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 ${\displaystyle {\tilde {D))_{n-1))$ for n ≥ 5, with ${\displaystyle {\tilde {D))_{4))$ = ${\displaystyle {\tilde {A))_{4))$ and for quarter n-cubic honeycombs ${\displaystyle {\tilde {D))_{5))$ = ${\displaystyle {\tilde {B))_{5))$ .^[1]

qδ_n	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{4,3²,4}	or or	h{4,3²}	h₃{4,3²}		{3,4}×{}
qδ₆	quarter 5-cubic honeycomb	q{4,3³,4}		h{4,3³}	h₄{4,3³}		Rectified 5-cell antiprism
qδ₇	quarter 6-cubic honeycomb	q{4,3⁴,4}		h{4,3⁴}	h₅{4,3⁴}	{3,3}×{3,3}
qδ₈	quarter 7-cubic honeycomb	q{4,3⁵,4}		h{4,3⁵}	h₆{4,3⁵}	{3,3}×{3,3^1,1}
qδ₉	quarter 8-cubic honeycomb	q{4,3⁶,4}		h{4,3⁶}	h₇{4,3⁶}	{3,3}×{3,3^2,1} {3,3^1,1}×{3,3^1,1}

qδ_n	quarter n-cubic honeycomb	q{4,3^n-3,4}	...	h{4,3^n-2}	h_n-2{4,3^n-2}	...

References

^ 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 [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]
Klitzing, Richard. "1D-8D Euclidean tesselations".

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A))_{n-1))$	${\displaystyle {\tilde {C))_{n-1))$	${\displaystyle {\tilde {B))_{n-1))$	${\displaystyle {\tilde {D))_{n-1))$	${\displaystyle {\tilde {G))_{2))$ / ${\displaystyle {\tilde {F))_{4))$ / ${\displaystyle {\tilde {E))_{n-1))$
E²	Uniform tiling	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	{3^[10]}	δ₁₀	hδ₁₀	qδ₁₀
E¹⁰	Uniform 10-honeycomb	{3^[11]}	δ₁₁	hδ₁₁	qδ₁₁
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

See also

References