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Only regular space-filling tessellation of the cube
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells
Cubic_honeycomb
Regular tiling of hyperbolic 3-space
hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With
Order-5_cubic_honeycomb
Tiling of five-dimensional space
geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space. Four 5-cubes meet
5-cubic_honeycomb
Regular tiling of hyperbolic 3-space
constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb. The order-4 dodecahedral honeycomb has the lowest sectional curvature relative to cell
Order-4 dodecahedral honeycomb
Order-4_dodecahedral_honeycomb
Space-filling tessellation
The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently,
Bitruncated_cubic_honeycomb
the quarter 5-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 5-demicubic honeycomb, and a quarter
Quarter_5-cubic_honeycomb
Quasiregular space-filling tesselation
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is
Tetrahedral-octahedral honeycomb
Tetrahedral-octahedral_honeycomb
honeycomb Order-4 dodecahedral honeycomb Order-5 dodecahedral honeycomb Order-5 cubic honeycomb Icosahedral honeycomb Order-3 icosahedral honeycomb Order-4
List_of_mathematical_shapes
Spatial tiling of convex uniform polyhedra
cells. Twenty-eight such honeycombs are known: the familiar cubic honeycomb and 7 truncations thereof; the alternated cubic honeycomb and 4 truncations thereof;
Convex_uniform_honeycomb
The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean
Quarter_cubic_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
Regular tessellation in 4D Euclidean space
tessellation by regular 16-cell honeycomb has Schläfli symbol {3,3,4,3}. Together with the tesseractic honeycomb (or 4-cubic honeycomb) these are the only regular
24-cell_honeycomb
hyperbolic 4-space, the cubic honeycomb honeycomb is one of two paracompact regular space-filling tessellations (or honeycombs). It is called paracompact
Cubic_honeycomb_honeycomb
In the geometry of hyperbolic 3-space, the cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from cube, octahedron, and cuboctahedron
Cubic-octahedral_honeycomb
In geometry, the 8-cubic honeycomb or octeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 8-space. It is analogous
8-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
Isogonal polytope with regular facets
Hyperbolic uniform honeycombs, 3D honeycombs: Alternated order-5 cubic honeycomb, ↔ (a quasiregular polytope) Tetrahedral-octahedral honeycomb, Tetrahedron-icosahedron
Semiregular_polytope
Geometric figure
4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells and
5-cell_honeycomb
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
Tiling of hyperbolic 3-space by uniform polyhedra
order-5 cubic honeycomb. The case p = 2 (with the pseudoicosahedron degenerating to a regular octahedron) degenerates to the Euclidean cubic honeycomb. Another
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
The 6-cubic honeycomb or hexeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 6-space. It is analogous to the
6-cubic_honeycomb
The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 7-space. It is analogous to
7-cubic_honeycomb
Tiling of euclidean or hyperbolic space of three or more dimensions
parallelepiped, copies can fill space, with the cubic honeycomb being special because it is the only regular honeycomb in ordinary (Euclidean) space. Another family
Honeycomb_(geometry)
quarter hypercubic honeycomb (or quarter n-cubic honeycomb) is a dimensional infinite series of honeycombs, based on the hypercube honeycomb. It is given a
Quarter_hypercubic_honeycomb
Tessellation in Euclidean geometry
the 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
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
the quarter 8-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 8-demicubic honeycomb, and a quarter
Quarter_8-cubic_honeycomb
the geometry of hyperbolic 3-space, the cubic-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from cube, triangular tiling
Cubic-triangular tiling honeycomb
Cubic-triangular_tiling_honeycomb
There are four regular star-honeycombs in H4 space, all compact: There is only one flat regular honeycomb of Euclidean 5-space: (previously listed above
List_of_regular_polytopes
Family of regular honeycombs in geometry
vertex in 1:3:3:1 counts. The orthotopic honeycombs are a family topologically equivalent to the cubic honeycombs but with lower symmetry, in which each
Hypercubic_honeycomb
the quarter 6-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 6-demicubic honeycomb, and a quarter
Quarter_6-cubic_honeycomb
Polyhedron with two kinds of faces
cuboctahedron, . In hyperbolic 3-space, one quasiregular honeycomb is the alternated order-5 cubic honeycomb, h{4,3,5}, Coxeter diagrams: = , composed of alternating
Quasiregular_polyhedron
In the geometry of hyperbolic 3-space, the cubic-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, cube, and cuboctahedron
Cubic-icosahedral_honeycomb
12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams: Regular and uniform honeycombs in 5-space: 5-cubic honeycomb
Cyclotruncated 5-simplex honeycomb
Cyclotruncated_5-simplex_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
Concept in euclidean geometry
tiling, {4,4}, of the plane and the cubic honeycomb, {4,3,4}, of 3-space. These are all part of the hypercubic honeycomb family of tessellations of the form
Tesseractic_honeycomb
Uniform 6-dimensional polytope
permutations, and 3 new ones. Coxeter calls the first one a quarter 5-cubic honeycomb, with symbols q{4,33,4}, = . The other two new ones are = , = . There
Uniform_6-polytope
Type of crystal structure
In crystallography, the diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the
Diamond_cubic
Operation in Euclidean geometry
is that self-dual 4-polytope {p,q,p} (and honeycombs) remain cell-transitive after bitruncation. There are 5 such forms corresponding to the five truncated
Bitruncation
Five dimensional space-filling tessellation
5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 5-simplex
Omnitruncated 5-simplex honeycomb
Omnitruncated_5-simplex_honeycomb
In the geometry of hyperbolic 3-space, the cubic-square tiling honeycomb is a paracompact uniform honeycomb, constructed from cube and square tiling cells
Cubic-square_tiling_honeycomb
Geometrical concept
regular Euclidean 4-space tesseractic honeycomb, {4,3,3,4}. It is analogous to the paracompact cubic honeycomb honeycomb, {4,3,4,3}, in 4-dimensional hyperbolic
Tesseractic honeycomb honeycomb
Tesseractic_honeycomb_honeycomb
5-dimensional hypercube
tesseractic honeycomb on a 4-sphere. It is related to the Euclidean 4-space (order-4) tesseractic honeycomb and paracompact hyperbolic honeycomb order-5 tesseractic
5-cube
edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure. A geometric honeycomb is a space-filling of polyhedral or higher-dimensional
Square_tiling_honeycomb
Uniform Euclidean 3D tessellations and their duals
Coxeters names are based on δ4 as a cubic honeycomb, hδ4 as an alternated cubic honeycomb, and qδ4 as a quarter cubic honeycomb. Hestenes, David; Holt, Jeremy
Architectonic and catoptric tessellation
Architectonic_and_catoptric_tessellation
form, and its dual, the order-6 cubic honeycomb. The order-4 hexagonal tiling honeycomb has a related alternated honeycomb, ↔ , with triangular tiling and
Order-4 hexagonal tiling honeycomb
Order-4_hexagonal_tiling_honeycomb
demitesseractic honeycomb, h{4,3,3,4}) it is related to the alternated cubic honeycomb. This honeycomb is one of 20 uniform honeycombs constructed by the D ~ 5 {\displaystyle
16-cell_honeycomb
Family of regular tessellations in geometry
alternated hypercube honeycomb (or demicubic honeycomb) is a dimensional infinite series of honeycombs, based on the hypercube honeycomb with an alternation
Alternated hypercubic honeycomb
Alternated_hypercubic_honeycomb
Geometric figure
Omnitruncated cubic honeycomb Regular and uniform honeycombs in 4-space: Tesseractic honeycomb 16-cell honeycomb 24-cell honeycomb Truncated 24-cell honeycomb Snub
Omnitruncated tesseractic honeycomb
Omnitruncated_tesseractic_honeycomb
Uniform 7-Honeycomb
Euclidean 7-space. It is constructed as an alternation of the regular 7-cubic honeycomb. It is composed of two different types of facets. The 7-cubes become
7-demicubic_honeycomb
Space-filling tessellation
trapezohedral honeycomb. A different orthogonal projection produces the quadrille where the rhombi are distorted into squares. It is dual to the quarter cubic honeycomb
Trigonal trapezohedral honeycomb
Trigonal_trapezohedral_honeycomb
Crystallographic system where the unit cell is in the shape of a cube
corresponding shape is the rhombic dodecahedral honeycomb or hexakis cubic honeycomb. The face-centered cubic lattice (cF) has lattice points on the faces
Cubic_crystal_system
Euclidean 8-space. It is constructed as an alternation of the regular 8-cubic honeycomb. It is composed of two different types of facets. The 8-cubes become
8-demicubic_honeycomb
Mathematical foam of equal-volume bubbles
Kelvin structure. His foam is based on the bitruncated cubic honeycomb, a convex uniform honeycomb formed by the truncated octahedron, a space-filling convex
Weaire–Phelan_structure
quasiregular polychora and honeycombs: The rectified order-4 hexagonal tiling honeycomb, t1{4,4,4}, has square tiling facets, with a cubic vertex figure. It is
Order-4 square tiling honeycomb
Order-4_square_tiling_honeycomb
Cube capped by two square pyramids
the rectified cubic honeycomb. Both honeycombs have a symmetry of [ 4 , 3 , 4 ] {\displaystyle [4,3,4]} . Cross-sections of the honeycomb, through cell
Elongated_square_bipyramid
geometry, runcination is an operation that cuts a regular polytope (or honeycomb) simultaneously along the faces, edges, and vertices, creating new facets
Runcination
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
2-dimensions, the honeycomb represents the trihexagonal tiling, with Coxeter graph . In 3-dimensions it represents the quarter cubic honeycomb, with Coxeter
Cyclotruncated simplicial honeycomb
Cyclotruncated_simplicial_honeycomb
Infinite regular skew polyhedron
Mucube: {4,6|4}: 6 squares about each vertex (related to cubic honeycomb, constructed by cubic cells, removing two opposite faces from each, and linking
Regular_skew_apeirohedron
Polytope whose facets are all simplices
convex uniform honeycombs: Disphenoid tetrahedral honeycomb Dual of cantitruncated cubic honeycomb Dual of omnitruncated cubic honeycomb Dual of cantitruncated
Simplicial_polytope
Polyhedron with 8 rhombic and 4 hexagonal faces
direction, the honeycomb looks like a square tiling with the rhombi projected into squares. The expanded dodecahedra can be distorted into cubic volumes, with
Elongated_dodecahedron
on each original edge. The bitruncated 5-orthoplex can tessellate space in the tritruncated 5-cubic honeycomb. Bitruncated pentacross Bitruncated triacontaditeron
Truncated_5-orthoplexes
Dense arrangement of congruent spheres in an infinite, regular arrangement
the duals of these honeycombs are produced: the rhombic dodecahedral honeycomb for FCC, and the trapezo-rhombic dodecahedral honeycomb for HCP. Spherical
Close-packing of equal spheres
Close-packing_of_equal_spheres
Shape made by slicing off a corner of a polytope
figure is a regular tetrahedron {3,3}. Also the vertex figure for a cubic honeycomb {4,3,4}, the vertex figure is a regular octahedron {3,4}. Since the
Vertex_figure
Generalization of a polytope in real space
apeirotopes: = . The first case is the R 3 {\displaystyle \mathbb {R} ^{3}} cubic honeycomb. There are 15 regular complex apeirotopes in C 4 {\displaystyle \mathbb
Complex_polytope
Four-dimensional analogue of the cube
4-polytopes. The tesseract is also called an 8-cell, C8, (regular) octachoron, or cubic prism. It is the four-dimensional measure polytope, taken as a unit for
Tesseract
Polyhedron with 2 faces
2} {5,2} ... {∞,2} Coxeter diagram ... Faces 2 {2} 2 {3} 2 {4} 2 {5} ... 2 {∞} Edges and vertices 2 3 4 5 ... ∞ Vertex config. 2.2 3.3 4.4 5.5 ... ∞.∞
Dihedron
Planar surface that forms part of the boundary of a solid object
2-face. The facets of a 4D polytope or 3-honeycomb are its 3-faces or cells. The facets of a 5D polytope or 4-honeycomb are its 4-faces. In related terminology
Face_(geometry)
Regular tiling of a two-dimensional space
for making honeycomb (or rather, soap bubbles) was investigated by Lord Kelvin, who believed that the Kelvin structure (or body-centered cubic lattice)
Hexagonal_tiling
Polytope contained by 7-polytope facets
a quarter 7-cubic honeycomb. , , , , , , , , , E ~ 7 {\displaystyle {\tilde {E}}_{7}} 143 uniquely ringed forms, including: 133 honeycomb: {3,33,3}, 331
Uniform_8-polytope
Four-dimensional geometrical object
each face. This honeycomb's Coxeter diagram is . Unlike the analogous honeycomb in three dimensions, the bitruncated cubic honeycomb which has three different
Runcinated_5-cell
Seven-dimensional geometric object
first one a quarter 6-cubic honeycomb. = = = = = = E ~ 6 {\displaystyle {\tilde {E}}_{6}} : [32,2,2], 39 forms Uniform 222 honeycomb: represented by symbols
Uniform_7-polytope
Four-dimensional shape
as well, it exists as the cells of the dual to the rectified 24-cell honeycomb. Triangular bipyramid - A lower dimensional analogy of the tetrahedral
Tetrahedral_bipyramid
Infinite polyhedron with non-planar faces
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
body centered cubic, the union of two 6-cube honeycombs in dual positions. ∪ ∪ ∪ = ∪ . The kissing number of the D6* lattice is 12 (2n for n≥5). and its Voronoi
6-demicubic_honeycomb
Catalan solid with 12 faces
all rhombic dodecahedra. It is dual to the tetroctahedrille or half cubic honeycomb, and it is described by two Coxeter diagrams: and . With D3d symmetry
Rhombic_dodecahedron
Removal of alternate vertices
Examples: Honeycombs An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb. An alternated hexagonal prismatic honeycomb is the gyrated
Alternation_(geometry)
Operation in Euclidean geometry
Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 John H. Conway, Heidi
Rectification_(geometry)
uniform honeycombs in 8-space: 8-cubic honeycomb 8-demicubic honeycomb 8-simplex honeycomb Truncated 8-simplex honeycomb 521 honeycomb 251 honeycomb 152 honeycomb
Omnitruncated 8-simplex honeycomb
Omnitruncated_8-simplex_honeycomb
Tessalating shape in four dimensional space
uniform honeycombs in 4-space: Tesseractic honeycomb Demitesseractic honeycomb 24-cell honeycomb Truncated 24-cell honeycomb Snub 24-cell honeycomb 5-cell
Rectified tesseractic honeycomb
Rectified_tesseractic_honeycomb
Heat dissipating material
a fiberglass honeycomb matrix. Density: 32 pounds per cubic foot (0.51 g/cm3) Post-ablation char-layer composition: 6.7 pounds per cubic foot (0.107 g/cm3)
AVCOAT
Prism with an 8-sided base
flicker-free images in movie projectors. It is an element of three uniform honeycombs: It is also an element of two four-dimensional uniform 4-polytopes: Weisstein
Octagonal_prism
uniform honeycombs in 6-space: 6-cubic honeycomb 6-demicubic honeycomb Truncated 6-simplex honeycomb Omnitruncated 6-simplex honeycomb 222 honeycomb "The
6-simplex_honeycomb
Prism with a 3-sided base
are gyroelongated alternated cubic honeycomb (with regular octahedra and tetrahedra), elongated alternated cubic honeycomb (with regular octahedra and
Triangular_prism
Space-filling polyhedron with 8 faces
polyhedron, and matches the geometry of the gyroelongated triangular prismatic honeycomb if the elongated gyrobifastigium is dissected back into cubes and triangular
Elongated_gyrobifastigium
Type of geometric object
8-cubic honeycomb, representing as q{4,36,4}, or qδ9. E ~ 8 {\displaystyle {\tilde {E}}_{8}} 511 forms 521 honeycomb: 251 honeycomb: 152 honeycomb: There
Uniform_9-polytope
honeycombs in hyperbolic space List of regular polytopes Tetrahedral-octahedral honeycomb - similar Euclidean honeycomb, Tetrahedral-cubic honeycomb Coxeter
Hyperbolic tetrahedral-octahedral honeycomb
Hyperbolic_tetrahedral-octahedral_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 related
4-polytope
Notation for polytopes and tessellations
regular tessellation of Euclidean 3-space: the cubic honeycomb, with a Schläfli symbol of {4,3,4}, made of cubic cells and 4 cubes around each edge. There
Schläfli_symbol
Solid with six equal square faces
centrally symmetric polygons. An example of a honeycomb with a cubic type only, called a cell, is a cubic honeycomb that consists of four cubes around its edges
Cube
Four-dimensional analog of the icosahedron
a sequence of 4-polytope and honeycombs with icosahedron vertex figures: The regular complex polygons 3{5}3, and 5{3}5, , in C 2 {\displaystyle \mathbb
600-cell
Archimedean solid with 8 faces
(brown) A solved Tetraminx Quarter cubic honeycomb – Fills space using truncated tetrahedra and smaller tetrahedra Truncated 5-cell – Similar uniform polytope
Truncated_tetrahedron
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
Polyhedron with 12 faces
of five Federov polyhedra or parallelohedra, generated to create its honeycomb. Armand Spitz used a dodecahedron as the "globe" equivalent for his Digital
Dodecahedron
and uniform honeycombs in 6-space: 6-cubic honeycomb 6-demicubic honeycomb 6-simplex honeycomb Truncated 6-simplex honeycomb 222 honeycomb * Weisstein
Omnitruncated 6-simplex honeycomb
Omnitruncated_6-simplex_honeycomb
and uniform honeycombs in 7-space: 7-cubic honeycomb 7-demicubic honeycomb 7-simplex honeycomb Truncated 7-simplex honeycomb 331 honeycomb Weisstein, Eric
Omnitruncated 7-simplex honeycomb
Omnitruncated_7-simplex_honeycomb
Math theorem about sphere packing
filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density
Kepler_conjecture
geometry, the truncated 16-cell honeycomb (or cantic tesseractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It
Truncated_16-cell_honeycomb
Convex polyhedron whose faces are almost regular polygons
coplanar misses from sections of the cubic honeycomb (alternatively convex polycubes) or alternated cubic honeycomb, ignoring any obscured faces. Examples:
Near-miss_Johnson_solid
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