Runcinated 120-cell
Encyclopedia
120-cell |
Runcinated 120-cell (Expanded 120-cell) |
Runcitruncated 120-cell |
600-cell |
Runcitruncated 600-cell (Expanded 600-cell) |
Omnitruncated 120-cell |
| Orthogonal projections in H3 Coxeter plane | ||
|---|---|---|
In four-dimensional geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a runcinated 120-cell (or runcinated 600-cell) is a convex uniform polychoron
Uniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....
, being a runcination
Runcination
In geometry, runcination is an operation that cuts a regular polytope simultaneously along the faces, edges and vertices, creating new facets in place of the original face, edge, and vertex centers....
(a 3rd order truncation) of the regular 120-cell.
There are 4 degrees of runcinations of the 120-cell including with permutations truncations and cantellations.
The runcinated 120-cell can be seen as an expansion
Expansion (geometry)
In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements...
applied to a regular polychoron, the 120-cell or 600-cell.
Runcinated 120-cell
| Runcinated 120-cell | |
|---|---|
| Type | Uniform polychoron Uniform polychoron In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra.... |
| Uniform index | 38 |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... |
|
| Cells | 2640 total: 120 5.5.5  720 4.4.5 Pentagonal prism In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices.- As a semiregular polyhedron :...  1200 4.4.3 Triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides....  600 3.3.3 Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...  |
| Faces | 7440: 2400{3}+3600{4}+ 1440{5} |
| Edges | 7200 |
| Vertices | 2400 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
 Equilateral-triangular antipodium |
| Schläfli symbol | t0,3{5,3,3} |
| Symmetry group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
H4, [3,3,5] |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
The runcinated 120-cell is a uniform polychoron
Uniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....
. It has 2640 cells: 120 dodecahedra, 720 pentagonal prism
Pentagonal prism
In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices.- As a semiregular polyhedron :...
s, 1200 triangular prism
Triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides....
s, and 600 tetrahedra
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
. Its vertex figure
Vertex figure
In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...
is a nonuniform triangular antiprism
Antiprism
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
(equilateral-triangular antipodium): its bases represent a dodecahedron and a tetrahedron, and its flanks represent three triangular prisms and three pentagonal prisms.
Alternate names
- Runcinated 120-cell / Runcinated 600-cell (Norman W. Johnson)
- Runcinated hecatonicosachoron / Runcinated hexacosichoron / Runcinated polydodecahedron / Runcinated polytetrahedron
- Small diprismatohexacosihecatonicosachoron (acronym: sidpixhi) (George OlshevskyGeorge OlshevskyGeorge Olshevsky is a freelance editor, writer, publisher, amateur paleontologist, and mathematician living in San Diego, California.Olshevsky maintains the comprehensive online Dinosaur Genera List...
, Jonathan Bowers)
Images
 |
|
| Schlegel diagram (Only tetrahedral cells shown) | |
|---|---|
H3 |
A2/B3 |
| Orthogonal projections in Coxeter planes | |
Runcitruncated 120-cell
| Runcitruncated 120-cell | |
|---|---|
| Type | Uniform polychoron Uniform polychoron In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra.... |
| Uniform index | 43 |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... |
|
| Cells | 2640 total: 120 (3.10.10) Truncated dodecahedron In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.- Geometric relations :...  720 (4.4.10) Decagonal prism In geometry, the decagonal prism is the eighth in an infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra.If faces are all regular, it is a semiregular polyhedron....  1200 (3.4.4) Triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides.... 600 (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...  |
| Faces | 13440: 4800{3}+7200{4}+ 1440{10} |
| Edges | 18000 |
| Vertices | 7200 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
 Irregular rectangular pyramid |
| Schläfli symbol | t0,1,3{5,3,3} |
| Symmetry group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
H4, [3,3,5] |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
The runciruncated 120-cell is a uniform polychoron
Uniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....
. It contains 2640 cells: 120 truncated dodecahedra
Truncated dodecahedron
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.- Geometric relations :...
, 720 decagonal prism
Decagonal prism
In geometry, the decagonal prism is the eighth in an infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra.If faces are all regular, it is a semiregular polyhedron....
s, 1200 triangular prism
Triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides....
s, and 600 cuboctahedra
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...
. Its vertex figure
Vertex figure
In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...
is an irregular rectangular pyramid, with one truncated dodecahedron, two decagonal prisms, one triangular prism, and one cuboctahedron.
Alternate names
- Runcicantellated 600-cell (Norman W. Johnson)
- Prismatorhombated hexacosichoron (Acronym: prix) (George OlshevskyGeorge OlshevskyGeorge Olshevsky is a freelance editor, writer, publisher, amateur paleontologist, and mathematician living in San Diego, California.Olshevsky maintains the comprehensive online Dinosaur Genera List...
, Jonathan Bowers)
Images
 |
|
| Schlegel diagram (Only triangular prisms shown) | |
|---|---|
H3 |
A2/B3 |
| Orthogonal projections in Coxeter planes | |
Runcitruncated 600-cell
| Runcitruncated 600-cell | |
|---|---|
| Type | Uniform polychoron Uniform polychoron In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra.... |
| Uniform index | 44 |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... |
|
| Cells | 2640 total: 120 3.4.5.4 Rhombicosidodecahedron In geometry, the rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces....  720 4.4.5 Pentagonal prism In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices.- As a semiregular polyhedron :... 1200 4.4.6 Hexagonal prism In geometry, the hexagonal prism is a prism with hexagonal base. The shape has 8 faces, 18 edges, and 12 vertices.Since it has eight faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces...  600 3.6.6 Truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...  |
| Faces | 13440: 2400{3}+7200{4}+ 1440{5}+2400{6} |
| Edges | 18000 |
| Vertices | 7200 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
 Trapezoidal pyramid |
| Schläfli symbol | t0,1,3{3,3,5} |
| Symmetry group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
H4, [3,3,5] |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
The runcitruncated 600-cell is a uniform polychoron
Uniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....
. It is composed of 2640 cells: 120 rhombicosidodecahedron
Rhombicosidodecahedron
In geometry, the rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces....
, 600 truncated tetrahedra
Truncated tetrahedron
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...
, 720 pentagonal prism
Pentagonal prism
In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices.- As a semiregular polyhedron :...
s, and 1200 hexagonal prism
Hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. The shape has 8 faces, 18 edges, and 12 vertices.Since it has eight faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces...
s. It has 7200 vertices, 18000 edges, and 13440 faces (2400 triangles, 7200 squares, and 2400 hexagons).
Alternate names
- Runcicantellated 120-cell (Norman W. Johnson)
- Prismatorhombated hecatonicosachoron (Acronym: prahi) (George OlshevskyGeorge OlshevskyGeorge Olshevsky is a freelance editor, writer, publisher, amateur paleontologist, and mathematician living in San Diego, California.Olshevsky maintains the comprehensive online Dinosaur Genera List...
, Jonathan Bowers)
Images
 |
|
| Schlegel diagram | |
|---|---|
H3 |
A2/B3 |
| Orthogonal projections in Coxeter planes | |
Omnitruncated 120-cell
| Omnitruncated 120-cell | |
|---|---|
| Type | Uniform polychoron Uniform polychoron In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra.... |
| Uniform index | 46 |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... |
|
| Cells | 2640 total: 120 4.6.10 Truncated icosidodecahedron In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces....  720 4.4.10 Decagonal prism In geometry, the decagonal prism is the eighth in an infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra.If faces are all regular, it is a semiregular polyhedron.... 1200 4.4.6 Hexagonal prism In geometry, the hexagonal prism is a prism with hexagonal base. The shape has 8 faces, 18 edges, and 12 vertices.Since it has eight faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces... 600 4.6.6 Truncated octahedron In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....  |
| Faces | 17040 total: 10800 {4} Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles... , 4800 {6} 1440 {10} Decagon In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and each internal angle equal to 144°... |
| Edges | 28800 |
| Vertices | 14400 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
 Chiral scalene tetrahedron |
| Schläfli symbol | t0,1,2,3{3,3,5} |
| Symmetry group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
H4, [3,3,5] |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
The omnitruncated 120-cell is a convex uniform polychoron
Uniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....
, composed of 2640 cells: 120 truncated icosidodecahedra
Truncated icosidodecahedron
In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces....
, 600 truncated octahedra
Truncated octahedron
In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....
, 720 decagonal prism
Decagonal prism
In geometry, the decagonal prism is the eighth in an infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra.If faces are all regular, it is a semiregular polyhedron....
s, and 1200 hexagonal prism
Hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. The shape has 8 faces, 18 edges, and 12 vertices.Since it has eight faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces...
s. It has 14400 vertices, 28800 edges, and 17040 faces (10800 squares, 4800 hexagons, and 1440 decagons). It is the largest nonprismatic convex uniform polychoron
Uniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....
.
The vertices and edges form the Cayley graph
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group...
of the Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
H4.
Alternate names
- Omnitruncated 120-cell / Omnitruncated 600-cell (Norman W. Johnson)
- Omnitruncated hecatonicosachoron / Omnitruncated hexacosichoron / Omnitruncated polydodecahedron / Omnitruncated polytetrahedron
- Great diprismatohexacosihecatonicosachoron (Acronym gidpixhi) (George OlshevskyGeorge OlshevskyGeorge Olshevsky is a freelance editor, writer, publisher, amateur paleontologist, and mathematician living in San Diego, California.Olshevsky maintains the comprehensive online Dinosaur Genera List...
, Jonathan Bowers)
Images
 |
 |
| Schlegel diagram (centered on truncated icosidodecahedron Truncated icosidodecahedron In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.... ) (Orthogonal view, centered on decagonal prism Decagonal prism In geometry, the decagonal prism is the eighth in an infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra.If faces are all regular, it is a semiregular polyhedron.... cell.) |
Stereographic projection Stereographic projection The stereographic projection, in geometry, is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it... (centered on truncated icosidodecahedron Truncated icosidodecahedron In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.... ) |
|---|---|
H3 |
A2/B3 |
| Orthogonal projections in Coxeter planes | |
 Polychoric net |
|
Models
The first complete physical model of a 3D projection of the omnitruncated 120-cell was built by a team led by Daniel Duddy and David Richter on August 9, 2006 using the (ZomeZome
The term zome is used in several related senses. A zome in the original sense is a building using unusual geometries ....
) system in the London Knowledge Lab for the 2006 Bridges Conference.
Related polytopes
These polytopes are a part of a set of 15 uniform polychoronUniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....
s with H4 symmetry: