# 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]

n Name Schläfli
symbol
Coxeter diagrams Facets Vertex figure
3
quarter square tiling
q{4,4} or

or

h{4}={2} { }×{ }
{ }×{ }
4
quarter cubic honeycomb
q{4,3,4} or
or

h{4,3}

h2{4,3}

Elongated
triangular antiprism
5 quarter tesseractic honeycomb q{4,32,4} or
or

h{4,32}

h3{4,32}

{3,4}×{}
6 quarter 5-cubic honeycomb q{4,33,4}

h{4,33}

h4{4,33}

Rectified 5-cell antiprism
7 quarter 6-cubic honeycomb q{4,34,4}

h{4,34}

h5{4,34}
{3,3}×{3,3}
8 quarter 7-cubic honeycomb q{4,35,4}

h{4,35}

h6{4,35}
{3,3}×{3,31,1}
9 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}

n 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 [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".
Fundamental convex regular and uniform honeycombs in dimensions 2-9
${\displaystyle {\tilde {A}}_{n-1}}$  ${\displaystyle {\tilde {C}}_{n-1}}$  ${\displaystyle {\tilde {B}}_{n-1}}$  ${\displaystyle {\tilde {D}}_{n-1}}$  ${\displaystyle {\tilde {G}}_{2}}$  / ${\displaystyle {\tilde {F}}_{4}}$  / ${\displaystyle {\tilde {E}}_{n-1}}$
{3[3]} δ3 3 3 Hexagonal
{3[4]} δ4 4 4
{3[5]} δ5 5 5 24-cell honeycomb
{3[6]} δ6 6 6
{3[7]} δ7 7 7 222
{3[8]} δ8 8 8 133331
{3[9]} δ9 9 9 152251521
{3[10]} δ10 10 10
{3[n]} δn n n 1k22k1k21