5demicubic honeycomb
The 5demicube honeycomb (or demipenteractic honeycomb) is a uniform spacefilling tessellation (or honeycomb) in Euclidean 5space. It is constructed as an alternation of the regular 5cube honeycomb.
Demipenteractic honeycomb  

(No image)  
Type  Uniform 5honeycomb 
Family  Alternated hypercubic honeycomb 
Schläfli symbols  h{4,3,3,3,4} h{4,3,3,3^{1,1}} ht_{0,5}{4,3,3,3,4} h{4,3,3,4}h{∞} h{4,3,3^{1,1}}h{∞} ht_{0,4}{4,3,3,4}h{∞} h{4,3,4}h{∞}h{∞} h{4,3^{1,1}}h{∞}h{∞} 
Coxeter diagrams 

Facets  {3,3,3,4} h{4,3,3,3} 
Vertex figure  t_{1}{3,3,3,4} 
Coxeter group  [4,3,3,3^{1,1}] [3^{1,1},3,3^{1,1}] 
It is the first tessellation in the demihypercube honeycomb family which, with all the next ones, is not regular, being composed of two different types of uniform facets. The 5cubes become alternated into 5demicubes h{4,3,3,3} and the alternated vertices create 5orthoplex {3,3,3,4} facets.
D5 lattice
The vertex arrangement of the 5demicubic honeycomb is the D_{5} lattice which is the densest known sphere packing in 5 dimensions.[1] The 40 vertices of the rectified 5orthoplex vertex figure of the 5demicubic honeycomb reflect the kissing number 40 of this lattice.[2]
The D^{+}
_{5} packing (also called D^{2}
_{5}) can be constructed by the union of two D_{5} lattices. The analogous packings form lattices only in even dimensions. The kissing number is 2^{4}=16 (2^{n1} for n<8, 240 for n=8, and 2n(n1) for n>8).[3]
∪
The D^{*}
_{5}[4] lattice (also called D^{4}
_{5} and C^{2}
_{5}) can be constructed by the union of all four 5demicubic lattices:[5] It is also the 5dimensional body centered cubic, the union of two 5cube honeycombs in dual positions.
∪ ∪ ∪ = ∪ .
The kissing number of the D^{*}
_{5} lattice is 10 (2n for n≥5) and it Voronoi tessellation is a tritruncated 5cubic honeycomb,
Symmetry constructions
There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 32 5demicube facets around each vertex.
Coxeter group  Schläfli symbol  CoxeterDynkin diagram  Vertex figure Symmetry 
Facets/verf 

= [3^{1,1},3,3,4] = [1^{+},4,3,3,4]  h{4,3,3,3,4}  [3,3,3,4] 
32: 5demicube 10: 5orthoplex  
= [3^{1,1},3,3^{1,1}] = [1^{+},4,3,3^{1,1}]  h{4,3,3,3^{1,1}}  [3^{2,1,1}] 
16+16: 5demicube 10: 5orthoplex  
2×½ = [[(4,3,3,3,4,2<sup>+</sup>)]]  ht_{0,5}{4,3,3,3,4}  16+8+8: 5demicube 10: 5orthoplex 
Related honeycombs
This honeycomb is one of 20 uniform honeycombs constructed by the Coxeter group, all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 20 permutations are listed with its highest extended symmetry relation:
D5 honeycombs  

Extended symmetry 
Extended diagram 
Extended group 
Honeycombs 
[3^{1,1},3,3^{1,1}]  
<[3^{1,1},3,3^{1,1}]> ↔ [3^{1,1},3,3,4] 
↔ 
×2_{1} =  
[[3^{1,1},3,3^{1,1}]]  ×2_{2}  
<2[3^{1,1},3,3^{1,1}]> ↔ [4,3,3,3,4] 
↔ 
×4_{1} =  
[<2[3^{1,1},3,3^{1,1}]>] ↔ [[4,3,3,3,4]] 
↔ 
×8 = ×2 
See also
Regular and uniform honeycombs in 5space:
References
 http://www.math.rwthaachen.de/~Gabriele.Nebe/LATTICES/D5.html
 Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai
 Conway (1998), p. 119
 http://www.math.rwthaachen.de/~Gabriele.Nebe/LATTICES/Ds5.html
 Conway (1998), p. 120
 Conway (1998), p. 466
 Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0486614808
 pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={3^{1,1},4}, h{4,3,3,4}={3,3,4,3}, ...
 Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0387985859.
External links
Fundamental convex regular and uniform honeycombs in dimensions 29  

Space  Family  / /  
E^{2}  Uniform tiling  {3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  {3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  {3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  {3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  {3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  {3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
E^{8}  Uniform 8honeycomb  {3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
E^{9}  Uniform 9honeycomb  {3^{[10]}}  δ_{10}  hδ_{10}  qδ_{10}  
E^{n1}  Uniform (n1)honeycomb  {3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 