geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the rectified

600-cell In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from ...

or rectified hexacosichoron 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 polyhedron, uniform polyhedra, and faces are regular polygons. There are 47 non-Prism (geometry), prism ...

composed of 600 regular octahedra and 120 icosahedra

cells Cell most often refers to: * Cell (biology), the functional basic unit of life * Cellphone, a phone connected to a cellular network * Clandestine cell, a penetration-resistant form of a secret or outlawed organization * Electrochemical cell, a d ...

. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icosahedra. In total it has 3600 triangle faces, 3600 edges, and 720 vertices. Containing the cell

realm A realm is a community or territory over which a sovereign rules. The term is commonly used to describe a monarchical or dynastic state. A realm may also be a subdivision within an empire, if it has its own monarch, e.g. the German Empire. Etymo ...

s of both 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, hec ...

and the regular

, it can be considered analogous to the polyhedron

icosidodecahedron In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (''icosi-'') triangular faces and twelve (''dodeca-'') pentagonal faces. An icosidodecahedron has 30 identical Vertex (geometry), vertices, with two triang ...

, which is a rectified

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...

and rectified

dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...

. The

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

of the rectified 600-cell is a uniform

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 ...

Semiregular polytope

It is one of three

semiregular 4-polytopes In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is isogonal figure, vertex-transitive and has all its facet (geometry), facets being regular polytopes. E.L. Elte compiled a Emanuel_Lodewij ...

made of two or more cells which are

Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

s, discovered by

Thorold Gosset John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, a ...

in his 1900 paper. He called it a ''octicosahedric'' for being made of

octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...

and

cells.

E. L. Elte Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibór) Emanuël Lodewijk Elte
...

identified it in 1912 as a semiregular polytope, labeling it as tC₆₀₀.

Alternate names

* octicosahedric (Thorold Gosset) * Icosahedral hexacosihecatonicosachoron * Rectified 600-cell (Norman W. Johnson) * Rectified hexacosichoron * Rectified polytetrahedron * Rox (Jonathan Bowers)

Images

Related polytopes

Diminished rectified 600-cell

A related

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face i ...

polytope can be constructed with equal edge lengths removes 120 vertices from the rectified 600-cell, but isn't uniform because it contains

square pyramid In geometry, a square pyramid is a Pyramid (geometry), pyramid with a square base and four triangles, having a total of five faces. If the Apex (geometry), apex of the pyramid is directly above the center of the square, it is a ''right square p ...

cells, discovered by George Olshevsky, calling it a ''swirlprismatodiminished rectified hexacosichoron'', with 840 cells (600 square pyramids, 120 pentagonal prisms, and 120 pentagonal antiprisms), 2640 faces (1800 triangles, 600 square, and 240 pentagons), 2400 edges, and 600 vertices. It has a chiral ''bi-diminished

. Each removed vertex creates a pentagonal prism cell, and diminishes two neighboring icosahedra into pentagonal antiprisms, and each octahedron into a square pyramid. This polytope can be partitioned into 12 rings of alternating 10 pentagonal prisms and 10 antiprisms, and 30 rings of square pyramids. Swirlprismatodiminished rectified hexacosichoron net

Swirlprismatodiminished rectified hexacosichoron net

Net

H4 family

Pentagonal prism vertex figures

References

* Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995
wiley.com
** (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 mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many br ...

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

External links

* *
Archimedisches Polychor Nr. 45 (rectified 600-cell)
Marco Möller's Archimedean polytopes in R⁴ (German) * H4 uniform polytopes with coordinates
r
{{Polytopes Uniform 4-polytopes