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 2₄₁ 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 2₄₁, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences. The rectified 2₄₁ is constructed by points at the mid-edges of the 2₄₁. The birectified 2₄₁ is constructed by points at the triangle face centers of the 2₄₁, and is the same as the rectified 1₄₂. 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, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

2₄₁ polytope

The 2₄₁ is composed of 17,520 facets (240 2₃₁ polytopes and 17,280 7-simplices), 144,960 ''6-faces'' (6,720 2₂₁ polytopes and 138,240 6-simplices), 544,320 5-faces (60,480 2₁₁ and 483,840 5-simplices), 1,209,600 ''4-faces'' ( 4-simplices), 1,209,600 cells (

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 ...

), 483,840

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 ...

(

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), 69,120

edges Enhanced Data rates for GSM Evolution (EDGE), also known as 2.75G and under various other names, is a 2G digital mobile phone technology for packet switched data transmission. It is a subset of General Packet Radio Service (GPRS) on the GS ...

, and 2160 vertices. Its

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 ...

is a

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 ...

. This polytope is a facet in the uniform tessellation, 2₅₁ with Coxeter-Dynkin diagram: :

Alternate names

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

named it V₂₁₆₀ (for its 2160 vertices) in his 1912 listing of semiregular polytopes. *It is named 2₄₁ by

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 ...

for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence. * Diacositetracont-myriaheptachiliadiacosioctaconta-zetton (Acronym Bay) - 240-17280 facetted polyzetton (Jonathan Bowers)

Coordinates

The 2160 vertices can be defined as follows: : 16 permutations of (±4,0,0,0,0,0,0,0) of (

8-orthoplex In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 Vertex (geometry), vertices, 112 Edge (geometry), edges, 448 triangle Face (geometry), faces, 1120 tetrahedron Cell (mathematics), cells, 1792 5-cell ''4-faces'', 179 ...

) : 1120 permutations of (±2,±2,±2,±2,0,0,0,0) of ( trirectified 8-orthoplex) : 1024 permutations of (±3,±1,±1,±1,±1,±1,±1,±1) ''with an odd number of minus-signs''

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 short branch leaves the

7-simplex In 7-dimensional geometry, a 7- simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos ...

: . There are 17280 of these facets Removing the node on the end of the 4-length branch leaves the 2₃₁, . There are 240 of these facets. They are centered at the positions of the 240 vertices in the 4₂₁ polytope. The

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

, 1₄₁, . 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.

Visualizations

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 reg ...

projections are 12, 18, or 30-sided based on the E6, E7, and E8 symmetries (respectively). The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.

Related polytopes and honeycombs

Rectified 2₄₁ polytope

The rectified 2₄₁ is 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 2₄₁ polytope, with vertices positioned at the mid-edges of the 2₄₁.

Alternate names

* Rectified Diacositetracont-myriaheptachiliadiacosioctaconta-zetton for rectified 240-17280 facetted polyzetton (known as robay for short)Klitzing, (o3x3o3o *c3o3o3o3o - robay)

Construction

It is created by a

upon a set of 8

mirrors in 8-dimensional space, defined by root vectors of the E₈

. The facet information can be extracted from its Coxeter-Dynkin diagram: . Removing the node on the short branch leaves the

rectified 7-simplex In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a Rectification (geometry), rectification of the regular 7-simplex. There are four unique degrees of rectifications, including the zeroth, the 7-simplex i ...

: . Removing the node on the end of the 4-length branch leaves the rectified 2₃₁, . Removing the node on the end of the 2-length branch leaves the

, 1₄₁. The

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

rectified 6-simplex In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex. There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the ''r ...

prism, .

Visualizations

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,

** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* x3o3o3o *c3o3o3o3o - bay, o3x3o3o *c3o3o3o3o - robay {{Polytopes 8-polytopes E8 (mathematics)

241 polytope

Alternate names

Coordinates

Construction

Visualizations

Related polytopes and honeycombs

Rectified 241 polytope

Alternate names

Construction

Visualizations

See also

Notes

References

2₄₁ polytope

Rectified 2₄₁ polytope