Hexagonal tiling honeycomb

In geometry, the quarter hypercube honeycomb (or quarter n-cubic honeycomb) is a dimensional infinite series of honeycombs, based on the hypercube honeycomb. It is given a Schläfli symbol q{4,3...3,4} or Coxeter symbol qδ₄ representing the regular form with quarter the vertices removed and containing the symmetry of Coxeter group ${\tilde{D}}_{n-1}$ for n ≥ 5, with ${\tilde{D}}_4$ = ${\tilde{A}}_4$ and for quarter n-cubic honeycombs ${\tilde{D}}_5$ = ${\tilde{B}}_5$.

qδn	Name	Schläfli symbol	Coxeter diagram	Facets			Vertex figure
qδ3	File:Square tiling uniform coloring 1.png quarter square tiling	q{4,4}	Template:CDD Template:CDD	h{4}={2}	{ }×{ }		File:Regular polygon 4 annotated.svg { }×{ }
qδ4	File:Bitruncated alternated cubic tiling.png quarter cubic honeycomb	q{4,3,4}	Template:CDD Template:CDD	File:Tetrahedron.png h{4,3}	File:Truncated tetrahedron.png h2{4,3}		Elongated triangular antiprism
qδ5	quarter tesseractic honeycomb	q{4,32,4}	Template:CDD = Template:CDD Template:CDD	h{4,32}	h3{4,32}		{3,4}×{}
qδ6	quarter 5-cubic honeycomb	q{4,33,4}	Template:CDD Template:CDD	h{4,33}	h4{4,33}		Rectified 5-cell antiprism
qδ7	quarter 6-cubic honeycomb	q{4,34,4}	Template:CDD Template:CDD	h{4,34}	h5{4,34}	{3,3}×{3,3}
qδ8	quarter 7-cubic honeycomb	q{4,35,4}	Template:CDD Template:CDD	h{4,35}	h6{4,35}	{3,3}×{3,31,1}
qδ9	quarter 8-cubic honeycomb	q{4,36,4}	Template:CDD Template:CDD	h{4,36}	h7{4,36}	{3,3}×{3,32,1} {3,31,1}×{3,31,1}
qδn	quarter n-cubic honeycomb	q{4,3n-3,4}	...	h{4,3n-2}	hn-2{4,3n-2}	...

See also

• Hypercubic honeycomb
• Alternated hypercubic honeycomb
• Simplectic honeycomb
• Truncated simplectic honeycomb
• Omnitruncated simplectic honeycomb

References

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• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
  1. pp. 122-123, 1973. (The lattice of hypercubes γ_n form the cubic honeycombs, δ_n+1)
  2. pp. 154-156: Partial truncation or alternation, represented by q prefix
  3. p. 296, Table II: Regular honeycombs, δ_n+1
• 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 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] See p318 [2]
• Template:KlitzingPolytopes

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