In four-dimensional

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

. 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

. 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''. ...

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 ...

. 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)

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. Ther ...

. 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), ...

is related to the regular

. 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 In geometry, this name may refer to: * Truncated icosidodecahedron - An Archimedean solid, with Schläfli symbol t0,1,2. * Nonconvex great rhombicosidodecahedron In geometry, the nonconvex great rhombicosidodecahedron is a nonconvex uniform polyh ...

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 ori ...

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)

Images

Cantellated 600-cell

The cantellated 600-cell is a

. It has 1440 cells: 120 icosidodecahedra, 600 cuboctahedra, 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 a ...

s. Its vertex figure is an isosceles

, 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)

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 and

s. There are 3600 squares between the cuboctahedra 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

. It is composed of 1440

: 120 truncated icosahedra, 720

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 hexagons and 6 squares), 36 ...

. 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...

, 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)

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 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, 1966
Four-dimensional Archimedean Polytopes
(German), Marco Möller, 2004 PhD dissertation
m63
* * 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 Br ...

Renaissance Banff 2005 Zome Project
a Zome model of a 3D orthogonal projection of the cantellated 600-cell. * H4 uniform polytopes with coordinates
rrrrtrtr
{{Polytopes 4-polytopes