In geometry, a truncated 5-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...

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 object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...

. There are two degrees of truncations, including a bitruncation.

Truncated 5-cell

The truncated 5-cell, truncated pentachoron or truncated 4-simplex is bounded by 10

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 ...

: 5 tetrahedra, and 5 truncated tetrahedra. Each vertex is surrounded by 3 truncated tetrahedra and one tetrahedron; the vertex figure is an elongated tetrahedron.

Construction

The truncated 5-cell may be constructed from the

by truncating its vertices at 1/3 of its edge length. This transforms the 5 tetrahedral cells into truncated tetrahedra, and introduces 5 new tetrahedral cells positioned near the original vertices.

Structure

The truncated tetrahedra are joined to each other at their hexagonal faces, and to the tetrahedra at their triangular faces. Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.

Projections

The truncated tetrahedron-first Schlegel diagram projection of the truncated 5-cell into 3-dimensional space has the following structure: * The projection envelope is a truncated tetrahedron. * One of the truncated tetrahedral cells project onto the entire envelope. * One of the tetrahedral cells project onto a tetrahedron lying at the center of the envelope. * Four flattened tetrahedra are joined to the triangular faces of the envelope, and connected to the central tetrahedron via 4 radial edges. These are the images of the remaining 4 tetrahedral cells. * Between the central tetrahedron and the 4 hexagonal faces of the envelope are 4 irregular truncated tetrahedral volumes, which are the images of the 4 remaining truncated tetrahedral cells. This layout of cells in projection is analogous to the layout of faces in the face-first projection of the truncated tetrahedron into 2-dimensional space. The truncated 5-cell is the 4-dimensional analogue of the truncated tetrahedron.

Images

Image:Truncated pentachoron net.png, net Image:Truncated simplex stereographic.png,

stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...

(centered on truncated tetrahedron)

Alternate names

* Truncated pentatope * Truncated

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

* Truncated pentachoron (Acronym: tip) (Jonathan Bowers)

Coordinates

The Cartesian coordinates for the vertices of an origin-centered truncated 5-cell having edge length 2 are: More simply, the vertices of the ''truncated 5-cell'' can be constructed on a

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

in 5-space as permutations of (0,0,0,1,2) ''or'' of (0,1,2,2,2). These coordinates come from positive orthant facets of the

truncated pentacross In five-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex. There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on ...

and

bitruncated penteract In five-dimensional geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube. There are four unique truncations of the 5-cube. Vertices of the truncated 5-cube are located as pairs on the edge of the 5 ...

respectively.

Related polytopes

The convex hull of the truncated 5-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 60 cells: 10 tetrahedra, 20 octahedra (as triangular antiprisms), 30 tetrahedra (as tetragonal disphenoids), and 40 vertices. Its vertex figure is a hexakis triangular cupola. Bitruncatodecachoron vertex figure

Vertex figure

Bitruncated 5-cell

The bitruncated

(also called a bitruncated pentachoron, decachoron and 10-cell) is a 4-dimensional polytope, or

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 ...

, composed of 10

in the shape of truncated tetrahedra. Topologically, under its highest symmetry,

3,3,3 In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is ...

, there is only one geometrical form, containing 10 uniform truncated tetrahedra. The hexagons are always regular because of the polychoron's inversion symmetry, of which the regular hexagon is the only such case among ditrigons (an isogonal hexagon with 3-fold symmetry).

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. Each hexagonal face of the truncated tetrahedra is joined in complementary orientation to the neighboring truncated tetrahedron. Each edge is shared by two hexagons and one triangle. Each vertex is surrounded by 4 truncated tetrahedral cells in a tetragonal disphenoid vertex figure. The bitruncated 5-cell is the

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...

of two pentachora in dual configuration. As such, it is also the intersection of a penteract with the hyperplane that bisects the penteract's long diagonal orthogonally. In this sense it is a 4-dimensional analog of the regular octahedron (intersection of regular tetrahedra in dual configuration / tesseract bisection on long diagonal) and the regular hexagon (equilateral triangles / cube). The 5-dimensional analog is the

birectified 5-simplex In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a Rectification (geometry), rectification of the regular 5-simplex. There are three unique degrees of rectifications, including the zeroth, the 5-simplex its ...

, and the

n

-dimensional analog is the polytope whose

Coxeter–Dynkin diagram In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describe ...

is linear with rings on the middle one or two nodes. The bitruncated 5-cell is one of the two non-regular convex uniform 4-polytopes which are

cell-transitive In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...

. The other is the

bitruncated 24-cell In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell. There are two degrees of truncations, including a bitruncation. Truncated 24-cell The truncated 24- ...

, which is composed of 48 truncated cubes.

Symmetry

This 4-polytope has a higher extended pentachoric symmetry (2×A₄,

), doubled to order 240, because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual.

Alternative names

* Bitruncated 5-cell (

Norman W. Johnson Norman Woodason Johnson () was a mathematician at Wheaton College, Massachusetts, Wheaton College, Norton, Massachusetts. Early life and education Norman Johnson was born on in Chicago. His father had a bookstore and published a local news ...

) * 10-cell as a

4-polytope * Bitruncated pentachoron * Bitruncated pentatope * Bitruncated

* Decachoron (Acronym: deca) (Jonathan Bowers)

Images

Coordinates

The Cartesian coordinates of an origin-centered bitruncated 5-cell having edge length 2 are: More simply, the vertices of the bitruncated 5-cell can be constructed on a

in 5-space as permutations of (0,0,1,2,2). These represent positive orthant facets of the

bitruncated pentacross In five-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex. There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on ...

. Another 5-space construction, centered on the origin are all 20 permutations of (-1,-1,0,1,1).

Related polytopes

The

bitruncated 5-cell In geometry, a truncated 5-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 5-cell. There are two degrees of truncations, including a bitruncation. Truncated 5-cell The truncated 5-cell, ...

can be seen as the intersection of two regular

s in dual positions. = ∩ .

Related regular skew polyhedron

The

regular skew polyhedron In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhe ...

, , exists in 4-space with 4 hexagonal around each vertex, in a zig-zagging nonplanar vertex figure. These hexagonal faces can be seen on the bitruncated 5-cell, using all 60 edges and 30 vertices. The 20 triangular faces of the bitruncated 5-cell can be seen as removed. The dual regular skew polyhedron, , is similarly related to the square faces of the

runcinated 5-cell In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation, up to face-planing) of the regular 5-cell. There are 3 unique degrees of runcinations of the 5-cell, including with pe ...

Disphenoidal 30-cell

The disphenoidal 30-cell is the

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

of the

. It is a 4-dimensional polytope (or

polychoron 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), a ...

) derived from the

. It is the convex hull of two

s in opposite orientations. Being the dual of a uniform polychoron, it is

, consisting of 30 congruent tetragonal disphenoids. In addition, it is

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 fa ...

under the group Aut(A₄).

Related polytopes

These polytope are from a set of 9 uniform 4-polytope 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 and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

: ** 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*

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

, ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, p. 88 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.) **Coxeter, H. S. M. ''Regular Skew Polyhedra in Three and Four Dimensions.'' Proc. London Math. Soc. 43, 33-62, 1937. * Norman Johnson ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966) * * x3x3o3o - tip, o3x3x3o - deca ;Specific {{Polytopes 4-polytopes