2 22 honeycomb
1_{33} honeycomb | |
---|---|
(no image) | |
Type | Uniform tessellation |
Schläfli symbol | {3,3^{3,3}} |
Coxeter symbol | 1_{33} |
Coxeter-Dynkin diagram | Template:CDD or Template:CDD |
7-face type | 1_{32} |
6-face types | 1_{22} 1_{31} |
5-face types | 1_{21} {3^{4}} |
4-face type | 1_{11} {3^{3}} |
Cell type | 1_{01} |
Face type | {3} |
Cell figure | Square |
Face figure | Triangular duoprism |
Edge figure | Tetrahedral duoprism |
Vertex figure | Trirectified 7-simplex |
Coxeter group | , [[3,3^{3,3}]] |
Properties | vertex-transitive, facet-transitive |
In 7-dimensional geometry, 1_{33} is a uniform honeycomb, also given by Schlafli symbol {3,3^{3,3}}, and is composed of 1_{32} facets.
Construction
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram.
Removing a node on the end of one of the 3-length branch leaves the 1_{32}, its only facet type.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex, 0_{33}.
The edge figure is determined by removing the ringed nodes of the vertex figure and ringing the neighboring node. This makes the tetrahedral duoprism, {3,3}×{3,3}.
Kissing number
Each vertex of this polytope corresponds to the center of a 6-sphere in a moderately dense sphere packing, in which each sphere is tangent to 70 others; the best known for 7 dimensions (the kissing number) is 126.
Geometric folding
The group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.
Template:CDD | Template:CDD |
{3,3^{3,3}} | {3,3,4,3} |
E_{7}^{*} lattice
contains as a subgroup of index 144.^{[1]} Both and can be seen as affine extension from from different nodes:
The E_{7}^{*} lattice (also called E_{7}^{2})^{[2]} has double the symmetry, represented by [[3,3^{3,3}]]. The Voronoi cell of the E_{7}^{*} lattice is the 1_{32} polytope, and voronoi tessellation the 1_{33} honeycomb.^{[3]} The E_{7}^{*} lattice is constructed by 2 copies of the E_{7} lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A_{7}^{*} lattices, also called A_{7}^{4}:
- Template:CDD + Template:CDD = Template:CDD + Template:CDD + Template:CDD + Template:CDD = dual of Template:CDD.
Related polytopes and honeycombs
The 1_{33} is fourth in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_{3k} series. The final is a noncompact hyperbolic honeycomb, 1_{34. 3) On the lane approach area, make sure that you are able to slide nicely and that the shoe opposite the sliding shoe does not slip when you take your steps.The If you are you looking for more info in regards to Ugg Outlet Store review our own internet site. Chromecast comes with an HDMI extender and a USB power cable as well as a power adaptor. Like the Roku streaming stick, it can be powered either through the USB port on your television or by plugging into an electrical outlet.http://www.restaurantcalcuta.com/outlet/ugg.asp?p=383 http://www.restaurantcalcuta.com/outlet/ugg.asp?p=280 http://www.restaurantcalcuta.com/outlet/ugg.asp?p=310 http://www.restaurantcalcuta.com/outlet/ugg.asp?p=133 http://www.restaurantcalcuta.com/outlet/ugg.asp?p=358 }
See also
Notes
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References
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
- 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]
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- ↑ N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 12: Euclidean symmetry groups, p 177
- ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es7.html
- ↑ The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin