, 121, . The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex
1_22_polytope
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
Runcinated_7-cubes
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
E8_polytope
Uniform 4-polytope bounded by 320 cells
tetrahedra joined to the triangular faces of each antiprism, and a circle of 5 tetrahedra between every pair of antiprisms, joining the 10 tetrahedra of
Grand_antiprism
7-polytope
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
Truncated_7-orthoplexes
Four-dimensional geometric objects
expressed in terms of the golden ratio φ = (1 + √5)/2 and σ = (3√5 + 1)/2. Coxeter expressed them as 5-dimensional coordinates. J.H. Conway and M.J.T.
H4_polytope
Family of regular tessellations in geometry
1} hδ6 5-demicube honeycomb h{4,33,4} {31,1,32,4} {31,1,3,31,1} hδ7 6-demicube honeycomb h{4,34,4} {31,1,33,4} {31,1,32,31,1} hδ8 7-demicube honeycomb
Alternated hypercubic honeycomb
Alternated_hypercubic_honeycomb
Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk
Cantellated_24-cells
Regular 7- polytope
vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cell 4-faces, 448 5-faces, and 128 6-faces. It has two constructed forms, the first
7-orthoplex
Uniform 6-polytope
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\\2&21&5&10&10&5
6-simplex
octeract Acronym: recto (Jonathan Bowers) Birectified octeract Rectified 8-demicube Acronym: bro (Jonathan Bowers) Trirectified octeract Acronym: tro (Jonathan
Rectified_8-cubes
geometry, a truncated 5-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 5-cell. There are two degrees
Truncated_5-cell
Regular object in four dimensional geometry
96 edges, and 24 vertices. Like other four-dimensional regular polytope, 5-cell, the 24-cell is self-dual. The 24-cell and the tesseract are the only
24-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
Truncated_8-simplexes
Type of 7-polytope
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 dihedral angle is cos−1(1/7)
7-simplex
Uniform 6-polytope
pentellated 6-simplex into two hexateral hypercupolas consisting of 7 5-simplexes, 21 5-cell prisms and 35 Tetrahedral-Triangular duoprisms each. The vertices
Pentellated_6-simplexes
Four-dimensional geometric object with flat sides
predecessor, enclosing more content within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity
4-polytope
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
Stericated_7-simplexes
uniform 6-polytope, being a truncation of the regular 6-orthoplex. There are 5 degrees of truncation for the 6-orthoplex. Vertices of the truncated 6-orthoplex
Truncated_6-orthoplexes
cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube. There are 6 unique cantellation for the 5-cube, including
Cantellated_5-cubes
five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex. There are unique 2 degrees
Truncated_5-simplexes
Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296
Runcinated_tesseracts
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