152 honeycomb
(No image)
Type Uniform tessellation
Family 1k2 polytope
Schläfli symbol {3,35,2}
Coxeter symbol 152
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
8-face types 142Gosset 1 42 polytope petrie.svg
151Demiocteract ortho petrie.svg
7-face types 132Gosset 1 32 petrie.svg
141Demihepteract ortho petrie.svg
6-face types 122Gosset 1 22 polytope.svg
{31,3,1}Demihexeract ortho petrie.svg
{35}6-simplex t0.svg
5-face types 121Demipenteract graph ortho.svg
{34}5-simplex t0.svg
4-face type 111Cross graph 4.svg
{33}4-simplex t0.svg
Cells {32}3-simplex t0.svg
Faces {3}2-simplex t0.svg
Vertex figure birectified 8-simplex:
t2{37} Birectified 8-simplex.png
Coxeter group , [35,2,1]

In geometry, the 152 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. It contains 142 and 151 facets, in a birectified 8-simplex vertex figure. It is the final figure in the 1k2 polytope 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 end of the 2-length branch leaves the 8-demicube, 151.


Removing the node on the end of the 5-length branch leaves the 142.


The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 8-simplex, 052.


Related polytopes and honeycombsEdit

See alsoEdit


  • 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] GoogleBook
    • (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