2 41 polytope

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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), 483,840 faces (triangles), 69,120 edges, and 2160 vertices. Its vertex figure is a 7-demicube.

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

:{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

• E. L. Elte named it V₂₁₆₀ (for its 2160 vertices) in his 1912 listing of semiregular polytopes.Elte, 1912
• It is named 2₄₁ by Coxeter 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)Klitzing, (x3o3o3o *c3o3o3o3o - bay)

The 2160 vertices can be defined as follows:

: 16 permutations of (±4,0,0,0,0,0,0,0) of (8-orthoplex)

: 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

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram: {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

Removing the node on the short branch leaves the 7-simplex: {{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}. There are 17280 of these facets

Removing the node on the end of the 4-length branch leaves the 2₃₁, {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}. There are 240 of these facets. They are centered at the positions of the 240 vertices in the 4₂₁ polytope.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 7-demicube, 1₄₁, {{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203

[[File:E8 241 Petrie Projection.png|right|thumb|290px|

The projection of 2₄₁ to the E₈ Coxeter plane (aka. the Petrie projection) with polytope radius $2\backslash sqrt\{2\}$ and 69120 edges of length $2\backslash sqrt\{2\}$]]

[[File:E8_241-3D.png|right|thumb|290px|Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry:

{{ubl

| u {{=}} (1, φ, 0, −1, φ, 0,0,0)

| v {{=}} (φ, 0, 1, φ, 0, −1,0,0)

| w {{=}} (0, 1, φ, 0, −1, φ,0,0)

}}

The 2160 projected 2₄₁ polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls for each set of tallied norms.

The overlapping vertices are color coded by overlap count. Also shown is a list of each hull group, the normed distance from the origin, and the number of vertices in the group.

]]

[[File:E8_241-3D_Concentric_Hulls_List.png|right|thumb|290px|The 2160 projected 2₄₁ polytope projected to 3D (as above) with each normed hull group listed individually with vertex counts. Notice the last two outer hulls are a combination of two overlapped Icosahedrons (24) and a Icosidodecahedron (30).

]]

Petrie polygon 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.

{{2 k1 polytopes}}

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

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

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space, defined by root vectors of the E₈ Coxeter group.

The facet information can be extracted from its Coxeter-Dynkin diagram: {{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

Removing the node on the short branch leaves the rectified 7-simplex: {{CDD|nodea|3a|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

Removing the node on the end of the 4-length branch leaves the rectified 2₃₁, {{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea}}.

Removing the node on the end of the 2-length branch leaves the 7-demicube, 1₄₁{{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the rectified 6-simplex prism, {{CDD|nodea_1|2|branch_10|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

Petrie polygon 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.

• List of E8 polytopes

{{reflist}}

• {{citation | last = Elte | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912}}
• 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, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• {{KlitzingPolytopes|polyzetta.htm|8D|Uniform polyzetta}} x3o3o3o *c3o3o3o3o - bay, o3x3o3o *c3o3o3o3o - robay

{{Polytopes}}

Category:8-polytopes

Category:E8 (mathematics)