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diagonals of a cube. The truncated octahedron tile a honeycomb known as a bitruncated cubic honeycomb. More generally, every permutohedron and parallelohedron...

dual. Bitruncated 5-cell (Norman W. Johnson) 10-cell as a cell-transitive 4-polytope Bitruncated pentachoron Bitruncated pentatope Bitruncated 4-simplex...

length √(2+√2). Bitruncated 24-cell (Norman W. Johnson) 48-cell as a cell-transitive 4-polytope Bitruncated icositetrachoron Bitruncated polyoctahedron...

In geometry, the truncated icosahedron is a polyhedron that can be constructed by truncating all of the regular icosahedron's vertices. Intuitively, it...

tesseract, is third in a sequence of truncated hypercubes: The bitruncated tesseract, bitruncated 16-cell, or tesseractihexadecachoron is constructed by a bitruncation...

The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently,...

original edge. The bitruncated 5-orthoplex can tessellate space in the tritruncated 5-cubic honeycomb. Bitruncated pentacross Bitruncated triacontaditeron...

around it. Bitruncated 120-cell / Bitruncated 600-cell (Norman W. Johnson) Bitruncated hecatonicosachoron / Bitruncated hexacosichoron / Bitruncated polydodecahedron...

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges...

calls this honeycomb a truncated cubille, and its dual pyramidille. The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean...

truncated hypercubes: Bitruncated penteract (Acronym: bittin) (Jonathan Bowers) The bitruncated 5-cube may be constructed by bitruncating the vertices of the...

6-orthoplex and bitruncated 6-cube respectively. Bitruncated hexateron (Acronym: bittix) (Jonathan Bowers) The vertices of the bitruncated 5-simplex can...

of the truncated 7-orthoplex. Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers) The vertices of the bitruncated 6-simplex can be most simply positioned...

proposed a foam called the Kelvin structure. His foam is based on the bitruncated cubic honeycomb, a convex uniform honeycomb formed by the truncated octahedron...

3-polytopes), a bitruncated form is the truncated dual. For example, a bitruncated cube is a truncated octahedron. For a regular 4-polytope, a bitruncated form is...

tetracomb Great prismatodispentachoric tetracomb The bitruncated 4-simplex honeycomb or bitruncated 5-cell honeycomb is a space-filling tessellation honeycomb...

hexagonal pyramid vertex figure. The bitruncated order-6 tetrahedral honeycomb is equivalent to the bitruncated hexagonal tiling honeycomb. The cantellated...

truncated hypercubes: Bitruncated hexeract (Acronym: botox) (Jonathan Bowers) The Cartesian coordinates of the vertices of a bitruncated 6-cube having edge...

with truncated pentagon and square faces: The bitruncated order-4 dodecahedral honeycomb, or bitruncated order-5 cubic honeycomb, , has truncated octahedron...

truncated 9-orthoplex. Bitruncated enneazetton (Acronym: batene) (Jonathan Bowers) The Cartesian coordinates of the vertices of the bitruncated 8-simplex can be...

Tetrahemihexacron Convex uniform honeycomb Cubic honeycomb Truncated cubic honeycomb Bitruncated cubic honeycomb Cantellated cubic honeycomb Cantitruncated cubic honeycomb...

coordinate permutations of (±2,±2,±2,±2,±2,±1,0,0) The bitruncated 8-cube is sixth in a sequence of bitruncated hypercubes: Tritruncated octeract (acronym: tato)...

lattice is the rectified 122 polytope, and the Voronoi tessellation is a bitruncated 222 honeycomb. It is constructed by 3 copies of the E6 lattice vertices...

3] Coxeter group family, including this regular form as well as the bitruncated form, t1,2{3,5,3}, , also called truncated dodecahedral honeycomb, each...

sign (4) and coordinate (30) permutations of (±2,±1,0,0,0,0) Bitruncated hexacross Bitruncated hexacontatetrapeton (Acronym: botag) (Jonathan Bowers) These...

sequence of truncated hypercubes: Bitruncated hepteract (Jonathan Bowers) Cartesian coordinates for the vertices of a bitruncated 7-cube, centered at the origin...

[5,3,5] Coxeter group family, including this regular form. Also the bitruncated form, t1,2{5,3,5}, , of this honeycomb has all truncated icosahedron...

of the truncated 8-orthoplex. Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers) The vertices of the bitruncated 7-simplex can be most simply positioned...

24-cell honeycomb, constructed by truncated tesseract and bitruncated 24-cell cells. Bitruncated icositetrachoric tetracomb/honeycomb Small tetracontaoctachoric...

t{∞,3} with apeirogonal and triangle faces: The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t1,2{6,3,3}, has...

5-cell honeycomb, with Coxeter graph , with 5-cell, truncated 5-cell, and bitruncated 5-cell facets. In 5-dimensions it is called a cyclotruncated 5-simplex...

5-cells in dual configuration is the disphenoidal 30-cell, dual of the bitruncated 5-cell. This configuration matrix represents the 5-cell. The rows and...

the cantitruncated order-4 square tiling honeycomb, tr{4,4,4}, . The bitruncated square tiling honeycomb, 2t{4,4,3}, has truncated cube and truncated...

[[3]] = [6] Tetrahedron centered 5 *runcinated 5-cell t0,3{3,3,3} 6 *bitruncated 5-cell decachoron 2t{3,3,3} 9 *omnitruncated 5-cell t0,1,2,3{3,3,3}...

In four-dimensional Euclidean geometry, the bitruncated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space...

h3{4,3,3} = = r{4,3,3} 2r{3,31,1} 4 runcicantic tesseract bitruncated 16-cell (Same as bitruncated tesseract) = = h2,3{4,3,3} = = 2t{4,3,3} 2t{3,31,1}...

hexagonal tiling facets, with a pentagonal pyramid vertex figure. The bitruncated order-5 hexagonal tiling honeycomb, t1,2{6,3,5}, has hexagonal tiling...

(±2,±1,0,0,0,0,0,0) Bitruncated octacross (acronym: batek) (Jonthan Bowers) Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered...

symmetry [[(p,r,q,r)]]. {2p,4|r} is represented by the {2p} faces of the bitruncated {r,p,r} uniform 4-polytope, and {4,2p|r} is represented by square faces...

The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean...

hexagonal tiling facets, with a hexagonal pyramid vertex figure. The bitruncated order-6 hexagonal tiling honeycomb is a lower symmetry construction of...

honeycomb Rhombic dodecahedral honeycomb Elongated dodecahedral honeycomb Bitruncated cubic honeycomb or truncated octahedra Other known examples of space-filling...

Coxeter group. Bitruncated heptacross Bitruncated hecatonicosaoctaexon (Jonathan Bowers) Cartesian coordinates for the vertices of a bitruncated 7-orthoplex...

hexagonal pyramid vertex figure. The bitruncated order-6 dodecahedral honeycomb is the same as the bitruncated order-5 hexagonal tiling honeycomb. The...

dodecahedron. Another is constructed with 2 antipodal vertices removed. The bitruncated and runcinated forms (5 and 6) contain the faces of two regular skew...

order-4 apeirogonal tiling, t{∞,4}, with apeirogonal and square faces: The bitruncated order-4 hexagonal tiling honeycomb, t1,2{6,3,4}, has truncated octahedron...

5} bitruncated 5-cell t1,2{3,3,3} bitruncated 16-cell 2t{3,3,4} bitruncated tesseract 2t{4,3,3} cantitruncated demitesseract 2t{3,31,1} bitruncated 24-cell...

t0,2{3,3,6} or rr{3,3,6} (1) (3.4.3.4) - (2) (4.4.6) (2) (3.6.3.6) 9 bitruncated hexagonal (tehexah) t1,2{6,3,3} or 2t{6,3,3} (2) (3.6.6) - - (2) (6.6...

The tiling of space generated by its translations has been called the bitruncated cubic honeycomb. The lengths of the segments within each zone can be...

truncated 24-cell snub 24-cell rectified 24-cell cantellated 24-cell bitruncated 24-cell cantitruncated 24-cell runcinated 24-cell runcitruncated 24-cell...

octahedral cells at the truncated vertices. The bitruncated order-5 cubic honeycomb is the same as the bitruncated order-4 dodecahedral honeycomb. The cantellated...

respectively, are the apeirogon, the regular hexagonal tiling, and the bitruncated cubic honeycomb. The dual tessellations contain all simplex facets, although...

2} 2.∞.2.∞ Birectified (dual) ∞ | 2 2 {2,∞} 2∞ Apeirogonal hosohedron Bitruncated 2 ∞ | 2 t{2,∞} 4.4.∞ Apeirogonal prism Cantellated ∞ 2 | 2 rr{∞,2} Omnitruncated...

truncated square tiling facets, with a square pyramid vertex figure. The bitruncated order-4 square tiling honeycomb, t1,2{4,4,4}, has truncated square tiling...

the disphenoid tetrahedral honeycomb, dual honeycomb of the uniform bitruncated cubic honeycomb: It is also the body centered cubic, the union of two...

runcinated 120-cell (also runcinated 600-cell) t0,3{5,3,3} 8 bitruncated 120-cell (also bitruncated 600-cell) t1,2{5,3,3} 9 cantellated 600-cell t0,2{3,3,5}...

3] Coxeter group family, including this regular form as well as the bitruncated form, t1,2{3,6,3}, with all truncated hexagonal tiling facets. The honeycomb...

In four-dimensional Euclidean geometry, the bitruncated 16-cell honeycomb (or runcicantic tesseractic honeycomb) is a uniform space-filling tessellation...

the dual. Bitruncated (2t) (also truncated dual) 2t{p,q} t1,2{p,q} A bitruncation can be seen as the truncation of the dual. A bitruncated cube is a truncated...

possible: in 1887, Lord Kelvin proposed a packing using only one solid, the bitruncated cubic honeycomb with very slightly curved faces. In 1993, Denis Weaire...

two new ones, including the quarter tesseractic honeycomb, = , and the bitruncated tesseractic honeycomb, = . Non-Wythoffian uniform tessellations in 4-space...

heptapeton (sril) (0,0,0,0,1,1,2) 35 210 560 805 525 105 6 Bitruncated 6-simplex bitruncated heptapeton (batal) (0,0,0,0,1,2,2) 14 84 245 385 315 105 7...

with a square pyramid vertex figure. The bitruncated order-4 octahedral honeycomb is the same as the bitruncated square tiling honeycomb. The cantellated...

interior part of the original faces. Example: a truncated octahedron is a bitruncated cube: t{3,4} = 2t{4,3}. A complete bitruncation, called a birectification...

(p q 2) Parent Truncated Rectified Bitruncated Birectified (dual) Cantellated Omnitruncated (Cantitruncated) Snub Wythoff symbol q | p 2 2 q | p 2 | p...

(or honeycomb). It is composed of 5-simplex, truncated 5-simplex, and bitruncated 5-simplex facets in a ratio of 1:1:1. Its vertex figure is an elongated...

Hosohedron Convex uniform honeycomb Cubic honeycomb Truncated cubic honeycomb Bitruncated cubic honeycomb Cantellated cubic honeycomb Cantitruncated cubic honeycomb...

rhombated hecatonicosaoctaexon (Sarz) 10 t1,2{3,3,3,3,3,4} Bitruncated 7-orthoplex Bitruncated hecatonicosaoctaexon (Botaz) 11 t0,3{3,3,3,3,3,4} Runcinated...

These are seen in the vertex figures of bitruncated and tritruncated regular 6-polytopes, like a bitruncated 6-cube and a tritruncated 6-simplex. The...

truncated 24-cell snub 24-cell rectified 24-cell cantellated 24-cell bitruncated 24-cell cantitruncated 24-cell runcinated 24-cell runcitruncated 24-cell...

4} (2) (4.4.4) (6) (4.4.4)   octahedron Cube J16 A3 W2 G28 t1,2δ4 O16 bitruncated cubic (batch) t1,2{4,3,4} 2t{4,3,4} (4) (4.6.6)     (disphenoid) Oblate...

equal volume as a foam in 1887; his solution uses just one solid, the bitruncated cubic honeycomb with very slightly curved faces to meet Plateau's laws...

diacosipentacontahexazetton (srek) 11 t1,2{3,3,3,3,3,3,4} Bitruncated 8-orthoplex Bitruncated diacosipentacontahexazetton (batek) 12 t0,3{3,3,3,3,3,3,4}...

600-cell rectified 600-cell truncated 600-cell cantellated 600-cell bitruncated 600-cell cantitruncated 600-cell runcitruncated 600-cell omnitruncated...

cells. The tetrahedral disphenoid honeycomb is the dual of the uniform bitruncated cubic honeycomb. Its vertices form the A* 3 / D* 3 lattice, which is...

2} 2.∞.2.∞ Birectified (dual) ∞ | 2 2 {2,∞} 2∞ Apeirogonal hosohedron Bitruncated 2 ∞ | 2 t{2,∞} 4.4.∞ Apeirogonal prism Cantellated ∞ 2 | 2 rr{∞,2} Omnitruncated...

tesseract truncated tesseract cantellated tesseract runcinated tesseract bitruncated tesseract cantitruncated tesseract runcitruncated tesseract omnitruncated...

4-honeycomb Schläfli symbol t0,2,3{4,3,3,4} Coxeter-Dynkin diagram 4-face type bitruncated tesseract truncated octahedral prism tesseract runcitruncated 16-cell...

Voronoi tessellation is a tritruncated 5-cubic honeycomb, , containing all bitruncated 5-orthoplex, Voronoi cells. There are three uniform construction symmetries...

conjectured that it was given by a structure combinatorially equivalent to the bitruncated cubic honeycomb, but this conjecture was disproved by the discovery of...

halfway between the center of a 24-cell and the boundary, one obtains a bitruncated cubic honeycomb. The cuboctahedra have shrunk, and the octahedra have...

6-simplex Small rhombated heptapeton (sril) 6 t1,2{3,3,3,3,3} Bitruncated 6-simplex Bitruncated heptapeton (batal) 7 t0,1,2{3,3,3,3,3} Cantitruncated 6-simplex...

The Wigner–Seitz cell of the body-centered cubic lattice is a truncated octahedron. In mathematics, it is known as the bitruncated cubic honeycomb....

5-cube Truncated penteract (tan) 8 t1,2{4,3,3,3} = 2t{4,3,3,3} Bitruncated 5-cube Bitruncated penteract (bittin) 9 t0,2{4,3,3,3} = rr{4,3,3,3} Cantellated...

tesseract, truncated 24-cell, truncated 120-cell, and duoprisms. All bitruncated, cantitruncated, or omnitruncated four-polytopes are simple. Simple polytopes...

(p q 2) Fund. triangles Parent Truncated Rectified Bitruncated Birectified (dual) Cantellated Omnitruncated (Cantitruncated) Snub Wythoff symbol q | p...

Name 5-cell truncated 5-cell rectified 5-cell cantellated 5-cell bitruncated 5-cell cantitruncated 5-cell runcinated 5-cell runcitruncated 5-cell omnitruncated...

{\displaystyle (0,0,0)} for the truncated octahedron, which tiles space as the bitruncated cubic honeycomb. ( 0 , 0 , − 30 s ) {\displaystyle (0,0,-30s)} for the...

r\end{array}}\right\}} r{p,q,r} t1{p,q,r} Rectified tesseract = Bitruncated 2t{p,q,r} t1,2{p,q,r} Bitruncated tesseract Birectified (Rectified dual) { q , p r } {\displaystyle...

honeycomb). The tessellation fills space by 8-simplex, truncated 8-simplex, bitruncated 8-simplex, tritruncated 8-simplex, and quadritruncated 8-simplex facets...

truncated 24-cell snub 24-cell rectified 24-cell cantellated 24-cell bitruncated 24-cell cantitruncated 24-cell runcinated 24-cell runcitruncated 24-cell...

(such as the tesseract and 16-cell, or the 120-cell and 600-cell), then bitruncating, runcinating or omnitruncating either produces the same figure as the...

600-cell rectified 600-cell truncated 600-cell cantellated 600-cell bitruncated 600-cell cantitruncated 600-cell runcitruncated 600-cell omnitruncated...

2} 2.∞.2.∞ Birectified (dual) ∞ | 2 2 {2,∞} 2∞ Apeirogonal hosohedron Bitruncated 2 ∞ | 2 t{2,∞} 4.4.∞ Apeirogonal prism Cantellated ∞ 2 | 2 rr{∞,2} Omnitruncated...

tesseract truncated tesseract cantellated tesseract runcinated tesseract bitruncated tesseract cantitruncated tesseract runcitruncated tesseract omnitruncated...

600-cell rectified 600-cell truncated 600-cell cantellated 600-cell bitruncated 600-cell cantitruncated 600-cell runcitruncated 600-cell omnitruncated...

is the 5-dimensional analog of the regular hexagon, octahedron, and bitruncated 5-cell. This characterization yields simple coordinates for the vertices...

2F4 Bitruncated 24-cell Runcinated 24-cell Omnitruncated 24-cell...

B3 [6] B2 [[4]] = [8] Octahedron centered 7 *runcinated 24-cell t0,3{3,4,3} 8 *bitruncated 24-cell 2t{3,4,3} 9 *omnitruncated 24-cell t0,1,2,3{3,4,3}...

2} 2.∞.2.∞ Birectified (dual) ∞ | 2 2 {2,∞} 2∞ Apeirogonal hosohedron Bitruncated 2 ∞ | 2 t{2,∞} 4.4.∞ Apeirogonal prism Cantellated ∞ 2 | 2 rr{∞,2} Omnitruncated...

7-simplex (saro) (0,0,0,0,0,1,1,2) 44 308 980 1750 1876 1008 168 7 t1,2 Bitruncated 7-simplex (bittoc) (0,0,0,0,0,1,2,2) 588 168 8 t0,3 Runcinated 7-simplex...

truncated 24-cell snub 24-cell rectified 24-cell cantellated 24-cell bitruncated 24-cell cantitruncated 24-cell runcinated 24-cell runcitruncated 24-cell...

polyhedron, {6,4|3}, is similarly related to the hexagonal faces of the bitruncated 5-cell. The runcitruncated 5-cell or prismatorhombated pentachoron is...

runcinated 8-cell (also runcinated 16-cell) = t03{4,3,3} 7 bitruncated 8-cell (also bitruncated 16-cell) = 2t{4,3,3} 8 truncated 16-cell = t{3,3,4} 9 cantitruncated...

figure in uniform polytopes of the form , including the bitruncated 5-orthoplex and bitruncated tesseractic honeycomb. Example 4-dimensional coordinates...

honeycomb). The tessellation fills space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets. These facet types occur...

diacosipentacontahexazetton (srek) 14784 1344 11 t1,2{36,4} Bitruncated 8-orthoplex Bitruncated diacosipentacontahexazetton (batek) 8064 1344 12 t0,3{36...

bipyramid , , J16 A3 W2 G28 t1,2δ4 bc [[4,3,4]] Truncated octahedrille (Bitruncated cubic honeycomb) Tetragonal disphenoid Oblate tetrahedrille Tetragonal...

tesseract truncated tesseract cantellated tesseract runcinated tesseract bitruncated tesseract cantitruncated tesseract runcitruncated tesseract omnitruncated...

cube.) Muoctahedron: {6,4|4}: 4 hexagons about each vertex (related to bitruncated cubic honeycomb, constructed by truncated octahedron with their square...

600-cell rectified 600-cell truncated 600-cell cantellated 600-cell bitruncated 600-cell cantitruncated 600-cell runcitruncated 600-cell omnitruncated...

3,3,3,3,3} Cantellated 10-simplex 4455 495 8 t1,2{3,3,3,3,3,3,3,3,3} Bitruncated 10-simplex 2475 495 9 t0,3{3,3,3,3,3,3,3,3,3} Runcinated 10-simplex 15840...

figures 4.2n.2n, extending into the hyperbolic plane: The 3-dimensional bitruncated cubic honeycomb projected into the plane shows two copies of a truncated...

predefined manner. A tritruncated 5-cubic honeycomb, , contains all bitruncated 5-orthoplex facets and is the Voronoi tessellation of the D5* lattice...

600-cell rectified 600-cell truncated 600-cell cantellated 600-cell bitruncated 600-cell cantitruncated 600-cell runcitruncated 600-cell omnitruncated...

Name 5-cell truncated 5-cell rectified 5-cell cantellated 5-cell bitruncated 5-cell cantitruncated 5-cell runcinated 5-cell runcitruncated 5-cell omnitruncated...

Equilateral triangle 3 hexagons 3 3 A ~ 3 {\displaystyle {\tilde {A}}_{3}} Bitruncated cubic honeycomb Truncated octahedron irr. tetrahedron 4 truncated octahedron...

t{3,3,3,3} Truncated 5-simplex (tix) 5 t1,2{3,3,3,3} or 2t{3,3,3,3} Bitruncated 5-simplex (bittix) 6 t0,2{3,3,3,3} or rr{3,3,3,3} Cantellated 5-simplex...

8-simplex 6 t0,2{3,3,3,3,3,3,3} Cantellated 8-simplex 7 t1,2{3,3,3,3,3,3,3} Bitruncated 8-simplex 8 t0,3{3,3,3,3,3,3,3} Runcinated 8-simplex 9 t1,3{3,3,3,3,3...

polyhedron, {8,4|3}, is similarly related to the octagonal faces of the bitruncated 24-cell. The runcitruncated 24-cell or prismatorhombated icositetrachoron...

7-simplex 6 t0,2{3,3,3,3,3,3} Cantellated 7-simplex 7 t1,2{3,3,3,3,3,3} Bitruncated 7-simplex 8 t0,3{3,3,3,3,3,3} Runcinated 7-simplex 9 t1,3{3,3,3,3,3,3}...

Small rhombated hexacontatetrapeton (srog) 9 t1,2{3,3,3,3,4} Bitruncated 6-orthoplex Bitruncated hexacontatetrapeton (botag) 10 t0,3{3,3,3,3,4} Runcinated...

wax-resist dyeing, sometimes spelled "batek" during the Dutch colonial period Bitruncated 8-orthoplex, a geometric object, also known by the abbreviation "batek"...

3,3,3,3,3,3} Cantellated 9-simplex 2880 360 8 t1,2{3,3,3,3,3,3,3,3} Bitruncated 9-simplex 1620 360 9 t0,3{3,3,3,3,3,3,3,3} Runcinated 9-simplex 8820...

honeycomb). The tessellation fills space by 7-simplex, truncated 7-simplex, bitruncated 7-simplex, and tritruncated 7-simplex facets. These facet types occur...

600-cell rectified 600-cell truncated 600-cell cantellated 600-cell bitruncated 600-cell cantitruncated 600-cell runcitruncated 600-cell omnitruncated...