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Isogonal honeycomb of uniform polytope facets
In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets
Uniform_honeycomb
Spatial tiling of convex uniform polyhedra
geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells
Convex_uniform_honeycomb
Tiling of euclidean or hyperbolic space of three or more dimensions
spaces, such as hyperbolic honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. There
Honeycomb_(geometry)
Only regular space-filling tessellation of the cube
such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space
Cubic_honeycomb
Quasiregular space-filling tesselation
number of dimensions. Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed
Tetrahedral-octahedral honeycomb
Tetrahedral-octahedral_honeycomb
Tiling of hyperbolic 3-space by uniform polyhedra
hyperbolic uniform honeycombs. More unsolved problems in mathematics In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
a triangular tiling extruded into prisms. It is one of 28 convex uniform honeycombs. It consists of 1 + 6 + 1 = 8 edges meeting at a vertex, There are
Triangular prismatic honeycomb
Triangular_prismatic_honeycomb
such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space
Order-4 hexagonal tiling honeycomb
Order-4_hexagonal_tiling_honeycomb
Seven-dimensional geometric object
Uniform 6-simplex honeycomb: {3[7]} Uniform Cyclotruncated 6-simplex honeycomb: t0,1{3[7]} Uniform Omnitruncated 6-simplex honeycomb: t0,1,2,3,4,5,6,7{3[7]}
Uniform_7-polytope
Regular tiling of hyperbolic 3-space
such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space
Order-4 dodecahedral honeycomb
Order-4_dodecahedral_honeycomb
number of dimensions. Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed
Order-6 hexagonal tiling honeycomb
Order-6_hexagonal_tiling_honeycomb
Tessellation of convex uniform polyhedron cells
In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23
Paracompact uniform honeycombs
Paracompact_uniform_honeycombs
Regular tiling of hyperbolic 3-space
such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space
Icosahedral_honeycomb
number of dimensions. Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed
Order-5 hexagonal tiling honeycomb
Order-5_hexagonal_tiling_honeycomb
Regular tiling of hyperbolic 3-space
such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space
Order-5_cubic_honeycomb
of hyperbolic 3-space, the tetrahedral-square tiling honeycomb is a paracompact uniform honeycomb, constructed from tetrahedron, cuboctahedron and square
Tetrahedral-square tiling honeycomb
Tetrahedral-square_tiling_honeycomb
such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space
Triangular_tiling_honeycomb
Regular and uniform honeycombs in 4-space: Tesseractic honeycomb 24-cell honeycomb Truncated 24-cell honeycomb Snub 24-cell honeycomb 5-cell honeycomb Truncated
16-cell_honeycomb
such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space
Square_tiling_honeycomb
Family of regular honeycombs in geometry
is a method for constructing a uniform polyhedron or plane tiling. The two general forms of the hypercube honeycombs are the regular form with identical
Hypercubic_honeycomb
such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space
Order-4 square tiling honeycomb
Order-4_square_tiling_honeycomb
geometry of hyperbolic 3-space, the cubic-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from cube, triangular tiling, and cuboctahedron
Cubic-triangular tiling honeycomb
Cubic-triangular_tiling_honeycomb
Polytope contained by 7-polytope facets
Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 Klitzing, Richard. "8D uniform polytopes (polyzetta)
Uniform_8-polytope
Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation (or honeycomb) of 4-dimensional Euclidean
24-cell_honeycomb
Space-filling tessellation
edge, and vertex-transitive. It is one of 28 uniform honeycombs. John Horton Conway calls this honeycomb a truncated octahedrille in his Architectonic
Bitruncated_cubic_honeycomb
Convex uniform honeycomb Cubic honeycomb Truncated cubic honeycomb Bitruncated cubic honeycomb Cantellated cubic honeycomb Cantitruncated cubic honeycomb Rectified
List_of_mathematical_shapes
cubic honeycomb. It is vertex-transitive with 6 truncated tetrahedra and 2 tetrahedra around each vertex. It is one of the 28 convex uniform honeycombs. The
Quarter_cubic_honeycomb
belong to 4 cells. The tetrahedral disphenoid honeycomb is the dual of the uniform bitruncated cubic honeycomb. Its vertices form the A* 3 / D* 3 lattice
Tetragonal disphenoid honeycomb
Tetragonal_disphenoid_honeycomb
Regular paracompact honeycomb
tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact. It is one of 15 uniform paracompact honeycombs in the
Hexagonal_tiling_honeycomb
hyperbolic 3-space, the tetrahedral-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling, tetrahedron, and
Tetrahedral-triangular tiling honeycomb
Tetrahedral-triangular_tiling_honeycomb
such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space
Order-4_octahedral_honeycomb
Five-dimensional geometric shape
Theory of Uniform Polytopes and Honeycombs, University of Toronto Non-convex uniform polytopes: 1966: Johnson describes two non-convex uniform antiprisms
Uniform_5-polytope
the geometry of hyperbolic 3-space, the cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from cube, octahedron, and cuboctahedron cells
Cubic-octahedral_honeycomb
such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space
Order-6_tetrahedral_honeycomb
Regular geometrical object in hyperbolic space
such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space
Order-6 dodecahedral honeycomb
Order-6_dodecahedral_honeycomb
Concept in euclidean geometry
and uniform honeycombs in 4-space: 16-cell honeycomb 24-cell honeycomb 5-cell honeycomb Truncated 5-cell honeycomb Omnitruncated 5-cell honeycomb Baake
Tesseractic_honeycomb
hyperbolic 3-space, the octahedron-hexagonal tiling honeycomb is a paracompact uniform honeycomb, constructed from octahedron, hexagonal tiling, and trihexagonal
Octahedral-hexagonal tiling honeycomb
Octahedral-hexagonal_tiling_honeycomb
Uniform 6-dimensional polytope
{C}}_{5}} There are 35 uniform honeycombs, including: Regular hypercube honeycomb of Euclidean 5-space, the 5-cube honeycomb, with symbols {4,33,4},
Uniform_6-polytope
Isogonal polytope with uniform facets
uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs (2-dimensional tilings and higher dimensional honeycombs)
Uniform_polytope
Regular tiling of a two-dimensional space
Hexagonal prismatic honeycomb Tilings of regular polygons List of uniform tilings List of regular polytopes Hexagonal tiling honeycomb Hex map board game
Hexagonal_tiling
In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,32,2}
2_22_honeycomb
hyperbolic 3-space, the hexagonal tiling-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling, hexagonal tiling
Hexagonal tiling-triangular tiling honeycomb
Hexagonal_tiling-triangular_tiling_honeycomb
Regular tiling of hyperbolic 3-space
such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space
Order-5 dodecahedral honeycomb
Order-5_dodecahedral_honeycomb
Geometric figure
geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells
5-cell_honeycomb
geometry of hyperbolic 3-space, the cubic-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, cube, and cuboctahedron
Cubic-icosahedral_honeycomb
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
Convex uniform honeycomb Cubic honeycomb Truncated cubic honeycomb Bitruncated cubic honeycomb Cantellated cubic honeycomb Cantitruncated cubic honeycomb Rectified
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Prism with an 8-sided base
projectors. It is an element of three uniform honeycombs: It is also an element of two four-dimensional uniform 4-polytopes: Weisstein, Eric W. "Octagonal
Octagonal_prism
geometry of hyperbolic 3-space, the tetrahedron-octahedron honeycomb is a compact uniform honeycomb, constructed from octahedron and tetrahedron cells, in
Hyperbolic tetrahedral-octahedral honeycomb
Hyperbolic_tetrahedral-octahedral_honeycomb
heptagonal tiling vertex arrangement. It has a second construction as a uniform honeycomb, Schläfli symbol {3,71,1}, Coxeter diagram, , with alternating types
Order-7-3 triangular honeycomb
Order-7-3_triangular_honeycomb
Honeycomb made from unique polyhedrons
geometry of hyperbolic 3-space, the dodecahedral-icosahedral honeycomb is a uniform honeycomb, constructed from dodecahedron, icosahedron, and icosidodecahedron
Dodecahedral-icosahedral honeycomb
Dodecahedral-icosahedral_honeycomb
triangular tiling vertex arrangement. It has a second construction as a uniform honeycomb, Schläfli symbol {3,(3,4,3)}, Coxeter diagram, , with alternating
Order-7_tetrahedral_honeycomb
geometry of hyperbolic 3-space, the tetrahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, tetrahedron, and octahedron
Tetrahedral-icosahedral honeycomb
Tetrahedral-icosahedral_honeycomb
geometry an omnitruncated simplicial honeycomb or omnitruncated n-simplex honeycomb is an n-dimensional uniform tessellation, based on the symmetry of
Omnitruncated simplicial honeycomb
Omnitruncated_simplicial_honeycomb
generate cyclotruncated 5-cell honeycomb divisions on each hyperplane. This honeycomb is one of 12 unique uniform honeycombs constructed by the A ~ 5 {\displaystyle
Cyclotruncated 5-simplex honeycomb
Cyclotruncated_5-simplex_honeycomb
Type of geometric object
Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 Klitzing, Richard. "9D uniform polytopes (polyyotta)
Uniform_9-polytope
such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space
Alternated hexagonal tiling honeycomb
Alternated_hexagonal_tiling_honeycomb
Tiling of n-dimensional space
In geometry, the simplicial honeycomb (or n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the A ~ n {\displaystyle {\tilde
Simplicial_honeycomb
pentagonal tiling vertex arrangement. It has a second construction as a uniform honeycomb, Schläfli symbol {3,51,1}, Coxeter diagram, , with alternating types
Order-4_icosahedral_honeycomb
Infinite polyhedron with non-planar faces
sponges. Many are directly related to a convex uniform honeycomb, being the polygonal surface of a honeycomb with some of the cells removed. Characteristically
Skew_apeirohedron
Covering by shapes without overlaps or gaps
mapping any vertex onto any other). A uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In three-dimensional
Tessellation
hexagonal tiling vertex arrangement. It has a second construction as a uniform honeycomb, Schläfli symbol {3,81,1}, Coxeter diagram, , with alternating types
Order-8-3 triangular honeycomb
Order-8-3_triangular_honeycomb
Class of 4-dimensional polytopes
dissertation The Theory of Uniform Polytopes and Honeycombs under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and
Uniform_4-polytope
(Schläfli–Hess)) Tessellation Tilings of regular polygons Convex uniform honeycomb Regular map (graph theory) (up to identity and idempotency) In a classification
List_of_regular_polytopes
Prism with a 6-sided base
forming the hexagonal prismatic honeycomb. The hexagonal prism also exists as cells of four prismatic uniform convex honeycombs in 3 dimensions: It also exists
Hexagonal_prism
Isogonal polytope with regular facets
Tetrahedron-icosahedron honeycomb, Paracompact uniform honeycombs, 3D honeycombs, which include uniform tilings as cells: Rectified order-6 tetrahedral honeycomb, Rectified
Semiregular_polytope
geometry of hyperbolic 3-space, the cubic-square tiling honeycomb is a paracompact uniform honeycomb, constructed from cube and square tiling cells, in a
Cubic-square_tiling_honeycomb
Type of geometrical object
and uniform tessellations in 9-space: Regular and uniform tessellations include: Regular 9-hypercubic honeycomb, with symbols {4,37,4}, Uniform alternated
Uniform_10-polytope
Compact uniform honeycomb
the geometry of hyperbolic 3-space, the tetrahedron-cube honeycomb is a compact uniform honeycomb, constructed from cube, tetrahedron, and cuboctahedron
Tetrahedral-cubic_honeycomb
the birectified 16-cell honeycomb (or runcic tesseractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. There
Birectified_16-cell_honeycomb
Four-dimensional geometric object with flat sides
Topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space; similarly the 3D cube is
4-polytope
Space-filling tessellation
In geometry, the trigonal trapezohedral honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. Cells are identical trigonal
Trigonal trapezohedral honeycomb
Trigonal_trapezohedral_honeycomb
below sections. Convex uniform honeycomb – The 28 uniform 3-dimensional tessellations, a parallel construction to the convex uniform Euclidean plane tilings
List of Euclidean uniform tilings
List_of_Euclidean_uniform_tilings
Tiling of five-dimensional space
Regular and uniform honeycombs in 5-space: 5-demicubic honeycomb 5-simplex honeycomb Truncated 5-simplex honeycomb Omnitruncated 5-simplex honeycomb de Bruijn
5-cubic_honeycomb
Geometric figure
four-dimensional Euclidean geometry, the omnitruncated tesseractic honeycomb is a uniform space-filling honeycomb. It has omnitruncated tesseract, truncated cuboctahedral
Omnitruncated tesseractic honeycomb
Omnitruncated_tesseractic_honeycomb
Type of uniform tessellation
In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the
5_21_honeycomb
Tessellation in Euclidean geometry
quarter 7-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 7-demicubic honeycomb, and a quarter
Quarter_7-cubic_honeycomb
In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,33,1} and is composed of 321 and 7-simplex facets
3_31_honeycomb
Regular space-filling tessellation
hyperbolic 3-space, the order-6-4 square honeycomb (or 4,6,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,6,4}. All
Order-6-4_square_honeycomb
cantitruncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a cantitruncation of the regular 24-cell honeycomb, containing truncated
Cantitruncated 24-cell honeycomb
Cantitruncated_24-cell_honeycomb
Prism with a 3-sided base
prismatic honeycomb, triangular-hexagonal prismatic honeycomb, truncated hexagonal prismatic honeycomb, rhombitriangular-hexagonal prismatic honeycomb, snub
Triangular_prism
Five dimensional space-filling tessellation
5-simplex. ∪ ∪ ∪ ∪ ∪ = dual of This honeycomb is one of 12 unique uniform honeycombs constructed by the A ~ 5 {\displaystyle {\tilde {A}}_{5}} Coxeter
Omnitruncated 5-simplex honeycomb
Omnitruncated_5-simplex_honeycomb
hexagonal tiling vertex arrangement. It has a second construction as a uniform honeycomb, Schläfli symbol {3,61,1}, Coxeter diagram, , with alternating types
Order-6-4 triangular honeycomb
Order-6-4_triangular_honeycomb
Topics referred to by the same term
Tetrahedral-octahedral honeycomb - Uniform honeycomb with regular tetrahedral and octahedral cells Tetragonal disphenoid honeycomb - Uniform dual honeycomb with tetragonal
Tetrahedral_honeycomb
triangular tiling vertex arrangement. It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,4,3)}, Coxeter diagram, , with alternating
Order-3-7_hexagonal_honeycomb
triangular tiling vertex arrangement. It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,4,3)}, Coxeter diagram, , with alternating
Order-7 dodecahedral honeycomb
Order-7_dodecahedral_honeycomb
Geometric object
Zeitschrift, Springer, Berlin, 1940 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 H.S.M. Coxeter:
Uniform_k_21_polytope
Regular space-filling tessellation with Schläfli symbol (7,3,7)
triangular tiling vertex arrangement. It has a second construction as a uniform honeycomb, Schläfli symbol {8,(3,4,3)}, Coxeter diagram, , with alternating
Order-3-7 heptagonal honeycomb
Order-3-7_heptagonal_honeycomb
This operation is dual-symmetric for regular uniform 4-polytopes and 3-space convex uniform honeycombs. For a regular {p,q,r} 4-polytope, the original
Runcination
runcinated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a runcination of the regular 24-cell honeycomb, containing runcinated
Runcinated_24-cell_honeycomb
511 permutations of uniform tessellations, 271 with unique symmetry and 270 with unique geometry. The expanded 8-cubic honeycomb is geometrically identical
8-cubic_honeycomb
hyperbolic 3-space, the order-5-4 square honeycomb (or 4,5,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,5,4}. All
Order-5-4_square_honeycomb
Uniform Euclidean 3D tessellations and their duals
defines architectonic and catoptric tessellations as the uniform tessellations (or honeycombs) of Euclidean 3-space with prime space groups and their duals
Architectonic and catoptric tessellation
Architectonic_and_catoptric_tessellation
5-simplex. ∪ ∪ ∪ ∪ ∪ = dual of This honeycomb is one of 12 unique uniform honeycombs constructed by the A ~ 5 {\displaystyle {\tilde {A}}_{5}} Coxeter
5-simplex_honeycomb
Natural number
hyperbolic Coxeter groups, or 4-prisms, of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams
5
hyperbolic 3-space, the order-3-6 heptagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a heptagonal
Order-3-6 heptagonal honeycomb
Order-3-6_heptagonal_honeycomb
truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract
Truncated_24-cell_honeycomb
the bitruncated 16-cell honeycomb (or runcicantic tesseractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. There
Bitruncated_16-cell_honeycomb
hyperbolic 3-space, the order-3-5 heptagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a heptagonal
Order-3-5 heptagonal honeycomb
Order-3-5_heptagonal_honeycomb
Regular tiling of the plane
using triangular tiling) List of uniform tilings Simplectic honeycomb Tilings of regular polygons Triangular tiling honeycomb Tilings and patterns, p.102-107
Triangular_tiling
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