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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 ...
, a 5- simplex is a self-dual regular
5-polytope In geometry, a five-dimensional polytope (or 5-polytope) is a polytope in five-dimensional space, bounded by ( 4-polytope) facets, pairs of which share a polyhedral cell. Definition A 5-polytope is a closed five-dimensional figure with vertic ...
. It has six vertices, 15
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 ...
s, 20 triangle
faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
, 15 tetrahedral cells, and 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 i ...
facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
. It has a
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...
of cos−1(), or approximately 78.46°. The 5-simplex is a solution to the problem: ''Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick.''


Alternate names

It can also be called a hexateron, or hexa-5-tope, as a 6- facetted polytope in 5-dimensions. The name ''hexateron'' is derived from ''hexa-'' for having six
facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
and '' teron'' (with ''ter-'' being a corruption of ''
tetra- Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: * unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-cyc ...
'') for having four-dimensional facets. By Jonathan Bowers, a hexateron is given the acronym hix.


As a configuration

This configuration matrix represents the 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 5-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation. \begin\begin6 & 5 & 10 & 10 & 5 \\ 2 & 15 & 4 & 6 & 4 \\ 3 & 3 & 20 & 3 & 3 \\ 4 & 6 & 4 & 15 & 2 \\ 5 & 10 & 10 & 5 & 6 \end\end


Regular hexateron cartesian coordinates

The ''hexateron'' can be constructed from a
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 ...
by adding a 6th vertex such that it is equidistant from all the other vertices of the 5-cell. The Cartesian coordinates for the vertices of an origin-centered regular hexateron having edge length 2 are: :\begin &\left(\tfrac\sqrt,\ \tfrac\sqrt,\ \tfrac\sqrt,\ \tfrac\sqrt,\ \pm1\right)\\ pt&\left(\tfrac\sqrt,\ \tfrac\sqrt,\ \tfrac\sqrt,\ -\tfrac\sqrt,\ 0\right)\\ pt&\left(\tfrac\sqrt,\ \tfrac\sqrt,\ -\tfrac\sqrt\sqrt,\ 0,\ 0\right)\\ pt&\left(\tfrac\sqrt,\ -\tfrac\sqrt,\ 0,\ 0,\ 0\right)\\ pt&\left(-\tfrac\sqrt\sqrt,\ 0,\ 0,\ 0,\ 0\right) \end The vertices of the ''5-simplex'' can be more simply positioned on a hyperplane in 6-space as permutations of (0,0,0,0,0,1) ''or'' (0,1,1,1,1,1). These construction can be seen as facets of the
6-orthoplex In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell ''4-faces'', and 64 ''5-faces''. It has two constructed forms, the first being regular wi ...
or rectified 6-cube respectively.


Projected images


Lower symmetry forms

A lower symmetry form is a ''5-cell pyramid'' ( )v, with ,3,3symmetry order 120, constructed as a
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 ...
base in a 4-space hyperplane, and an
apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics), a teenaged super villainess in the Marvel Universe * Ape-X, a super-intelligent ape in the Squadron Supreme universe *Apex, ...
point ''above'' the hyperplane. The five ''sides'' of the pyramid are made of 5-cell cells. These are seen as
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 line ...
s of truncated regular 6-polytopes, like a
truncated 6-cube In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube. There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the ...
. Another form is ∨, with ,3,3symmetry order 48, the joining of an orthogonal digon and a tetrahedron, orthogonally offset, with all pairs of vertices connected between. Another form is ∨, with ,2,3symmetry order 36, and extended symmetry 3,2,3, order 72. It represents joining of 2 orthogonal triangles, orthogonally offset, with all pairs of vertices connected between. The form ∨∨ has symmetry [2,2,2,_order_48. These_are_seen_in_the_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_line_...
s_of_Bitruncation.html" ;"title=",2,2.html" ;"title="[2,2,2">[2,2,2, order 48. These are seen in the
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 line ...
s of Bitruncation">bitruncated and tritruncated regular 6-polytopes, like a bitruncated 6-cube and a tritruncated 6-simplex. The edge labels here represent the types of face along that direction, and thus represent different edge lengths. The vertex figure of the omnitruncated 5-simplex honeycomb, , is a 5-simplx with a
petrie polygon In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a ...
cycle of 5 long edges.


Compound

The compound of two 5-simplexes in dual configurations can be seen in this A6
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 ar ...
projection, with a red and blue 5-simplex vertices and edges. This compound has 3,3,3,3 symmetry, order 1440. The intersection of these two 5-simplexes is a uniform birectified 5-simplex. = ∩ . :


Related uniform 5-polytopes

It is first in a dimensional series of uniform polytopes and honeycombs, 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 13k series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral
hosohedron In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune hav ...
. It is first in a dimensional series of uniform polytopes and honeycombs, 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 3k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral
dihedron A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...
. The 5-simplex, as 220 polytope is first in dimensional series 22k. The regular 5-simplex is 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 refle ...
, 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 ar ...
orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)


See also

*
11-cell In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli symbol , with 3 hemi-icosahedra ...


Notes


References

* * Coxeter, H.S.M.: ** ** *** (Paper 22) *** (Paper 23) *** (Paper 24) * * **


External links

*
Polytopes of Various Dimensions
Jonathan Bowers

{{Polytopes 5-polytopes