Burgers vector
| 5-cubic honeycomb | |
|---|---|
| (no image) | |
| Type | Regular 5-space honeycomb |
| Family | Hypercube honeycomb |
| Schläfli symbol | {4,33,4} t0,5{4,33,4} {4,3,3,31,1} {4,3,4}x{∞} {4,3,4}x{4,4} {4,3,4}x{∞}2 {4,4}2x{∞} {∞}5 |
| Coxeter-Dynkin diagrams |
Template:CDD |
| 5-face type | {4,33} |
| 4-face type | {4,3,3} |
| Cell type | {4,3} |
| Face type | {4} |
| Face figure | {4,3} (octahedron) |
| Edge figure | 8 {4,3,3} (16-cell) |
| Vertex figure | 32 {4,33} (5-orthoplex) |
| Coxeter group | , [4,33,4] |
| Dual | self-dual |
| Properties | vertex-transitive, edge-transitive, face-transitive, cell-transitive |
The 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an order-4 penteractic honeycomb.
It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic honeycomb of 4-space.
Constructions
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,33,4}. Another form has two alternating 5-cube facets (like a checkerboard) with Schläfli symbol {4,3,3,31,1}. The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol {∞}5.
Related polytopes and honeycombs
The [4,33,4], Template:CDD, Coxeter group generates 63 permutations of uniform tessellations, 35 with unique symmetry and 34 with unique geometry. The expanded 5-cubic honeycomb is geometrically identical to the 5-cubic honeycomb.
The 5-cubic honeycomb can be alternated into the 5-demicubic honeycomb, replacing the 5-cubes with 5-demicubes, and the alternated gaps are filled by 5-orthoplex facets.
It is also related to the regular 6-cube which exists in 6-space with 3 5-cubes on each cell. This could be considered as a tessellation on the 5-sphere, an order-3 penteractic honeycomb, {4,34}.
Tritruncated 5-cubic honeycomb
A tritruncated 5-cubic honeycomb, Template:CDD, containins all bitruncated 5-orthoplex facets and is the Voronoi tessellation of the D5* lattice. Facets can be identically colored from a doubled ×2, [[4,33,4]] symmetry, alternately colored from , [4,33,4] symmetry, three colors from , [4,3,3,31,1] symmetry, and 4 colors from , [31,1,3,31,1] symmetry.
See also
Regular and uniform honeycombs in 5-space:
- 5-demicubic honeycomb
- 5-simplex honeycomb
- Truncated 5-simplex honeycomb
- Omnitruncated 5-simplex honeycomb
References
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
- 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 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
(I'm no baseball guy, but if you increase the opponent's chance of winning by intentionally putting a runner on first in that situation, how come EVERY manager does it?
The citizens of Summerside and the surrounding area derserved this facility, for too long we had to use outdated decrepit, unsafe and embarassing facilities, I applaud ethe Mayor and the Council at the time who had the vision, and the and the community spirit to push the building of this facility thru all the politics and legal stuff to get this built. And it continues to grow, The addition of a the skateboarding facility tp CUP is proving to be a well used and appreciated park for our young citizens.
http://southfloridanfp.org/coach/?key=cheap-coach-outlet-24
http://southfloridanfp.org/coach/?key=coach-gilroy-outlet-90
http://southfloridanfp.org/coach/?key=coach-sneakers-outlet-25
http://southfloridanfp.org/coach/?key=coach-bags-on-sale-at-outlet-33
http://southfloridanfp.org/coach/?key=coach-pocketbooks-outlet-64
If you adored this post and you would certainly such as to receive additional facts concerning Cheap Uggs Boots kindly go to the web site.