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 stericated 5-simplex 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 facets. The complete set of convex uniform 5-polytopes has not been deter ...

with fourth-order truncations (

sterication In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...

) of the regular

5-simplex In five-dimensional geometry, a 5- simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°. The 5 ...

. There are six unique sterications of the 5-simplex, including permutations of truncations, cantellations, and runcinations. The simplest stericated 5-simplex is also called an expanded 5-simplex, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-simplex. The highest form, the ''steriruncicantitruncated 5-simplex'' is more simply called an

omnitruncated 5-simplex In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations ( sterication) of the regular 5-simplex. There are six unique sterications of the 5-simplex, including permutations of truncations, ...

with all of the nodes ringed.

Stericated 5-simplex

A stericated 5-simplex can be constructed by an expansion operation applied to the regular

, and thus is also sometimes called an expanded 5-simplex. It has 30 vertices, 120 edges, 210

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 ...

(120

triangle 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- colline ...

s and 90

squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

), 180 cells (60

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 o ...

and 120

triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A ...

s) and 62 4-faces (12

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 ...

s, 30

tetrahedral prism In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedral cells: 2 tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vertices. It is one of 18 u ...

s and 20

3-3 duoprism In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. It can be constructed as the Cartesian product of two triangles and is the simplest of an infinite family of four-dimensional polytopes ...

s).

Alternate names

* Expanded 5-simplex * Stericated hexateron * Small cellated dodecateron (Acronym: scad) (Jonathan Bowers)

Cross-sections

The maximal cross-section of the stericated hexateron with a 4-dimensional hyperplane is a

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 ...

. This cross-section divides the stericated hexateron into two pentachoral hypercupolas consisting of 6

s, 15

s and 10

s each.

Coordinates

The vertices of the ''stericated 5-simplex'' can be constructed on a hyperplane in 6-space as permutations of (0,1,1,1,1,2). This represents 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'' mutua ...

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 cut ...

of the

stericated 6-orthoplex In six-dimensional geometry, a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-orthoplex. There are 16 unique sterications for the 6-orthoplex with permutations of trunca ...

. A second construction in 6-space, from the center of a rectified 6-orthoplex is given by coordinate permutations of: : (1,-1,0,0,0,0) The Cartesian coordinates in 5-space for the normalized vertices of an origin-centered stericated hexateron are: :

\left(\pm1,\ 0,\ 0,\ 0,\ 0\right)

\left(0,\ \pm1,\ 0,\ 0,\ 0\right)

\left(0,\ 0,\ \pm1,\ 0,\ 0\right)

\left(\pm1/2,\      0,\ \pm1/2,\ -\sqrt,\ -\sqrt\right)

\left(\pm1/2,\      0,\ \pm1/2,\  \sqrt,\  \sqrt\right)

\left(     0,\ \pm1/2,\ \pm1/2,\ -\sqrt,\  \sqrt\right)

\left(     0,\ \pm1/2,\ \pm1/2,\  \sqrt,\ -\sqrt\right)

\left(\pm1/2,\ \pm1/2,\ 0,\ \pm\sqrt,\ 0\right)

Root system

Its 30 vertices represent the root vectors of the

simple Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...

A₅. It is also 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 ...

of the 5-simplex honeycomb.

Images

Steritruncated 5-simplex

Alternate names

* Steritruncated hexateron * Celliprismated hexateron (Acronym: cappix) (Jonathan Bowers)

Coordinates

The coordinates can be made in 6-space, as 180 permutations of: : (0,1,1,1,2,3) This construction exists as one of 64

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 ...

of the

steritruncated 6-orthoplex In six-dimensional geometry, a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-orthoplex. There are 16 unique sterications for the 6-orthoplex with permutations of trunca ...

Images

Stericantellated 5-simplex

Alternate names

* Stericantellated hexateron * Cellirhombated dodecateron (Acronym: card) (Jonathan Bowers)

Coordinates

The coordinates can be made in 6-space, as permutations of: : (0,1,1,2,2,3) This construction exists as one of 64

of the stericantellated 6-orthoplex.

Images

Stericantitruncated 5-simplex

Alternate names

* Stericantitruncated hexateron * Celligreatorhombated hexateron (Acronym: cograx) (Jonathan Bowers)

Coordinates

The coordinates can be made in 6-space, as 360 permutations of: : (0,1,1,2,3,4) This construction exists as one of 64

of the stericantitruncated 6-orthoplex.

Images

Steriruncitruncated 5-simplex

Alternate names

* Steriruncitruncated hexateron * Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers)

Coordinates

The coordinates can be made in 6-space, as 360 permutations of: : (0,1,2,2,3,4) This construction exists as one of 64

of the steriruncitruncated 6-orthoplex.

Images

Omnitruncated 5-simplex

The omnitruncated 5-simplex has 720 vertices, 1800

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, 1560

(480

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

s and 1080

), 540 cells (360

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 ...

, 90 cubes, and 90

hexagonal prism In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices.. Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used ...

s), and 62 4-faces (12

omnitruncated 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 ...

s, 30

truncated octahedral prism In 4-dimensional geometry, a truncated octahedral prism or omnitruncated tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 16 cells (2 truncated octahedra connected by 6 cubes, 8 hexagonal prisms.) It has 64 faces (48 squares ...

s, and 20 6-6

duoprism In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, wher ...

s).

Alternate names

* Steriruncicantitruncated 5-simplex (Full description of

omnitruncation In geometry, an omnitruncation is an operation applied to a regular polytope (or honeycomb) in a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed. It is a ''shor ...

for 5-polytopes by Johnson) * Omnitruncated hexateron * Great cellated dodecateron (Acronym: gocad) (Jonathan Bowers)Klitizing, (x3x3x3x3x - gocad)

Coordinates

The vertices of the ''omnitruncated 5-simplex'' can be most simply constructed on a hyperplane in 6-space as permutations of (0,1,2,3,4,5). These coordinates come from the positive

of the steriruncicantitruncated 6-orthoplex, t_0,1,2,3,4, .

Images

Permutohedron

The omnitruncated 5-simplex is the permutohedron of order 6. It is also a

zonotope In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments i ...

, the

Minkowski sum In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski ...

of six line segments parallel to the six lines through the origin and the six vertices of the 5-simplex.

Related honeycomb

The

omnitruncated 5-simplex honeycomb In five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 5-simplex facets. The facets of all omn ...

is constructed by omnitruncated 5-simplex facets with 3

around each ridge. It has Coxeter-Dynkin diagram of .

Full snub 5-simplex

The full snub 5-simplex or omnisnub 5-simplex, defined as an alternation of the omnitruncated 5-simplex is not uniform, but it can be given Coxeter diagram and symmetry ⁺, and constructed from 12 snub 5-cells, 30 snub tetrahedral antiprisms, 20 3-3 duoantiprisms, and 360 irregular

s filling the gaps at the deleted vertices.

Related uniform polytopes

These polytopes are a part of 19 uniform 5-polytopes 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)

Notes

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. * x3o3o3o3x - scad, x3x3o3o3x - cappix, x3o3x3o3x - card, x3x3x3o3x - cograx, x3x3o3x3x - captid, x3x3x3x3x - gocad

External links

*
Polytopes of Various Dimensions

{{Polytopes 5-polytopes