Simplectic honeycomb
A~2{displaystyle {tilde {A}}_{2}}  | A~3{displaystyle {tilde {A}}_{3}}  |
|---|---|
Triangular tiling | Tetrahedral-octahedral honeycomb |
 With red and yellow equilateral triangles |  With cyan and yellow tetrahedra, and red rectified tetrahedra (octahedra) |
   |    |
In geometry, the simplectic honeycomb (or n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the A~n{displaystyle {tilde {A}}_{n}}
In 2 dimensions, the honeycomb represents the triangular tiling, with Coxeter graph filling the plane with alternately colored triangles. In 3 dimensions it represents the tetrahedral-octahedral honeycomb, with Coxeter graph filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the 5-cell honeycomb, with Coxeter graph 
, with 5-cell and rectified 5-cell facets. In 5 dimensions it is called the 5-simplex honeycomb, with Coxeter graph , filling space by 5-simplex, rectified 5-simplex, and birectified 5-simplex facets. In 6 dimensions it is called the 6-simplex honeycomb, with Coxeter graph , filling space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets.
Contents
1 By dimension
2 Projection by folding
3 Kissing number
4 See also
5 References
By dimension
| n | A~2+{displaystyle {tilde {A}}_{2+}}  | Tessellation | Vertex figure | Facets per vertex figure | Vertices per vertex figure | Edge figure |
|---|---|---|---|---|---|---|
| 1 | A~1{displaystyle {tilde {A}}_{1}}  |  Apeirogon  | | 1 | 2 | - |
2 | A~2{displaystyle {tilde {A}}_{2}} |  Triangular tiling 2-simplex honeycomb |  Hexagon (Truncated triangle)  | 3+3 triangles | 6 | Line segment |
3 | A~3{displaystyle {tilde {A}}_{3}} |  Tetrahedral-octahedral honeycomb 3-simplex honeycomb |  Cuboctahedron (Cantellated tetrahedron) | 4+4 tetrahedron 6 rectified tetrahedra | 12 |  Rectangle  |
4 | A~4{displaystyle {tilde {A}}_{4}}  | 4-simplex honeycomb |  Runcinated 5-cell | 5+5 5-cells 10+10 rectified 5-cells | 20 |  Triangular antiprism   |
5 | A~5{displaystyle {tilde {A}}_{5}}  | 5-simplex honeycomb |  Stericated 5-simplex | 6+6 5-simplex 15+15 rectified 5-simplex 20 birectified 5-simplex | 30 |  Tetrahedral antiprism  |
6 | A~6{displaystyle {tilde {A}}_{6}}  | 6-simplex honeycomb |  Pentellated 6-simplex | 7+7 6-simplex 21+21 rectified 6-simplex 35+35 birectified 6-simplex | 42 | 4-simplex antiprism |
7 | A~7{displaystyle {tilde {A}}_{7}}  | 7-simplex honeycomb |  Hexicated 7-simplex | 8+8 7-simplex 28+28 rectified 7-simplex 56+56 birectified 7-simplex 70 trirectified 7-simplex | 56 | 5-simplex antiprism |
8 | A~8{displaystyle {tilde {A}}_{8}}  | 8-simplex honeycomb |  Heptellated 8-simplex | 9+9 8-simplex 36+36 rectified 8-simplex 84+84 birectified 8-simplex 126+126 trirectified 8-simplex | 72 | 6-simplex antiprism |
| 9 | A~9{displaystyle {tilde {A}}_{9}}  | 9-simplex honeycomb |  Octellated 9-simplex | 10+10 9-simplex 45+45 rectified 9-simplex 120+120 birectified 9-simplex 210+210 trirectified 9-simplex 252 quadrirectified 9-simplex | 90 | 7-simplex antiprism |
| 10 | A~10{displaystyle {tilde {A}}_{10}}  | 10-simplex honeycomb |  Ennecated 10-simplex | 11+11 10-simplex 55+55 rectified 10-simplex 165+165 birectified 10-simplex 330+330 trirectified 10-simplex 462+462 quadrirectified 10-simplex | 110 | 8-simplex antiprism |
| 11 | A~11{displaystyle {tilde {A}}_{11}}  | 11-simplex honeycomb | ... | ... | ... | ... |
Projection by folding
The (2n-1)-simplex honeycombs and 2n-simplex honeycombs can be projected into the n-dimensional hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
A~2{displaystyle {tilde {A}}_{2}} | | A~4{displaystyle {tilde {A}}_{4}} | | A~6{displaystyle {tilde {A}}_{6}} | | A~8{displaystyle {tilde {A}}_{8}} | | A~10{displaystyle {tilde {A}}_{10}} | | ... |
|---|---|---|---|---|---|---|---|---|---|---|
A~3{displaystyle {tilde {A}}_{3}} |   | A~3{displaystyle {tilde {A}}_{3}} | | A~5{displaystyle {tilde {A}}_{5}} | | A~7{displaystyle {tilde {A}}_{7}} | | A~9{displaystyle {tilde {A}}_{9}} | | ... |
C~1{displaystyle {tilde {C}}_{1}}  | | C~2{displaystyle {tilde {C}}_{2}}  | | C~3{displaystyle {tilde {C}}_{3}}  | | C~4{displaystyle {tilde {C}}_{4}}  | | C~5{displaystyle {tilde {C}}_{5}}  | | ... |
Kissing number
These honeycombs, seen as tangent n-spheres located at the center of each honeycomb vertex have a fixed number of contacting spheres and correspond to the number of vertices in the vertex figure. For 2 and 3 dimensions, this represents the highest kissing number for 2 and 3 dimensions, but fall short on higher dimensions. In 2-dimensions, the triangular tiling defines a circle packing of 6 tangent spheres arranged in a regular hexagon, and for 3 dimensions there are 12 tangent spheres arranged in a cuboctahedral configuration. For 4 to 8 dimensions, the kissing numbers are 20, 30, 42, 56, and 72 spheres, while the greatest solutions are 24, 40, 72, 126, and 240 spheres respectively.
See also
- Hypercubic honeycomb
- Alternated hypercubic honeycomb
- Quarter hypercubic honeycomb
- Truncated simplectic honeycomb
- Omnitruncated simplectic honeycomb
References
- George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
Norman Johnson Uniform Polytopes, Manuscript (1991)
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
ISBN 0-486-61480-8
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 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
- (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 | ||||||
|---|---|---|---|---|---|---|
| Space | Family | A~n−1{displaystyle {tilde {A}}_{n-1}}  | C~n−1{displaystyle {tilde {C}}_{n-1}}  | B~n−1{displaystyle {tilde {B}}_{n-1}}  | D~n−1{displaystyle {tilde {D}}_{n-1}}  | G~2{displaystyle {tilde {G}}_{2}}  / F~4{displaystyle {tilde {F}}_{4}} / E~n−1{displaystyle {tilde {E}}_{n-1}} |
| E2 | Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
| E3 | Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
| E4 | Uniform 4-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
| E5 | Uniform 5-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
| E6 | Uniform 6-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
| E7 | Uniform 7-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
| E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| E9 | Uniform 9-honeycomb | {3[10]} | δ10 | hδ10 | qδ10 | |
| En-1 | Uniform (n-1)-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |
