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In four-dimensional
The cantellated 120-cell is a
The cantitruncated 120-cell is a
The cantellated 600-cell is a
The cantitruncated 600-cell is a
Archimedisches Polychor Nr. 57 (cantellated 120-cell)
Marco Möller's Archimedean polytopes in R4 (German) *
', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ** (Paper 22) H.S.M. Coxeter
Regular and Semi Regular Polytopes I
ath. Zeit. 46 (1940) 380-407, MR 2,10** (Paper 23) H.S.M. Coxeter
Regular and Semi-Regular Polytopes II
ath. Zeit. 188 (1985) 559-591** (Paper 24) H.S.M. Coxeter
Regular and Semi-Regular Polytopes III
ath. Zeit. 200 (1988) 3-45*
Four-dimensional Archimedean Polytopes
(German), Marco Möller, 2004 PhD dissertation
m63
* * o3x3o5x - srahi, o3x3x5x - grahi, x3o3x5o - srix, x3x3x5o - grix
Four-Dimensional Polytope Projection Barn Raisings
(A Zometool construction of the ''cantitruncated 120-cell''),
Renaissance Banff 2005 Zome Project
a
rrrrtrtr
{{Polytopes 4-polytopes
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a cantellated 120-cell is a convex uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. There ...
, being a cantellation
In geometry, a cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tiling ...
(a 2nd order truncation) of the regular 120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...
.
There are four degrees of cantellations of the 120-cell including with permutations truncations. Two are expressed relative to the dual 600-cell.
Cantellated 120-cell
The cantellated 120-cell is a uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. There ...
. It is named by its construction as a Cantellation
In geometry, a cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tiling ...
operation applied to the regular 120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...
. It contains 1920 cells
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
, including 120 rhombicosidodecahedra, 1200 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. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A unif ...
s, 600 octahedra
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
. Its vertex figure is a wedge
A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...
, with two rhombicosidodecahedra, two triangular prisms, and one octahedron meeting at each vertex.
Alternative names
*Cantellated 120-cell Norman Johnson *Cantellated hecatonicosachoron / Cantellated dodecacontachoron / Cantellated polydodecahedron *Small rhombated hecatonicosachoron (Acronym srahi) ( George Olshevsky and Jonathan Bowers) *Ambo-02 polydodecahedron (John Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
)
Images
Cantitruncated 120-cell
The cantitruncated 120-cell is a uniform polychoron
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. There ...
.
This 4-polytope
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...
is related to the regular 120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...
. The cantitruncation operation create new truncated tetrahedral cells at the vertices, and triangular prisms at the edges. The original dodecahedron cells are cantitruncated into great rhombicosidodecahedron cells.
The image shows the 4-polytope drawn as a Schlegel diagram
In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the origina ...
which projects the 4-dimensional figure into 3-space, distorting the sizes of the cells. In addition, the decagon
In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°.
A self-intersecting ''regular decagon'' i ...
al faces are hidden, allowing us to see the elemented projected inside.
Alternative names
*Cantitruncated 120-cell Norman Johnson *Cantitruncated hecatonicosachoron / Cantitruncated polydodecahedron *Great rhombated hecatonicosachoron (Acronym grahi) ( George Olshevsky and Jonthan Bowers) *Ambo-012 polydodecahedron (John Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
)
Images
Cantellated 600-cell
The cantellated 600-cell is a uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. There ...
. It has 1440 cells: 120 icosidodecahedra, 600 cuboctahedra
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a t ...
, and 720 pentagonal prism
In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices. As a semiregular (or uniform) polyhedron
If faces are all regular, the pentagonal prism is ...
s. Its vertex figure is an isosceles 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. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A unif ...
, defined by one icosidodecahedron, two cuboctahedra, and two pentagonal prisms.
Alternative names
*Cantellated 600-cell Norman Johnson *Cantellated hexacosichoron / Cantellated tetraplex *Small rhombihexacosichoron (Acronym srix) ( George Olshevsky and Jonathan Bowers) *Ambo-02 tetraplex (John Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
)
Construction
This 4-polytope has cells at 3 of 4 positions in the fundamental domain, extracted from the Coxeter diagram by removing one node at a time: There are 1440 pentagonal faces between the icosidodecahedra andpentagonal prism
In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices. As a semiregular (or uniform) polyhedron
If faces are all regular, the pentagonal prism is ...
s. There are 3600 squares between the cuboctahedra
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a t ...
and pentagonal prisms. There are 2400 triangular faces between the icosidodecahedra and cuboctahedra, and 1200 triangular faces between pairs of cuboctahedra.
There are two classes of edges: 3-4-4, 3-4-5: 3600 have two squares and a triangle around it, and 7200 have one triangle, one square, and one pentagon.
Images
Cantitruncated 600-cell
The cantitruncated 600-cell is a uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. There ...
. It is composed of 1440 cells
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
: 120 truncated icosahedra, 720 pentagonal prism
In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices. As a semiregular (or uniform) polyhedron
If faces are all regular, the pentagonal prism is ...
s and 600 truncated octahedra
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
. It has 7200 vertices, 14400 edges, and 8640 faces (3600 squares, 1440 pentagons, and 3600 hexagons). It has an irregular tetrahedral vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
, filled by one truncated icosahedron, one pentagonal prism and two truncated octahedra.
Alternative names
*Cantitruncated 600-cell ( Norman Johnson) *Cantitruncated hexacosichoron / Cantitruncated polydodecahedron *Great rhombated hexacosichoron (acronym ''grix'') ( George Olshevsky and Jonathan Bowers)Klitzing, (x3x3x5o - grix) *Ambo-012 polytetrahedron (John Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
)
Images
Related polytopes
Notes
References
*Archimedisches Polychor Nr. 57 (cantellated 120-cell)
Marco Möller's Archimedean polytopes in R4 (German) *
', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ** (Paper 22) H.S.M. Coxeter
Regular and Semi Regular Polytopes I
ath. Zeit. 46 (1940) 380-407, MR 2,10** (Paper 23) H.S.M. Coxeter
Regular and Semi-Regular Polytopes II
ath. Zeit. 188 (1985) 559-591** (Paper 24) H.S.M. Coxeter
Regular and Semi-Regular Polytopes III
ath. Zeit. 200 (1988) 3-45*
J.H. Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...
and M.J.T. Guy: ''Four-Dimensional Archimedean Polytopes'', Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
* N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966Four-dimensional Archimedean Polytopes
(German), Marco Möller, 2004 PhD dissertation
* * o3x3o5x - srahi, o3x3x5x - grahi, x3o3x5o - srix, x3x3x5o - grix
External links
Four-Dimensional Polytope Projection Barn Raisings
(A Zometool construction of the ''cantitruncated 120-cell''),
George W. Hart
George William Hart (born 1955) is an American sculptor and geometer. Before retiring, he was an associate professor of Electrical Engineering at Columbia University in New York City and then an interdepartmental research professor at Stony B ...
Renaissance Banff 2005 Zome Project
a
Zome
The term ''zome'' is used in several related senses. A zome in the original sense is a building using unusual geometries (different from the standard house or other building which is essentially one or a series of rectangular boxes).
The word "zome ...
model of a 3D orthogonal projection of the cantellated 600-cell.
* H4 uniform polytopes with coordinatesrr
{{Polytopes 4-polytopes