In 6-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 ...

, the 1₂₂ polytope is a

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

, constructed from the E₆ group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V₇₂ (for its 72 vertices). Its

Coxeter symbol 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 ...

is 1₂₂, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 1₂₂, constructed by positions points on the elements of 1₂₂. The rectified 1₂₂ is constructed by points at the mid-edges of the 1₂₂. The birectified 1₂₂ is constructed by points at the triangle face centers of the 1₂₂. These polytopes are from a family of 39 convex uniform polytopes in 6-dimensions, made of

facets and

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, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

1₂₂ polytope

The 1₂₂ polytope contains 72 vertices, and 54 5-demicubic facets. It has a

birectified 5-simplex In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex. There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the ''rec ...

. Its 72 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 ...

E₆.

Alternate names

* Pentacontatetra-peton (Acronym Mo) - 54-facetted polypeton (Jonathan Bowers)

Images

Construction

It is created by a

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 ...

upon a set of 6

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 ...

mirrors in 6-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram, . Removing the node on either of 2-length branches leaves the

5-demicube In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' (penteract) with alternated vertices removed. It was discovered by Thorold Gosset. Since it was the only semiregular 5- ...

, 1₃₁, . The

is determined by removing the ringed node and ringing the neighboring node. This makes the

, 0₂₂, . Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of

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 ...

orders.

Related complex polyhedron

The regular complex polyhedron ₃₃₂, , in

\mathbb^2

has a real representation as the ''1₂₂'' polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 33 faces. Its complex reflection group is ₃ sub>3 sub>2, order 1296. It has a half-symmetry quasiregular construction as , as 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

Hessian polyhedron In geometry, the Hessian polyhedron is a regular complex polyhedron 333, , in \mathbb^3. It has 27 vertices, 72 3 edges, and 27 33 faces. It is self-dual. Coxeter named it after Ludwig Otto Hesse for sharing the '' Hessian configuration'' \ ...

, .

Related polytopes and honeycomb

Along with the semiregular polytope, 2₂₁, it is also one of a family of 39 convex

s in 6-dimensions, made of

facets and

s, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

Geometric folding

The 1₂₂ is related to the

24-cell In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, o ...

by a geometric folding E6 → F4 of Coxeter-Dynkin diagrams, E6 corresponding to 1₂₂ in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the

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 ...

projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 1₂₂.

Tessellations

This polytope is the

for a uniform tessellation of 6-dimensional space, 2₂₂, .

Rectified 1₂₂ polytope

The rectified 1₂₂ polytope (also called 0₂₂₁) can tessellate 6-dimensional space as the

Voronoi cell In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed ...

of the E6* honeycomb lattice (dual of E6 lattice).

Alternate names

* Birectified 2₂₁ polytope * Rectified pentacontatetrapeton (acronym ''Ram'') - rectified 54-facetted polypeton (Jonathan Bowers)

Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

Construction

Its construction is based on the E₆ group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: . Removing the ring on the short branch leaves the

, . Removing the ring on the either 2-length branch leaves the

birectified 5-orthoplex In five-dimensional geometry, a rectified 5-cube is a convex uniform 5-polytope, being a rectification of the regular 5-cube. There are 5 degrees of rectifications of a 5-polytope, the zeroth here being the 5-cube, and the 4th and last being the ...

in its alternated form: t₂(2₁₁), . The

is determined by removing the ringed node and ringing the neighboring ring. This makes 3-3 duoprism prism, ××, . Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of

orders.

Truncated 1₂₂ polytope

Alternate names

* Truncated 1₂₂ polytope

Construction

Its construction is based on the E₆ group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .

Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

Birectified 1₂₂ polytope

Alternate names

* Bicantellated 2₂₁ * Birectified pentacontitetrapeton (barm) (Jonathan Bowers)

Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

Trirectified 1₂₂ polytope

Alternate names

* Tricantellated 2₂₁ * Trirectified pentacontitetrapeton (trim or cacam) (Jonathan Bowers)Klitzing, (x3o3o3o3x *c3o
cacam
/ref>

Notes

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 ...

, ''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 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45See p334 (figure 3.6a) by Peter mcMullen: (12-gonal node-edge graph of 1₂₂) * o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3x3o3x3o *c3o - barm {{Polytopes 6-polytopes

122 polytope

Alternate names

Images

Construction

Related complex polyhedron

Related polytopes and honeycomb

Geometric folding

Tessellations

Rectified 122 polytope

Alternate names

Images

Construction

Truncated 122 polytope

Alternate names

Construction

Images

Birectified 122 polytope

Alternate names

Images

Trirectified 122 polytope

Alternate names

See also

Notes

References

1₂₂ polytope

Rectified 1₂₂ polytope

Truncated 1₂₂ polytope

Birectified 1₂₂ polytope

Trirectified 1₂₂ polytope