1 32 polytope

This polytope can tessellate 7-dimensional space, with symbol 1₃₃, and Coxeter-Dynkin diagram, {{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea}}. It is the Voronoi cell of the dual E₇^* lattice.[http://home.digital.net/~pervin/publications/vermont.html The Voronoi Cells of the E₆^* and E₇^* Lattices] {{Webarchive|url=https://web.archive.org/web/20160130193811/http://home.digital.net/~pervin/publications/vermont.html |date=2016-01-30 }}, Edward Pervin

• Emanuel Lodewijk Elte named it V₅₇₆ (for its 576 vertices) in his 1912 listing of semiregular polytopes.Elte, 1912
• Coxeter called it 1₃₂ for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
• Pentacontahexa-hecatonicosihexa-exon (Acronym: lin) - 56-126 facetted polyexon (Jonathan Bowers)Klitzing, (o3o3o3x *c3o3o3o - lin)

It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram, {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea}}

Removing the node on the end of the 2-length branch leaves the 6-demicube, 1₃₁, {{CDD|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea}}

Removing the node on the end of the 3-length branch leaves the 1₂₂, {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 6-simplex, 0₃₂, {{CDD|nodea|3a|nodea|3a|nodea_1|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

The 1₃₂ is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_3k series. The next figure is the Euclidean honeycomb 1₃₃ and the final is a noncompact hyperbolic honeycomb, 1₃₄.

{{1 3k polytopes}}

{{1 k2 polytopes}}

The rectified 1₃₂ (also called 0₃₂₁) is a rectification of the 1₃₂ polytope, creating new vertices on the center of edge of the 1₃₂. Its vertex figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: {3,3}×{3}×{}.

• Rectified pentacontahexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (Acronym: rolin) (Jonathan Bowers)Klitzing, (o3o3x3o *c3o3o3o - rolin)

It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. These mirrors are represented by its Coxeter-Dynkin diagram, {{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea|3a|nodea}}, and the ring represents the position of the active mirror(s).

Removing the node on the end of the 3-length branch leaves the rectified 1₂₂ polytope, {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}

Removing the node on the end of the 2-length branch leaves the demihexeract, 1₃₁, {{CDD|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea}}

Removing the node on the end of the 1-length branch leaves the birectified 6-simplex, {{CDD|nodea|3a|nodea|3a|nodea_1|3a|nodea|3a|nodea|3a|nodea}}

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the tetrahedron-triangle duoprism prism, {3,3}×{3}×{}, {{CDD|nodea|3a|nodea_1|2|nodea_1|2|nodea_1|3a|nodea|3a|nodea}}

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

• List of E7 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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{isbn|978-0-471-01003-6}}
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• {{KlitzingPolytopes|polyexa.htm|7D uniform polytopes (polyexa)}} o3o3o3x *c3o3o3o - lin, o3o3x3o *c3o3o3o - rolin

{{Polytopes}}

Category:7-polytopes