251 honeycomb
(No image)
Type Uniform tessellation
Family 2k1 polytope
Schläfli symbol {3,3,35,1}
Coxeter symbol 251
Coxeter-Dynkin diagram CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
8-face types 241Gosset 2 41 petrie.svg
{37}8-simplex t0.svg
7-face types 231Gosset 2 31 polytope.svg
{36}7-simplex t0.svg
6-face types 221E6 graph.svg
{35}6-simplex t0.svg
5-face types 211Cross graph 5.svg
{34}5-simplex t0.svg
4-face type {33}4-simplex t0.svg
Cells {32}3-simplex t0.svg
Faces {3}2-simplex t0.svg
Edge figure 051 6-simplex t1.svg
Vertex figure 151 8-demicube.svg
Edge figure 051 7-simplex t1.svg
Coxeter group , [35,2,1]

In 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation. It is composed of 241 polytope and 8-simplex facets arranged in an 8-demicube vertex figure. It is the final figure in the 2k1 family.


It is created by a Wythoff construction upon a set of 9 hyperplane 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 8-simplex.


Removing the node on the end of the 5-length branch leaves the 241.


The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 8-demicube, 151.


The edge figure is the vertex figure of the vertex figure. This makes the rectified 7-simplex, 051.


Related polytopes and honeycombsEdit


  • Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
  • Coxeter Regular Polytopes (1963), Macmillan Company
    • Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter 5: The Kaleidoscope)
  • 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 [1]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Fundamental convex regular and uniform honeycombs in dimensions 2-9
          /   /  
{3[3]} δ3 3 3 Hexagonal
{3[4]} δ4 4 4
{3[5]} δ5 5 5 24-cell honeycomb
{3[6]} δ6 6 6
{3[7]} δ7 7 7 222
{3[8]} δ8 8 8 133331
{3[9]} δ9 9 9 152251521
{3[10]} δ10 10 10
{3[n]} δn n n 1k22k1k21