In five-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 runcinated 5-cube is a convex

uniform 5-polytope In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope Facet (geometry), facets. The complete set of convex uniform 5-polytopes ...

that is a

runcination In geometry, runcination is an operation that cuts a regular polytope (or honeycomb) simultaneously along the faces, edges, and vertices, creating new facets in place of the original face, edge, and vertex centers. It is a higher order truncatio ...

(a 3rd order truncation) of the regular 5-cube. There are 8 unique degrees of runcinations of the 5-cube, along with permutations of truncations and cantellations. Four are more simply constructed relative to the

5-orthoplex In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces. It has two constructed forms, the first being regular with ...

Runcinated 5-cube

Alternate names

* Small prismated penteract (Acronym: span) (Jonathan Bowers)

Coordinates

The

Cartesian coordinate A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

s of the vertices of a ''runcinated 5-cube'' having edge length 2 are all permutations of: :

\left(\pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt),\ \pm(1+\sqrt)\right)

Images

Runcitruncated 5-cube

Alternate names

* Runcitruncated penteract * Prismatotruncated penteract (Acronym: pattin) (Jonathan Bowers)

Construction and coordinates

The

s of the vertices of a ''runcitruncated 5-cube'' having edge length 2 are all permutations of: :

\left(\pm1,\ \pm(1+\sqrt),\ \pm(1+\sqrt),\ \pm(1+2\sqrt),\ \pm(1+2\sqrt)\right)

Images

Runcicantellated 5-cube

Alternate names

* Runcicantellated penteract * Prismatorhombated penteract (Acronym: prin) (Jonathan Bowers)

Coordinates

The

s of the vertices of a ''runcicantellated 5-cube'' having edge length 2 are all permutations of: :

\left(\pm1,\ \pm1,\ \pm(1+\sqrt),\ \pm(1+2\sqrt),\ \pm(1+2\sqrt)\right)

Images

Runcicantitruncated 5-cube

Alternate names

* Runcicantitruncated penteract * Biruncicantitruncated pentacross * great prismated penteract () (Jonathan Bowers)

Coordinates

The

s of the vertices of a runcicantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of: :

\left(1,\ 1+\sqrt,\ 1+2\sqrt,\ 1+3\sqrt,\ 1+3\sqrt\right)

Images

Related polytopes

These polytopes are a part of a set of 31 uniform polytera generated from the regular 5-cube or

References

* H.S.M. Coxeter: ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 ** 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,

*** (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* Norman Johnson ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. * o3x3o3o4x - span, o3x3o3x4x - pattin, o3x3x3o4x - prin, o3x3x3x4x - gippin

External links

*
Polytopes of Various Dimensions
Jonathan Bowers *

(spid), Jonathan Bowers

{{Polytopes 5-polytopes