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Four-dimensional geometrical object
as a semiregular polytope. Runcinated 5-cell (Norman Johnson) Runcinated pentachoron Runcinated 4-simplex Expanded 5-cell/4-simplex/pentachoron Small
Runcinated_5-cell
runcinated 5-cell. The disphenoidal 30-cell is the dual of the bitruncated 5-cell. It is a 4-dimensional polytope (or polychoron) derived from the 5-cell
Truncated_5-cell
six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex. There are 4
Runcinated_5-simplexes
five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex. There are
Runcinated_5-orthoplexes
5-cube, Runcinated 5-cube, Stericated 5-cube 5-orthoplex, Rectified 5-orthoplex, Truncated 5-orthoplex, Cantellated 5-orthoplex, Runcinated 5-orthoplex
List_of_mathematical_shapes
Uniform polychoron
four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron
Rectified_5-cell
geometry, the runcinated 16-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a runcination of the regular 16-cell honeycomb, containing
Runcinated_16-cell_honeycomb
prisms. Runcinated 120-cell / Runcinated 600-cell (Norman W. Johnson) Runcinated hecatonicosachoron / Runcinated dodecacontachoron / Runcinated hexacosichoron
Runcinated_120-cells
five-dimensional 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
semiregular polytope. Runcinated 24-cell (Norman W. Johnson) Runcinated icositetrachoron Runcinated polyoctahedron Small prismatotetracontoctachoron (spic)
Runcinated_24-cells
cantellated 5-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation, up to edge-planing) of the regular 5-cell. The cantellated 5-cell
Cantellated_5-cell
Geometric figure
arrangement of the 5-cell honeycomb is called the A4 lattice, or 4-simplex lattice. The 20 vertices of its vertex figure, the runcinated 5-cell represent the
5-cell_honeycomb
regular 24-cell honeycomb, containing runcinated 24-cell, 24-cell, octahedral prism, and 3-3 duoprism cells. Runcinated icositetrachoric tetracomb/honeycomb
Runcinated_24-cell_honeycomb
In four-dimensional geometry, a runcinated tesseract (or runcinated 16-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation)
Runcinated_tesseracts
6-simplex, Runcinated 6-simplex, Stericated 6-simplex, Pentellated 6-simplex 6-demicube, Truncated 6-demicube, Cantellated 6-demicube, Runcinated 6-demicube
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
triangular prism cells of the runcinated 5-cell. The cupola can be seen in A2 and A3 Coxeter plane orthogonal projection of the runcinated 5-cell: Tetrahedral
Tetrahedral_cupola
Regular tiling of hyperbolic 3-space
4}: The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb , has cube, dodecahedron, and pentagonal prism cells, with an
Order-5_cubic_honeycomb
Class of 4-dimensional polytopes
that all cells, faces, edges, and vertices are congruent: 6 regular convex 4-polytopes: 5-cell {3,3,3}, 8-cell {4,3,3}, 16-cell {3,3,4}, 24-cell {3,4,3}
Uniform_4-polytope
Four-dimensional analog of the icosahedron
cells (5-cell, 16-cell, 600-cell) have twisted cell rings, and the others (whose cells have opposing faces) do not. Separately, he categorized cell rings
600-cell
Polyhedron with non-planar faces
uniform 4-polytope, and {4,2p|r} is represented by square faces of the runcinated {r,p,r}. {4,4|n} produces a n-n duoprism, and specifically {4,4|4} fits
Regular_skew_polyhedron
rectified 5-cells cells on opposite sides, and 20 vertices of a runcinated 5-cell passing through the center. When combined with the 10 vertices of the 5-orthoplex
Rectified_5-orthoplexes
Five-dimensional geometric shape
Snub 24-cell honeycomb, with symbols s{3,4,3,3}, and constructed by four snub 24-cell, one 16-cell, and five 5-cells at each vertex. 16-cell honeycomb
Uniform_5-polytope
pentagonal prism facets, with a mirrored sphenoid vertex figure. The runcinated order-5 hexagonal tiling honeycomb, t0,3{6,3,5}, has dodecahedron, hexagonal
Order-5 hexagonal tiling honeycomb
Order-5_hexagonal_tiling_honeycomb
Regular tiling of hyperbolic 3-space
prism cells, with a mirrored sphenoid vertex figure. The runcinated icosahedral honeycomb, t0,3{3,5,3}, , has icosahedron and triangular prism cells, with
Icosahedral_honeycomb
moving the cells of the regular form away from the center, and filling in new faces in the gaps for each opened vertex and edge. Runcinated 4-polytopes/honeycombs
Runcination
Regular paracompact honeycomb
truncated trihexagonal tiling, and triangular prism cells, with a mirrored sphenoid vertex figure. The runcinated hexagonal tiling honeycomb, t0,3{6,3,3}, has
Hexagonal_tiling_honeycomb
5-cube. Steric 5-cubes have half the vertices of stericated 5-cubes. Steric penteract, runcinated demipenteract Small prismated hemipenteract (siphin) (Jonathan
Steric_5-cubes
Regular object in four dimensional geometry
vertices. Like other four-dimensional regular polytope, 5-cell, the 24-cell is self-dual. The 24-cell and the tesseract are the only convex regular 4-polytopes
24-cell
hyperplane is a runcinated 5-cell. This cross-section divides the stericated hexateron into two pentachoral hypercupolas consisting of 6 5-cells, 15 tetrahedral
Stericated_5-simplexes
Solid made by joining an n- and 2n-gon with triangles and squares
covers empty space. Hence the {5/2}- and {7/2}-cupoloids pictured above have membranes (not filled in), while the {5/4}- and {7/4}-cupoloids pictured
Cupola_(geometry)
Regular tiling of hyperbolic 3-space
prism cells, with a mirrored sphenoid vertex figure. The runcinated order-5 dodecahedral honeycomb, , has dodecahedron and pentagonal prism cells, with
Order-5 dodecahedral honeycomb
Order-5_dodecahedral_honeycomb
prism facets, with an isosceles triangular pyramid vertex figure. The runcinated square tiling honeycomb, t0,3{4,4,3}, has octahedron, triangular prism
Square_tiling_honeycomb
rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has
Rectified_600-cell
Uniform 4-polytope
In geometry, a truncated 120-cell is a uniform 4-polytope formed as the truncation of the regular 120-cell. There are three truncations, including a bitruncation
Truncated_120-cells
rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and
Rectified_24-cell
In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination (3rd order truncations) of the regular
Runcinated_6-simplexes
In six-dimensional geometry, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-cube. There are
Runcinated_6-cubes
polytopes with A4 symmetry. There is one self-dual regular form, the 5-cell with 5 vertices. A4 symmetry, or [3,3,3] is order 120, with Conway quaternion
A4_polytope
Tiling of n-dimensional space
and octahedral cells. In 4 dimensions it is called the 5-cell honeycomb, with Coxeter graph , with 5-cell and rectified 5-cell facets. In 5 dimensions it
Simplicial_honeycomb
Isogonal polytope with uniform facets
applied to Uniform 5-polytopes and higher. Sterication truncates vertices, edges, faces, and cells, replacing each with new facets. 5-faces are replaced
Uniform_polytope
In four-dimensional Euclidean geometry, the runcinated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space
Runcinated tesseractic honeycomb
Runcinated_tesseractic_honeycomb
In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex
Runcinated_7-simplexes
vertices of a runcinated tesseract, with its construction, called a runcic tesseract. It has two uniform constructions, as a rectified 8-cell r{4,3,3} and
Rectified_tesseract
tesseractic cupolae and a runcinated tesseract between them. This dissection can be seen as analogous to the 4D runcinated tesseract being dissected into
Stericated_5-cubes
Type of tesseract
a tritruncation, which creates the truncated 16-cell. The truncated tesseract is bounded by 24 cells: 8 truncated cubes, and 16 tetrahedra. Truncated
Truncated_tesseract
related to the square faces of the runcinated 24-cell. The disphenoidal 288-cell is the dual of the bitruncated 24-cell. It is a 4-dimensional polytope (or
Truncated_24-cells
prism cells, with a mirrored sphenoid vertex figure. The runcinated triangular tiling honeycomb, , has triangular tiling and triangular prism cells, with
Triangular_tiling_honeycomb
geometry, a cantellated 24-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 24-cell. There are 2 unique degrees
Cantellated_24-cells
16-cell honeycomb 24-cell honeycomb Rectified 24-cell honeycomb Snub 24-cell honeycomb 5-cell honeycomb Truncated 5-cell honeycomb Omnitruncated 5-cell honeycomb
Stericated_24-cell_honeycomb
4D geometry item
geometry, a cantellated 120-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 120-cell. There are four degrees
Cantellated_120-cell
In seven-dimensional geometry, a stericated 7-cube (or runcinated 7-demicube) is a convex uniform 7-polytope, being a runcination of the uniform 7-demicube
Steric_7-cubes
Four-dimensional analog of the dodecahedron
instances of the first in the sequence, the 5-cell, which is not found in any of the others. The 120-cell contains examples of every relationship among
120-cell
Regular tiling of hyperbolic 3-space
and cube cells, with a mirrored sphenoid vertex figure. The runcinated order-4 dodecahedral honeycomb is the same as the runcinated order-5 cubic honeycomb
Order-4 dodecahedral honeycomb
Order-4_dodecahedral_honeycomb
Regular geometrical object in hyperbolic space
mirrored sphenoid vertex figure. The runcinated order-6 dodecahedral honeycomb is the same as the runcinated order-5 hexagonal tiling honeycomb. The runcitruncated
Order-6 dodecahedral honeycomb
Order-6_dodecahedral_honeycomb
Runcinated enneazetton Small prismated enneazetton (Acronym: spene) (Jonathan Bowers) The Cartesian coordinates of the vertices of the runcinated 8-simplex
Runcinated_8-simplexes
octahedron, cube, and truncated trihexagonal tiling cells, with a mirrored sphenoid vertex figure. The runcinated order-4 hexagonal tiling honeycomb, t0,3{6,3
Order-4 hexagonal tiling honeycomb
Order-4_hexagonal_tiling_honeycomb
Convex uniform 4-polytope
truncations. Two are also derived from the 24-cell family. The cantellated tesseract, bicantellated 16-cell, or small rhombated tesseract is a convex uniform
Cantellated_tesseract
Concept in geometry
for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes. Cantellated 5-demicube/demipenteract Small rhombated hemipenteract (sirhin)
Runcic_5-cubes
Quasiregular space-filling tesselation
alternated cubic honeycomb. The expanded cubic honeycomb (also known as the runcinated tesseractic honeycomb) is geometrically identical to the cubic honeycomb
Tetrahedral-octahedral honeycomb
Tetrahedral-octahedral_honeycomb
Uniform 6-dimensional polytope
polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes. The complete set of convex uniform 6-polytopes has not been determined
Uniform_6-polytope
Four-dimensional geometric objects
Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at position 0 are shown solid. The coordinates
H4_polytope
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
figure. It is the same as the truncated square tiling honeycomb, . The runcinated order-4 square tiling honeycomb, t0,3{4,4,4}, has square tiling and cube
Order-4 square tiling honeycomb
Order-4_square_tiling_honeycomb
Tiling of hyperbolic 3-space by uniform polyhedra
5] or The bitruncated and runcinated forms (29 and 30) contain the faces of two regular skew polyhedra: {4,6|5} and {6,4|5}. There are 11 forms (and only
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
In seven-dimensional geometry, a runcinated 7-cube is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-cube. There
Runcinated_7-cubes
4-polytopes with B4 symmetry. There are two regular forms, the tesseract and 16-cell, with 16 and 8 vertices respectively. They can be visualized as symmetric
B4_polytope
Only regular space-filling tessellation of the cube
space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex
Cubic_honeycomb
Notation for polytopes and tessellations
( ) ∨ {p}. In 4D: A p-q-hedral pyramid is represented as ( ) ∨ {p,q}. A 5-cell is represented as ( ) ∨ [( ) ∨ {3}] or [( ) ∨ ( )] ∨ {3} = { } ∨ {3}. A
Schläfli_symbol
being made of tetrahedron and icosahedron cells. (The other two are the rectified 5-cell and rectified 600-cell.) Snub icositetrachoron Snub demitesseract
Snub_24-cell
Prism with a 5-sided base
image through a right angle without changing its chirality. It exists as cells of four nonprismatic uniform 4-polytopes in four dimensions: Weisstein,
Pentagonal_prism
120-cell, including the zeroth, the 120-cell itself. The birectified 120-cell is more easily seen as a rectified 600-cell, and the trirectified 120-cell is
Rectified_120-cell
In 5-dimensional geometry, there are 31 uniform polytopes with B5 symmetry. There are two regular forms, the 5-orthoplex, and 5-cube with 10 and 32 vertices
B5_polytope
Space-filling tessellation
alternated cubic honeycomb. The expanded cubic honeycomb (also known as the runcinated tesseractic honeycomb) is geometrically identical to the cubic honeycomb
Bitruncated_cubic_honeycomb
6-polytope. There are unique 4 steric forms of the 6-cube. Runcinated demihexeract Runcinated 6-demicube Small prismated hemihexeract (Acronym: sophax)
Steric_6-cubes
In six-dimensional geometry, a runcinated 6-orthplex is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-orthoplex
Runcinated_6-orthoplexes
Polygon with 24 edges
{24/5}, {24/7}, and {24/11}. There are also 7 regular star figures using the same vertex arrangement: 2{12}, 3{8}, 4{6}, 6{4}, 8{3}, 3{8/3}, and 2{12/5}
Icositetragon
a mirrored sphenoid vertex figure. The runcinated order-4 octahedral honeycomb is the same as the runcinated square tiling honeycomb. The runcitruncated
Order-4_octahedral_honeycomb
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
hyperspace, George Olshevsky. Polytopes of Various Dimensions, Jonathan Bowers Runcinated uniform polytera (spid), Jonathan Bowers Multi-dimensional Glossary
Cantellated_5-cubes
In seven-dimensional geometry, a runcinated 7-orthoplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-orthoplex
Runcinated_7-orthoplexes
Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at position 0 are shown solid. Vertex coordinates
List_of_F4_polytopes
Tessellation of convex uniform polyhedron cells
honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families of
Paracompact uniform honeycombs
Paracompact_uniform_honeycombs
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{3,3,3,3,3,3,3} Bicantellated 8-simplex 10 t2,3{3,3,3
A8_polytope
In 5-dimensional geometry, there are 19 uniform polytopes with A5 symmetry. There is one self-dual regular form, the 5-simplex with 6 vertices. Each can
A5_polytope
geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical
Order-6_tetrahedral_honeycomb
Type of geometrical object
10-simplex Truncated 10-simplex Rectified 10-simplex Cantellated 10-simplex Runcinated 10-simplex Stericated 10-simplex Pentellated 10-simplex Hexicated 10-simplex
Uniform_10-polytope
truncated trihexagonal tiling, and hexagonal prism cells, with a mirrored sphenoid vertex figure. The runcinated order-6 hexagonal tiling honeycomb, t0,3{6,3
Order-6 hexagonal tiling honeycomb
Order-6_hexagonal_tiling_honeycomb
Spatial tiling of convex uniform polyhedra
uniform honeycombs via truncation operations. (One redundant form, the runcinated cubic honeycomb, is included for completeness though identical to the
Convex_uniform_honeycomb
rootstock. sclereid A cell with a thick, lignified, cell wall that is shorter than a fiber cell and dies soon after the thickening of its cell wall. sclerenchyma
Glossary_of_botanical_terms
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} Bicantellated 7-simplex 10 t2,3{3,3,3,3
A7_polytope
Seven-dimensional geometric object
geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets. A uniform 7-polytope
Uniform_7-polytope
6-orthoplex Bitruncated hexacontatetrapeton (botag) 10 t0,3{3,3,3,3,4} Runcinated 6-orthoplex Small prismated hexacontatetrapeton (spog) 11 t1,3{3,3,3,3
B6_polytope
Type of geometric object
9-simplex Rectified 9-simplex Truncated 9-simplex Cantellated 9-simplex Runcinated 9-simplex Stericated 9-simplex Pentellated 9-simplex Hexicated 9-simplex
Uniform_9-polytope
2 unique forms. A runcic 7-cube, h3{4,35}, has half the vertices of a runcinated 7-cube, t0,3{4,35}. Small rhombated hemihepteract (acronym: sirhesa) (Jonathan
Runcic_7-cubes
Group of polytopes
Bitruncated diacosipentacontahexazetton (batek) 12 t0,3{3,3,3,3,3,3,4} Runcinated 8-orthoplex Small prismated diacosipentacontahexazetton (spek) 13 t1,3{3
B8_polytope
symmetry doubles to [2(k+1)]. These 35 polytopes are each shown in these 5 symmetry planes, with vertices and edges drawn, and vertices colored by the
A6_polytope
Polytope contained by 7-polytope facets
Truncated 8-orthoplex Cantellated 8-orthoplex Runcinated 8-orthoplex Hexicated 8-orthoplex Cantellated 8-cube Runcinated 8-cube Stericated 8-cube Pentellated 8-cube
Uniform_8-polytope
7-orthoplex Bitruncated hecatonicosaoctaexon (Botaz) 11 t0,3{3,3,3,3,3,4} Runcinated 7-orthoplex Small prismated hecatonicosaoctaexon (Spaz) 12 t1,3{3,3,3
B7_polytope
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