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 rectified 5-simplex is a convex uniform 5-polytope, being a

rectification Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Recti ...

of the regular 5-simplex. There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the ''rectified 5-simplex'' are located at the edge-centers of the ''5-simplex''. Vertices of the ''birectified 5-simplex'' are located in the triangular face centers of the ''5-simplex''.

Rectified 5-simplex

In five-dimensional

, a rectified 5-simplex is a uniform 5-polytope with 15 vertices, 60 edges, 80

triangular A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, ...

faces, 45 cells (30

tetrahedral In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

, and 15

octahedral 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 ea ...

), and 12 4-faces (6

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 is ...

and 6 rectified 5-cells). It is also called 0_3,1 for its branching Coxeter-Dynkin diagram, shown as .

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 S.

Alternate names

* Rectified hexateron (Acronym: rix) (Jonathan Bowers)

Coordinates

The vertices of the rectified 5-simplex can be more simply positioned 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 hype ...

in 6-space as permutations of (0,0,0,0,1,1) ''or'' (0,0,1,1,1,1). These construction can be seen as facets of the rectified 6-orthoplex or birectified 6-cube respectively.

As a configuration

This configuration matrix represents the rectified 5-simplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole rectified 5-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. The diagonal f-vector 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

Related polytopes

The rectified 5-simplex, 0₃₁, is second in a dimensional series of uniform polytopes, expressed by

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 ...

as 1_3k series. The fifth figure is a Euclidean honeycomb, 3₃₁, and the final is a noncompact hyperbolic honeycomb, 4₃₁. Each progressive uniform polytope is constructed from the previous as its

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 ...

Birectified 5-simplex

The birectified 5-simplex is isotopic, with all 12 of its facets as rectified 5-cells. It has 20 vertices, 90 edges, 120

faces, 60 cells (30

, and 30

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 S. It is also called 0_2,2 for its branching Coxeter-Dynkin diagram, shown as . It is seen in the

of the 6-dimensional 1₂₂, .

Alternate names

* Birectified hexateron * dodecateron (Acronym: dot) (For 12-facetted polyteron) (Jonathan Bowers)

Construction

The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts ( f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The diagonal f-vector numbers are derived through the

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

Images

The A5 projection has an identical appearance to ''Metatron's Cube''. p.160 Figure 6-12

Intersection of two 5-simplices

The ''birectified 5-simplex'' 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, thei ...

of two regular 5-simplexes in

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 ...

configuration. The vertices of a

birectification In Euclidean geometry, rectification, also known as critical truncation or complete-truncation, is the process of truncating a polytope by marking the midpoints of all its Edge (geometry), edges, and cutting off its Vertex (geometry), vertices ...

exist at the center of the faces of the original polytope(s). This intersection is analogous to the 3D stellated octahedron, seen as a compound of two regular

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

and intersected in a central

octahedron 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 e ...

, while that is a first

where vertices are at the center of the original edges. It is also the intersection of a 6-cube with the hyperplane that bisects the 6-cube's long diagonal orthogonally. In this sense it is the 5-dimensional analog of the regular hexagon,

, and

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, ...

. This characterization yields simple coordinates for the vertices of a birectified 5-simplex in 6-space: the 20 distinct permutations of (1,1,1,−1,−1,−1). The vertices of the ''birectified 5-simplex'' can also be positioned on a

in 6-space as permutations of (0,0,0,1,1,1). This construction can be seen as facets of the birectified 6-orthoplex.

Related polytopes

k_22 polytopes

The ''birectified 5-simplex'', 0₂₂, is second in a dimensional series of uniform polytopes, expressed by

as k₂₂ series. The ''birectified 5-simplex'' is the vertex figure for the third, the 1₂₂. The fourth figure is a Euclidean honeycomb, 2₂₂, and the final is a noncompact hyperbolic honeycomb, 3₂₂. Each progressive uniform polytope is constructed from the previous as its

Isotopics polytopes

Related uniform 5-polytopes

This polytope is the

of the 6-demicube, and the edge figure of the uniform 2₃₁ polytope. It is also one of 19 uniform polytera based on the ,3,3,3

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 ...

, all shown here in A₅

Coxeter plane In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...

orthographic projection Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal ...

s. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

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* Norman Johnson ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. * o3x3o3o3o - rix, o3o3x3o3o - dot

External links

*
Polytopes of Various Dimensions
Jonathan Bowers *

(Rix), Jonathan Bowers

{{Polytopes 5-polytopes