7-demicubic honeycomb
(No image)
Type	Uniform honeycomb
Family	Alternated hypercube honeycomb
Schläfli symbol	h{4,3,3,3,3,3,4}
Coxeter-Dynkin diagram	Template:CDD or Template:CDD Template:CDD Template:CDD
Facets	{3,3,3,3,3,4} h{4,3,3,3,3,3}
Vertex figure	Rectified heptacross
Coxeter group	${\displaystyle {\tilde {B}}_{7}}$ [4,3,3,3,3,31,1] ${\displaystyle {\tilde {D}}_{7}}$, [31,1,3,3,3,31,1]

The 7-demicubic honeycomb, or demihepteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 7-space. It is constructed as an alternation of the regular 7-cubic honeycomb.

It is composed of two different types of facets. The 7-cubes become alternated into 7-demicubes h{4,3,3,3,3,3} and the alternated vertices create 7-orthoplex {3,3,3,3,3,4} facets.

D7 lattice

The vertex arrangement of the 7-demicubic honeycomb is the D₇ lattice.^[1] The 84 vertices of the rectified 7-orthoplex vertex figure of the 7-demicubic honeycomb reflect the kissing number 84 of this lattice.^[2] The best known is 126, from the E₇ lattice and the 3₃₁ honeycomb.

The DTemplate:Sup sub packing (also called DTemplate:Sup sub) can be constructed by the union of two D₇ lattices. The DTemplate:Sup sub packings form lattices only in even dimensions. The kissing number is 2⁶=64 (2^n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).^[3]

Template:CDD + Template:CDD

The DTemplate:Sup sub lattice (also called DTemplate:Sup sub and CTemplate:Sup sub) can be constructed by the union of all four 7-demicubic lattices:^[4] It is also the 7-dimensional body centered cubic, the union of two 7-cube honeycombs in dual positions.

Template:CDD + Template:CDD + Template:CDD + Template:CDD = Template:CDD + Template:CDD.

The kissing number of the DTemplate:Sup sub lattice is 14 (2n for n≥5) and its Voronoi tessellation is a quadritruncated 7-cubic honeycomb, Template:CDD, containing all with tritruncated 7-orthoplex, Template:CDD Voronoi cells.^[5]

See also

• 8-cubic honeycomb

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
  • 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, 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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Notes

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External links

• Template:GlossaryForHyperspace

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Summary

DescriptionFudge dice tree.svg	English: A tree (made with w:Graphviz describing the process of throwing four Fudge dice and computing their sum. The uppermost row is the starting position (${\displaystyle 0}$ dice thrown), the next row contains the possible states after one die has been thrown, etc. Each node contains the sum so far (above) and the probability of getting to that node (below).
Date	7 October 2007
Source	Own work
Author	w:User:Bromskloss

The following source code was compiled with dot -Tsvg -o "Fudge dice tree.svg" "Fudge dice tree.dot" to produce the image:

digraph G {
	"0_0" [label="0\n1"]
	"0_0" -> "1_-1"
	"0_0" -> "1_0"
	"0_0" -> "1_1"

	"1_-1" [label="-1\n1/3"]
	"1_0" [label="0\n1/3"]
	"1_1" [label="1\n1/3"]
	"1_-1" -> "2_-2"
	"1_-1" -> "2_-1"
	"1_-1" -> "2_0"
	"1_0" -> "2_-1"
	"1_0" -> "2_0"
	"1_0" -> "2_1"
	"1_1" -> "2_0"
	"1_1" -> "2_1"
	"1_1" -> "2_2"

	"2_-2" [label="-2\n1/9"]
	"2_-1" [label="-1\n2/9"]
	"2_0" [label="0\n3/9"]
	"2_1" [label="1\n2/9"]
	"2_2" [label="2\n1/9"]
	"2_-2" -> "3_-3"
	"2_-2" -> "3_-2"
	"2_-2" -> "3_-1"
	"2_-1" -> "3_-2"
	"2_-1" -> "3_-1"
	"2_-1" -> "3_0"
	"2_0" -> "3_-1"
	"2_0" -> "3_0"
	"2_0" -> "3_1"
	"2_1" -> "3_0"
	"2_1" -> "3_1"
	"2_1" -> "3_2"
	"2_2" -> "3_1"
	"2_2" -> "3_2"
	"2_2" -> "3_3"

	"3_-3" [label="-3\n1/27"]
	"3_-2" [label="-2\n3/27"]
	"3_-1" [label="-1\n6/27"]
	"3_0" [label="0\n7/27"]
	"3_1" [label="1\n6/27"]
	"3_2" [label="2\n3/27"]
	"3_3" [label="3\n1/27"]
	"3_-3" -> "4_-4"
	"3_-3" -> "4_-3"
	"3_-3" -> "4_-2"
	"3_-2" -> "4_-3"
	"3_-2" -> "4_-2"
	"3_-2" -> "4_-1"
	"3_-1" -> "4_-2"
	"3_-1" -> "4_-1"
	"3_-1" -> "4_0"
	"3_0" -> "4_-1"
	"3_0" -> "4_0"
	"3_0" -> "4_1"
	"3_1" -> "4_0"
	"3_1" -> "4_1"
	"3_1" -> "4_2"
	"3_2" -> "4_1"
	"3_2" -> "4_2"
	"3_2" -> "4_3"
	"3_3" -> "4_2"
	"3_3" -> "4_3"
	"3_3" -> "4_4"

	"4_-4" [label="-4\n1/81"]
	"4_-3" [label="-3\n4/81"]
	"4_-2" [label="-2\n10/81"]
	"4_-1" [label="-1\n16/81"]
	"4_0" [label="0\n19/81"]
	"4_1" [label="1\n16/81"]
	"4_2" [label="2\n10/81"]
	"4_3" [label="3\n4/81"]
	"4_4" [label="4\n1/81"]
	
}

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