In four-dimensional

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 ...

, a cantellated 5-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 polyhedron, uniform polyhedra, and faces are regular polygons. There are 47 non-Prism (geometry), prism ...

, 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, up to

edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

-planing) of the regular

5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional space, four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, hypertetrahedron, pentachoron, pentatope, pe ...

Cantellated 5-cell

The cantellated

or small rhombated pentachoron is a

. It has 30 vertices, 90 edges, 80 faces, and 20 cells. The cells are 5 cuboctahedra, 5

octahedra 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 10

triangular prism In geometry, a triangular prism or trigonal prism is a Prism (geometry), prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a ''right triangular prism''. A right triangul ...

s. Each vertex is surrounded by 2 cuboctahedra, 2 triangular prisms, and 1 octahedron; 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 ...

is a nonuniform triangular prism.

Alternate names

* Cantellated pentachoron * Cantellated 4-simplex * (small) prismatodispentachoron * Rectified dispentachoron * Small rhombated pentachoron (Acronym: Srip) (Jonathan Bowers)

Configuration

Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal

f-vector Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral com ...

numbers are derived through the

Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

, dividing the full group order of a subgroup order by removing one mirror at a time.

Images

Coordinates

The

Cartesian coordinate In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

s of the vertices of the origin-centered cantellated 5-cell having edge length 2 are: The vertices of the ''cantellated 5-cell'' can be most simply positioned in 5-space as permutations of: : (0,0,1,1,2) This construction is from the positive

orthant In geometry, an orthant or hyperoctant is the analogue in ''n''-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions. In general an orthant in ''n''-dimensions can be considered the intersection of ''n'' mutu ...

facet of the

cantellated 5-orthoplex In five-dimensional geometry, a cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex. There are 6 cantellation for the 5-orthoplex, including truncations. Some of them are more easily constructed ...

Related polytopes

The convex hull of two cantellated 5-cells in opposite positions is a nonuniform polychoron composed of 100 cells: three kinds of 70

(10 rectified tetrahedra, 20 triangular antiprisms, 40 triangular antipodiums), 30

tetrahedra In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

(as tetragonal disphenoids), and 60 vertices. Its vertex figure is a shape topologically equivalent to a

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

with a

attached to one of its square faces. Birhombatodecachoron vertex figure

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 ...

Cantitruncated 5-cell

The cantitruncated

or great rhombated pentachoron is a

. It is composed of 60 vertices, 120 edges, 80 faces, and 20 cells. The cells are: 5

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 ...

, 10

s, and 5

truncated tetrahedra In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron. ...

. Each vertex is surrounded by 2 truncated octahedra, one triangular prism, and one truncated tetrahedron.

Configuration

Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal

numbers are derived through the

, dividing the full group order of a subgroup order by removing one mirror at a time.

Alternative names

* Cantitruncated pentachoron * Cantitruncated 4-simplex * Great prismatodispentachoron * Truncated dispentachoron * Great rhombated pentachoron (Acronym: grip) (Jonathan Bowers)

Images

Cartesian coordinates

The

s of an origin-centered cantitruncated 5-cell having edge length 2 are: These vertices can be more simply constructed on a

hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

in 5-space, as the

permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...

s of: : (0,0,1,2,3) This construction is from the positive

facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cu ...

of the cantitruncated 5-orthoplex.

Related polytopes

A double symmetry construction can be made by placing truncated tetrahedra on the truncated octahedra, resulting in a nonuniform polychoron with 10

, 20

hexagonal prism In geometry, the hexagonal prism is a Prism (geometry), prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 face (geometry), faces, 18 Edge (geometry), edges, and 12 vertex (geometry), vertices.. As a semiregular polyhedro ...

s (as ditrigonal trapezoprisms), two kinds of 80

s (20 with ''D_3h'' symmetry and 60 ''C_2v''-symmetric wedges), and 30

(as tetragonal disphenoids). Its vertex figure is topologically equivalent to the

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 ...

Related 4-polytopes

These polytopes are art of a set of 9

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 ...

s constructed from the ,3,3

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 reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

References

H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

: ** 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
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* Norman Johnson ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966) * * x3o3x3o - srip, x3x3x3o - grip {{Polytopes Uniform 4-polytopes