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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 D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} for n ≥ 5, with D ~ 4 {\displaystyle {\tilde {D}}_{4}} = A ~ 4 {\displaystyle {\tilde {A}}_{4}} and for quarter n-cubic honeycombs D ~ 5 {\displaystyle {\tilde {D}}_{5}} = B ~ 5 {\displaystyle {\tilde {B}}_{5}} .

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{4,3,4} or
or

h{4,3}
h3{4,3}
{3,4}×{}
qδ6 quarter 5-cubic honeycomb q{4,3,4}

h{4,3}
h4{4,3}
Rectified 5-cell antiprism
qδ7 quarter 6-cubic honeycomb q{4,3,4}

h{4,3}
h5{4,3} 		{3,3}×{3,3}
qδ8 quarter 7-cubic honeycomb q{4,3,4}

h{4,3}
h6{4,3} 		{3,3}×{3,3}
qδ9 quarter 8-cubic honeycomb q{4,3,4}

h{4,3}
h7{4,3} 		{3,3}×{3,3}
{3,3}×{3,3}
 
qδn quarter n-cubic honeycomb q{4,3,4} ... h{4,3} hn-2{4,3} ...

See also

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, (1.9 Uniform space-fillings)
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, See p318
  • Klitzing, Richard. "1D-8D Euclidean tesselations".
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family A ~ n − 1 {\displaystyle {\tilde {A}}_{n-1}} C ~ n − 1 {\displaystyle {\tilde {C}}_{n-1}} B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}} D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n − 1 {\displaystyle {\tilde {E}}_{n-1}}
E Uniform tiling 0 δ3 hδ3 qδ3 Hexagonal
E Uniform convex honeycomb 0 δ4 hδ4 qδ4
E Uniform 4-honeycomb 0 δ5 hδ5 qδ5 24-cell honeycomb
E Uniform 5-honeycomb 0 δ6 hδ6 qδ6
E Uniform 6-honeycomb 0 δ7 hδ7 qδ7 222
E Uniform 7-honeycomb 0 δ8 hδ8 qδ8 133 • 331
E Uniform 8-honeycomb 0 δ9 hδ9 qδ9 152 • 251 • 521
E Uniform 9-honeycomb 0 δ10 hδ10 qδ10
E Uniform 10-honeycomb 0 δ11 hδ11 qδ11
E Uniform (n-1)-honeycomb 0 δn hδn qδn 1k2 • 2k1 • k21
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