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 qδ_{4} 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]
qδ_{n} | Name | Schläfli symbol |
Coxeter diagrams | Facets | Vertex figure | ||
---|---|---|---|---|---|---|---|
qδ_{3} | quarter square tiling |
q{4,4} | h{4}={2} | { }×{ } | { }×{ } | ||
qδ_{4} | quarter cubic honeycomb |
q{4,3,4} | h{4,3} | h_{2}{4,3} |
Elongated triangular antiprism | ||
qδ_{5} | quarter tesseractic honeycomb | q{4,3^{2},4} | h{4,3^{2}} | h_{3}{4,3^{2}} |
{3,4}×{} | ||
qδ_{6} | quarter 5-cubic honeycomb | q{4,3^{3},4} | h{4,3^{3}} | h_{4}{4,3^{3}} |
Rectified 5-cell antiprism | ||
qδ_{7} | quarter 6-cubic honeycomb | q{4,3^{4},4} | h{4,3^{4}} | h_{5}{4,3^{4}} |
{3,3}×{3,3} | ||
qδ_{8} | quarter 7-cubic honeycomb | q{4,3^{5},4} | h{4,3^{5}} | h_{6}{4,3^{5}} |
{3,3}×{3,3^{1,1}} | ||
qδ_{9} | quarter 8-cubic honeycomb | q{4,3^{6},4} | h{4,3^{6}} | h_{7}{4,3^{6}} |
{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}} | ... |
See also
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
- pp. 122–123, 1973. (The lattice of hypercubes γ_{n} form the cubic honeycombs, δ_{n+1})
- pp. 154–156: Partial truncation or alternation, represented by q prefix
- 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
- Klitzing, Richard. "1D-8D Euclidean tesselations".
Fundamental convex regular and uniform honeycombs in dimensions 2-9 | ||||||
---|---|---|---|---|---|---|
Space | Family | / / | ||||
E^{2} | Uniform tiling | {3^{[3]}} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |
E^{3} | Uniform convex honeycomb | {3^{[4]}} | δ_{4} | hδ_{4} | qδ_{4} | |
E^{4} | Uniform 4-honeycomb | {3^{[5]}} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |
E^{5} | Uniform 5-honeycomb | {3^{[6]}} | δ_{6} | hδ_{6} | qδ_{6} | |
E^{6} | Uniform 6-honeycomb | {3^{[7]}} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |
E^{7} | Uniform 7-honeycomb | {3^{[8]}} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |
E^{8} | Uniform 8-honeycomb | {3^{[9]}} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |
E^{9} | Uniform 9-honeycomb | {3^{[10]}} | δ_{10} | hδ_{10} | qδ_{10} | |
E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |
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