In five-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 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 Facet (geometry), facets. The complete set of convex uniform 5-polytopes ...

with fourth-order truncations (

sterication In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the sam ...

) 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 ...

. 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 Expansion may refer to: Arts, entertainment and media * ''L'Expansion'', a French monthly business magazine * ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004 * ''Expansions'' (McCoy Tyner album), 1970 * ''Expansi ...

operation applied to the regular 5-simplex. The highest form, the ''steriruncicantitruncated 5-simplex'' is more simply called an omnitruncated 5-simplex with all of the nodes ringed.

Stericated 5-simplex

A stericated 5-simplex can be constructed by an

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 the 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 affect the ...

(120

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

s and 90

squares In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

), 180

cells Cell most often refers to: * Cell (biology), the functional basic unit of life * Cellphone, a phone connected to a cellular network * Clandestine cell, a penetration-resistant form of a secret or outlawed organization * Electrochemical cell, a d ...

(60

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 ...

and 120

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) and 62

4-face In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object. For example, a cube has six faces in this sense. In more modern treatments of the geometry of polyhedra and higher-dimensional polyto ...

s (12

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 ...

s, 30

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

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. Descriptions The duoprism is a 4-polytope that can be constructed using Cartesian product of two polygons. In the case of 3-3 duopr ...

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 (geometry), face-planing) of the regular 5-cell. There are 3 unique degrees of runcinations of the 5-cell, i ...

. 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 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 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'' mutu ...

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

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 In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex. There are unique 6 degrees of rectifications, the zeroth being the 6-orthoplex, and the 6th and last being the ...

is given by coordinate permutations of: : (1,-1,0,0,0,0) The

Cartesian coordinates 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 ...

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 general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

of the

5-simplex honeycomb In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. ...

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 of the steritruncated 6-orthoplex.

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

facets of the

stericantellated 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 truncat ...

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

facets of the

stericantitruncated 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 truncat ...

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

facets of the

steriruncitruncated 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 truncat ...

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 by ...

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 is de ...

s and 1080

), 540

(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

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 ...

s, and 90

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), and 62

s (12 omnitruncated 5-cells, 30

truncated octahedral prism 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, 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 of a convex polytope is a simple polytope of the same dimension, having a vertex for each flag of the original polytope and a facet for each face of any dimension of the original polytope. Omnitruncation is the dual ...

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

in 6-space as permutations of (0,1,2,3,4,5). These coordinates come from the positive

of the

steriruncicantitruncated 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 truncat ...

, 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 point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...

, the

Minkowski sum In geometry, the Minkowski sum 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'': A + B = \ The Minkowski difference (also ''Minkowski subtraction'', ''Minkowsk ...

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 facets around each

ridge A ridge is a long, narrow, elevated geomorphologic landform, structural feature, or a combination of both separated from the surrounding terrain by steep sides. The sides of a ridge slope away from a narrow top, the crest or ridgecrest, wi ...

. 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 Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

⁺, and constructed from 12 snub 5-cells, 30

snub tetrahedral antiprism 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, 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 ref ...

, all shown here in A₅

Coxeter plane In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which hav ...

orthographic projection Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Plane (mathematics), two dimensions. Orthographic projection is a form of parallel projection in ...

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

Notes

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

External links

*
Polytopes of Various Dimensions

{{Polytopes 5-polytopes