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E9 honeycomb

January 01, 1970

In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E10) is a paracompact hyperbolic group, so either facets or vertex figures will not be bounded.

E10 is last of the series of Coxeter groups with a bifurcated Coxeter-Dynkin diagram of lengths 6,2,1. There are 1023 unique E10 honeycombs by all combinations of its Coxeter-Dynkin diagram. There are no regular honeycombs in the family since its Coxeter diagram is a nonlinear graph, but there are three simplest ones, with a single ring at the end of its 3 branches: 621, 261, 162.

621 honeycomb edit

621 honeycomb
Family k21 polytope
Schläfli symbol {3,3,3,3,3,3,32,1}
Coxeter symbol 621
Coxeter-Dynkin diagram                  
9-faces 611  
{38}  
8-faces {37}  
7-faces {36}  
6-faces {35}  
5-faces {34}  
4-faces {33}  
Cells {32}  
Faces {3}  
Vertex figure 521
Symmetry group  , [36,2,1]

The 621 honeycomb is constructed from alternating 9-simplex and 9-orthoplex facets within the symmetry of the E10 Coxeter group.

This honeycomb is highly regular in the sense that its symmetry group (the affine E9 Weyl group) acts transitively on the k-faces for k ≤ 7. All of the k-faces for k ≤ 8 are simplices.

This honeycomb is last in the series of k21 polytopes, enumerated by Thorold Gosset in 1900, listing polytopes and honeycombs constructed entirely of regular facets, although his list ended with the 8-dimensional the Euclidean honeycomb, 521.[1]

Construction edit

It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9-dimensional hyperbolic 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 9-orthoplex, 711.

               

Removing the node on the end of the 1-length branch leaves the 9-simplex.

               

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 521 honeycomb.

               

The edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. This makes the 421 polytope.

             

The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 321 polytope.

           

The cell figure is determined from the face figure by removing the ringed node and ringing the neighboring node. This makes the 221 polytope.

         

Related polytopes and honeycombs edit

The 621 is last in a dimensional series of semiregular polytopes and honeycombs, identified in 1900 by Thorold Gosset. Each member of the sequence has the previous member as its vertex figure. All facets of these polytopes are regular polytopes, namely simplexes and orthoplexes.

k21 figures in n dimensions
Space Finite Euclidean Hyperbolic
En 3 4 5 6 7 8 9 10
Coxeter
group 		E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 =   = E8+ E10 =   = E8++
Coxeter
diagram 		                                                                                         
Symmetry [3−1,2,1] [30,2,1] [31,2,1] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 1,920 51,840 2,903,040 696,729,600 ∞
Graph             - -
Name −121 021 121 221 321 421 521 621

261 honeycomb edit

261 honeycomb
Family 2k1 polytope
Schläfli symbol {3,3,36,1}
Coxeter symbol 261
Coxeter-Dynkin diagram                  
9-face types 251
{37} 
8-face types 241 , {37} 
7-face types 231 , {36} 
6-face types 221 , {35} 
5-face types 211 , {34} 
4-face type {33} 
Cells {32} 
Faces {3} 
Vertex figure 161  
Coxeter group  , [36,2,1]

The 261 honeycomb is composed of 251 9-honeycomb and 9-simplex facets. It is the final figure in the 2k1 family.

Construction edit

It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9-dimensional hyperbolic space.

The facet information can be extracted from its Coxeter-Dynkin diagram.

                 

Removing the node on the short branch leaves the 9-simplex.

                 

Removing the node on the end of the 6-length branch leaves the 251 honeycomb. This is an infinite facet because E10 is a paracompact hyperbolic group.

               

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 9-demicube, 161.

               

The edge figure is the vertex figure of the edge figure. This makes the rectified 8-simplex, 051.

           

The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 5-simplex prism.

           

Related polytopes and honeycombs edit

The 261 is last in a dimensional series of uniform polytopes and honeycombs.

2k1 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group 		E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 =   = E8+ E10 =   = E8++
Coxeter
diagram 		                                                                                         
Symmetry [3−1,2,1] [30,2,1] [[31,2,1]] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 384 51,840 2,903,040 696,729,600 ∞
Graph             - -
Name 2−1,1 201 211 221 231 241 251 261

162 honeycomb edit

162 honeycomb
Family 1k2 polytope
Schläfli symbol {3,36,2}
Coxeter symbol 162
Coxeter-Dynkin diagram                  
9-face types 152, 161 
8-face types 142 , 151 
7-face types 132 , 141 
6-face types 122 , {31,3,1} 
{35} 
5-face types 121 , {34} 
4-face type 111 , {33} 
Cells {32} 
Faces {3} 
Vertex figure t2{38}  
Coxeter group  , [36,2,1]

The 162 honeycomb contains 152 (9-honeycomb) and 161 9-demicube facets. It is the final figure in the 1k2 polytope family.

Construction edit

It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9-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 9-demicube, 161.

               

Removing the node on the end of the 6-length branch leaves the 152 honeycomb.

               

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 9-simplex, 062.

                 

Related polytopes and honeycombs edit

The 162 is last in a dimensional series of uniform polytopes and honeycombs.

1k2 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group 		E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 =   = E8+ E10 =   = E8++
Coxeter
diagram 		                                                                                         
Symmetry
(order) 		[3−1,2,1] [30,2,1] [31,2,1] [[32,2,1]] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 1,920 103,680 2,903,040 696,729,600 ∞
Graph             - -
Name 1−1,2 102 112 122 132 142 152 162

Notes edit

  1. ^ Conway, 2008, The Gosset series, p 413

References edit

  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN 978-1-56881-220-5
  • 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 [2]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

January 01, 1970
honeycomb, geometry, honeycomb, tessellation, uniform, polytopes, hyperbolic, dimensional, space, displaystyle, also, paracompact, hyperbolic, group, either, facets, vertex, figures, will, bounded, last, series, coxeter, groups, with, bifurcated, coxeter, dynk. In geometry an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9 dimensional space T 9 displaystyle bar T 9 also E10 is a paracompact hyperbolic group so either facets or vertex figures will not be bounded E10 is last of the series of Coxeter groups with a bifurcated Coxeter Dynkin diagram of lengths 6 2 1 There are 1023 unique E10 honeycombs by all combinations of its Coxeter Dynkin diagram There are no regular honeycombs in the family since its Coxeter diagram is a nonlinear graph but there are three simplest ones with a single ring at the end of its 3 branches 621 261 162 Contents 1 621 honeycomb 1 1 Construction 1 2 Related polytopes and honeycombs 2 261 honeycomb 2 1 Construction 2 2 Related polytopes and honeycombs 3 162 honeycomb 3 1 Construction 3 2 Related polytopes and honeycombs 4 Notes 5 References621 honeycomb edit621 honeycomb Family k21 polytope Schlafli symbol 3 3 3 3 3 3 32 1 Coxeter symbol 621 Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 9 faces 611 nbsp 38 nbsp 8 faces 37 nbsp 7 faces 36 nbsp 6 faces 35 nbsp 5 faces 34 nbsp 4 faces 33 nbsp Cells 32 nbsp Faces 3 nbsp Vertex figure 521 Symmetry group T 9 displaystyle bar T 9 nbsp 36 2 1 The 621 honeycomb is constructed from alternating 9 simplex and 9 orthoplex facets within the symmetry of the E10 Coxeter group This honeycomb is highly regular in the sense that its symmetry group the affine E9 Weyl group acts transitively on the k faces for k 7 All of the k faces for k 8 are simplices This honeycomb is last in the series of k21 polytopes enumerated by Thorold Gosset in 1900 listing polytopes and honeycombs constructed entirely of regular facets although his list ended with the 8 dimensional the Euclidean honeycomb 521 1 Construction edit It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9 dimensional hyperbolic space The facet information can be extracted from its Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Removing the node on the end of the 2 length branch leaves the 9 orthoplex 711 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Removing the node on the end of the 1 length branch leaves the 9 simplex nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp The vertex figure is determined by removing the ringed node and ringing the neighboring node This makes the 521 honeycomb nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp The edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node This makes the 421 polytope nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node This makes the 321 polytope nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp The cell figure is determined from the face figure by removing the ringed node and ringing the neighboring node This makes the 221 polytope nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Related polytopes and honeycombs edit The 621 is last in a dimensional series of semiregular polytopes and honeycombs identified in 1900 by Thorold Gosset Each member of the sequence has the previous member as its vertex figure All facets of these polytopes are regular polytopes namely simplexes and orthoplexes k21 figures in n dimensions Space Finite Euclidean Hyperbolic En 3 4 5 6 7 8 9 10 Coxetergroup E3 A2A1 E4 A4 E5 D5 E6 E7 E8 E9 E 8 displaystyle tilde E 8 nbsp E8 E10 T 8 displaystyle bar T 8 nbsp E8 Coxeterdiagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Symmetry 3 1 2 1 30 2 1 31 2 1 32 2 1 33 2 1 34 2 1 35 2 1 36 2 1 Order 12 120 1 920 51 840 2 903 040 696 729 600 Graph nbsp nbsp nbsp nbsp nbsp nbsp Name 121 021 121 221 321 421 521 621261 honeycomb edit261 honeycomb Family 2k1 polytope Schlafli symbol 3 3 36 1 Coxeter symbol 261 Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 9 face types 251 37 nbsp 8 face types 241 nbsp 37 nbsp 7 face types 231 nbsp 36 nbsp 6 face types 221 nbsp 35 nbsp 5 face types 211 nbsp 34 nbsp 4 face type 33 nbsp Cells 32 nbsp Faces 3 nbsp Vertex figure 161 nbsp Coxeter group T 9 displaystyle bar T 9 nbsp 36 2 1 The 261 honeycomb is composed of 251 9 honeycomb and 9 simplex facets It is the final figure in the 2k1 family Construction edit It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9 dimensional hyperbolic space The facet information can be extracted from its Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Removing the node on the short branch leaves the 9 simplex nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Removing the node on the end of the 6 length branch leaves the 251 honeycomb This is an infinite facet because E10 is a paracompact hyperbolic group nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp The vertex figure is determined by removing the ringed node and ringing the neighboring node This makes the 9 demicube 161 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp The edge figure is the vertex figure of the edge figure This makes the rectified 8 simplex 051 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node This makes the 5 simplex prism nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Related polytopes and honeycombs edit The 261 is last in a dimensional series of uniform polytopes and honeycombs 2k1 figures in n dimensions Space Finite Euclidean Hyperbolic n 3 4 5 6 7 8 9 10 Coxetergroup E3 A2A1 E4 A4 E5 D5 E6 E7 E8 E9 E 8 displaystyle tilde E 8 nbsp E8 E10 T 8 displaystyle bar T 8 nbsp E8 Coxeterdiagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Symmetry 3 1 2 1 30 2 1 31 2 1 32 2 1 33 2 1 34 2 1 35 2 1 36 2 1 Order 12 120 384 51 840 2 903 040 696 729 600 Graph nbsp nbsp nbsp nbsp nbsp nbsp Name 2 1 1 201 211 221 231 241 251 261162 honeycomb edit162 honeycomb Family 1k2 polytope Schlafli symbol 3 36 2 Coxeter symbol 162 Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 9 face types 152 161 nbsp 8 face types 142 nbsp 151 nbsp 7 face types 132 nbsp 141 nbsp 6 face types 122 nbsp 31 3 1 nbsp 35 nbsp 5 face types 121 nbsp 34 nbsp 4 face type 111 nbsp 33 nbsp Cells 32 nbsp Faces 3 nbsp Vertex figure t2 38 nbsp Coxeter group T 9 displaystyle bar T 9 nbsp 36 2 1 The 162 honeycomb contains 152 9 honeycomb and 161 9 demicube facets It is the final figure in the 1k2 polytope family Construction edit It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9 dimensional space The facet information can be extracted from its Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Removing the node on the end of the 2 length branch leaves the 9 demicube 161 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Removing the node on the end of the 6 length branch leaves the 152 honeycomb nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp The vertex figure is determined by removing the ringed node and ringing the neighboring node This makes the birectified 9 simplex 062 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Related polytopes and honeycombs edit The 162 is last in a dimensional series of uniform polytopes and honeycombs 1k2 figures in n dimensions Space Finite Euclidean Hyperbolic n 3 4 5 6 7 8 9 10 Coxetergroup E3 A2A1 E4 A4 E5 D5 E6 E7 E8 E9 E 8 displaystyle tilde E 8 nbsp E8 E10 T 8 displaystyle bar T 8 nbsp E8 Coxeterdiagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Symmetry order 3 1 2 1 30 2 1 31 2 1 32 2 1 33 2 1 34 2 1 35 2 1 36 2 1 Order 12 120 1 920 103 680 2 903 040 696 729 600 Graph nbsp nbsp nbsp nbsp nbsp nbsp Name 1 1 2 102 112 122 132 142 152 162Notes edit Conway 2008 The Gosset series p 413References editThe Symmetries of Things 2008 John H Conway Heidi Burgiel Chaim Goodman Strauss ISBN 978 1 56881 220 5 1 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 2 Paper 24 H S M Coxeter Regular and Semi Regular Polytopes III Math Zeit 200 1988 3 45 vteFundamental convex regular and uniform polytopes in dimensions 2 10 Family An Bn I2 p Dn E6 E7 E8 F4 G2 Hn Regular polygon Triangle Square p gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron Cube Demicube Dodecahedron Icosahedron Uniform polychoron Pentachoron 16 cell Tesseract Demitesseract 24 cell 120 cell 600 cell Uniform 5 polytope 5 simplex 5 orthoplex 5 cube 5 demicube Uniform 6 polytope 6 simplex 6 orthoplex 6 cube 6 demicube 122 221 Uniform 7 polytope 7 simplex 7 orthoplex 7 cube 7 demicube 132 231 321 Uniform 8 polytope 8 simplex 8 orthoplex 8 cube 8 demicube 142 241 421 Uniform 9 polytope 9 simplex 9 orthoplex 9 cube 9 demicube Uniform 10 polytope 10 simplex 10 orthoplex 10 cube 10 demicube Uniform n polytope n simplex n orthoplex n cube n demicube 1k2 2k1 k21 n pentagonal polytope Topics Polytope families Regular polytope List of regular polytopes and compounds Retrieved from https en 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