# 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 ${\displaystyle {\tilde {D}}_{n-1}}$ 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]

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{4,32,4} 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}×{3,31,1}
qδ9 quarter 8-cubic honeycomb q{4,36,4}

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} ...

## 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 γn form the cubic honeycombs, δn+1)
2. pp. 154–156: Partial truncation or alternation, represented by q prefix
3. p. 296, Table II: Regular honeycombs, δn+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 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 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21