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Quarter hypercubic honeycomb

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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 for n ≥ 5, with = and for quarter n-cubic honeycombs = .[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} ...

See also

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References

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  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".
Space Family / /
E2 Uniform tiling 0[3] δ3 3 3 Hexagonal
E3 Uniform convex honeycomb 0[4] δ4 4 4
E4 Uniform 4-honeycomb 0[5] δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb 0[6] δ6 6 6
E6 Uniform 6-honeycomb 0[7] δ7 7 7 222
E7 Uniform 7-honeycomb 0[8] δ8 8 8 133331
E8 Uniform 8-honeycomb 0[9] δ9 9 9 152251521
E9 Uniform 9-honeycomb 0[10] δ10 10 10
E10 Uniform 10-honeycomb 0[11] δ11 11 11
En-1 Uniform (n-1)-honeycomb 0[n] δn n n 1k22k1k21