In 8-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 ...

, the 1₄₂ is a

uniform 8-polytope In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets. A uniform 8-polytope is one which is vertex-transitive ...

, constructed within the symmetry of the E₈ group. Its

Coxeter symbol 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 ...

is 1₄₂, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences. The rectified 1₄₂ is constructed by points at the mid-edges of the 1₄₂ and is the same as the birectified 2₄₁, and the quadrirectified 4₂₁. These polytopes are part of a family of 255 (2⁸ − 1) convex

uniform polytope 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 ...

s in 8 dimensions, made of

facets and

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 ...

s, defined by all non-empty combinations of rings in this Coxeter-Dynkin diagram: .

1₄₂ polytope

The 1₄₂ is composed of 2400 facets: 240 1₃₂ polytopes, and 2160

7-demicube In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube ( hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L ...

s (1₄₁). Its

is a birectified 7-simplex. This polytope, along with the demiocteract, can

tessellate A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...

8-dimensional space, represented by the symbol 1₅₂, and Coxeter-Dynkin diagram: .

Alternate names

E. L. Elte Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibór) Emanuël Lodewijk Elte
...

(1912) excluded this polytope from his listing of semiregular polytopes, because it has more than two types of 6-faces, but under his naming scheme it would be called V₁₇₂₈₀ for its 17280 vertices. *

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 ...

named it 1₄₂ for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch. * Diacositetracont-dischiliahectohexaconta-zetton (acronym ''bif'') - 240-2160 facetted polyzetton (Jonathan Bowers)

Coordinates

The 17280 vertices can be defined as sign and location permutations of: All sign combinations (32): (280×32=8960 vertices) : (4, 2, 2, 2, 2, 0, 0, 0) Half of the sign combinations (128): ((1+8+56)×128=8320 vertices) : (2, 2, 2, 2, 2, 2, 2, 2) : (5, 1, 1, 1, 1, 1, 1, 1) : (3, 3, 3, 1, 1, 1, 1, 1) The edge length is 2 in this coordinate set, and the polytope radius is 4.

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 8

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 ...

mirrors in 8-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram: . Removing the node on the end of the 2-length branch leaves the

, 1₄₁, . Removing the node on the end of the 4-length branch leaves the 1₃₂, . The

is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 7-simplex, 0₄₂, . 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 ref ...

orders.Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203

Projections

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 are shown for the sub-symmetries of E₈: E₇, E₆, B₈, B₇, B₆, B₅, B₄, B₃, B₂, A₇, and 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 ...

s, as well as two more symmetry planes of order 20 and 24. Vertices are shown as circles, colored by their order of overlap in each projective plane.

Related polytopes and honeycombs

Rectified 1₄₂ polytope

The rectified 1₄₂ is named from 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 1₄₂ polytope, with vertices positioned at the mid-edges of the 1₄₂. It can also be called a 0₄₂₁ polytope with the ring at the center of 3 branches of length 4, 2, and 1.

Alternate names

* 0₄₂₁ polytope * Birectified 2₄₁ polytope * Quadrirectified 4₂₁ polytope * Rectified diacositetracont-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (acronym ''buffy'') (Jonathan Bowers)Klitzing, (o3o3o3x *c3o3o3o3o - buffy)

Construction

It is created by a

upon a set of 8

mirrors in 8-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram: . Removing the node on the end of the 1-length branch leaves the birectified 7-simplex, Removing the node on the end of the 2-length branch leaves the birectified 7-cube, . Removing the node on the end of the 3-length branch leaves the rectified 1₃₂, . The

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

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 ...

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 ...

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

orders.

Projections

s are shown for the sub-symmetries of B₆, B₅, B₄, B₃, B₂, A₇, and A₅

s. Vertices are shown as circles, colored by their order of overlap in each projective plane. (Planes for E₈: E₇, E₆, B₈, B₇, 4are not shown for being too large to display.)

Notes

References

* 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,
Kaleidoscopes: Selected Writings of H.S.M. Coxeter , Wiley
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* o3o3o3x *c3o3o3o3o - bif, o3o3o3x *c3o3o3o3o - buffy {{Polytopes 8-polytopes E8 (mathematics)

142 polytope

Alternate names

Coordinates

Construction

Projections

Related polytopes and honeycombs

Rectified 142 polytope

Alternate names

Construction

Projections

See also

Notes

References

1₄₂ polytope

Rectified 1₄₂ polytope