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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}} affine Coxeter group symmetry. It is given a Schläfli symbol {3^[n+1]}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of n+1 nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes x+y+...∈Z{displaystyle x+y+...in mathbb {Z} }, then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an n-simplex honeycomb is an expanded n-simplex.

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.

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]
  
  
  
  

v t e 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