5-simplex honeycomb
| 5-simplex honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform 5-honeycomb |
| Family | Simplectic honeycomb |
| Schläfli symbol | {3[6]} |
| Coxeter diagram |       |
| 5-face types | {34}  , t1{34}  t2{34}  |
| 4-face types | {33}  , t1{33}  |
| Cell types | {3,3}  , t1{3,3}  |
| Face types | {3}  |
| Vertex figure | t0,4{34}  |
| Coxeter groups | A~5{displaystyle {tilde {A}}_{5}}  ×2, <[3[6]]> |
| Properties | vertex-transitive |
In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. These facet types occur in proportions of 2:2:1 respectively in the whole honeycomb.
Contents
1 A5 lattice
2 Related polytopes and honeycombs
3 Projection by folding
4 See also
5 Notes
6 References
A5 lattice
This vertex arrangement is called the A5 lattice or 5-simplex lattice. The 30 vertices of the stericated 5-simplex vertex figure represent the 30 roots of the A~5{displaystyle {tilde {A}}_{5}}
The A2
5 lattice is the union of two A5 lattices:
∪
The A3
5 is the union of three A5 lattices:
∪ 
∪ 
.
The A*
5 lattice (also called A6
5) is the union of six A5 lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex.
∪
∪
∪
∪
∪
= dual of 
Related polytopes and honeycombs
This honeycomb is one of 12 unique uniform honeycombs[2] constructed by the A~5{displaystyle {tilde {A}}_{5}}
| A5 honeycombs | ||||
|---|---|---|---|---|
Hexagon symmetry | Extended symmetry | Extended diagram | Extended group | Honeycomb diagrams |
a1 | [3[6]] | | A~5{displaystyle {tilde {A}}_{5}} | |
d2 | <[3[6]]> |     | A~5{displaystyle {tilde {A}}_{5}} ×21 | 1, , , , |
p2 | [[3[6]]] |   | A~5{displaystyle {tilde {A}}_{5}} ×22 | 2, |
i4 | [<[3[6]]>] | | A~5{displaystyle {tilde {A}}_{5}} ×21×22 | , |
d6 | <3[3[6]]> |   | A~5{displaystyle {tilde {A}}_{5}} ×61 | |
r12 | [6[3[6]]] | | A~5{displaystyle {tilde {A}}_{5}} ×12 | 3 |
Projection by folding
The 5-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
A~5{displaystyle {tilde {A}}_{5}} | |
|---|---|
C~3{displaystyle {tilde {C}}_{3}}  |   |
See also
Regular and uniform honeycombs in 5-space:
- 5-cubic honeycomb
- 5-demicube honeycomb
- Truncated 5-simplex honeycomb
- Omnitruncated 5-simplex honeycomb
Notes
^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A5.html
^ mathworld: Necklace, OEIS sequence A000029 13-1 cases, skipping one with zero marks
References
Norman Johnson Uniform Polytopes, Manuscript (1991)
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, .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .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 .cs1-lock-limited a,.mw-parser-output .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 .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-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.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 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 |
