Quarter hypercubic honeycomb

In geometry, the quarter hypercubic 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 three quarters of 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 ${\tilde {D}}_{5}$ = ${{\tilde {B}}}_{5}$.^[1]

qδn	Name	Schläfli symbol	Coxeter diagrams	Facets			Vertex figure
qδ3	quarter square tiling	q{4,4}	or or	h{4}={2}	{ }×{ }		{ }×{ }
qδ4	quarter cubic honeycomb	q{4,3,4}	or or	h{4,3}	h2{4,3}		Elongated triangular antiprism
qδ5	quarter tesseractic honeycomb	q{4,3^2,4}	or or	h{4,3^2}	h3{4,3^2}		{3,4}×{}
qδ6	quarter 5-cubic honeycomb	q{4,3^3,4}		h{4,3^3}	h4{4,3^3}		Rectified 5-cell antiprism
qδ7	quarter 6-cubic honeycomb	q{4,3^4,4}		h{4,3^4}	h5{4,3^4}	{3,3}×{3,3}
qδ8	quarter 7-cubic honeycomb	q{4,3^5,4}		h{4,3^5}	h6{4,3^5}	{3,3}×{3,3^1,1}
qδ9	quarter 8-cubic honeycomb	q{4,3^6,4}		h{4,3^6}	h7{4,3^6}	{3,3}×{3,3^2,1} {3,3^1,1}×{3,3^1,1}
qδn	quarter n-cubic honeycomb	q{4,3^n-3,4}	...	h{4,3^n-2}	hn-2{4,3^n-2}	...

See also

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

References

1. Coxeter, Regular and semi-regular honeycoms, 1988, p.318-319

• 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
  • (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
• Klitzing, Richard. "1D-8D Euclidean tesselations".

Fundamental convex regular and uniform honeycombs in dimensions 3–10 (or 2-9)
Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
Uniform tiling	{3^[3]}	δ3	hδ3	qδ3	Hexagonal
Uniform convex honeycomb	{3^[4]}	δ4	hδ4	qδ4
Uniform 5-honeycomb	{3^[5]}	δ5	hδ5	qδ5	24-cell honeycomb
Uniform 6-honeycomb	{3^[6]}	δ6	hδ6	qδ6
Uniform 7-honeycomb	{3^[7]}	δ7	hδ7	qδ7	222
Uniform 8-honeycomb	{3^[8]}	δ8	hδ8	qδ8	133 • 331
Uniform 9-honeycomb	{3^[9]}	δ9	hδ9	qδ9	152 • 251 • 521
Uniform 10-honeycomb	{3^[10]}	δ10	hδ10	qδ10
Uniform n-honeycomb	{3^[n]}	δn	hδn	qδn	1k2 • 2k1 • k21