In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the quarter hypercubic honeycomb (or quarter n-cubic honeycomb) is a dimensional infinite series of
honeycombs, based on the
hypercube honeycomb
In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in -dimensional spaces with the Schläfli symbols and containing the symmetry of Coxeter group (or ) for .
The tessellation is constructed from 4 -hypercube ...
. It is given a
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
q 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
=
.
[Coxeter, Regular and semi-regular honeycoms, 1988, p.318-319]
See also
*
Hypercubic honeycomb
In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in -dimensional spaces with the Schläfli symbols and containing the symmetry of Coxeter group (or ) for .
The tessellation is constructed from 4 -hypercube ...
*
Alternated hypercubic honeycomb
In geometry, the alternated hypercube honeycomb (or demicubic honeycomb) is a dimensional infinite series of Honeycomb (geometry), honeycombs, based on the hypercube honeycomb with an Alternation (geometry), alternation operation. It is given a Sc ...
*
Simplectic honeycomb
*
Truncated simplectic honeycomb In geometry, the cyclotruncated simplicial honeycomb (or cyclotruncated n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the symmetry of the _n affine Coxeter group. It is given a Schläfli symbol t0,1, and is represen ...
*
Omnitruncated simplectic honeycomb
References
*
Coxeter, H.S.M. ''
Regular Polytopes'', (3rd edition, 1973), Dover edition,
*# 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
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'',
ath. 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'',
ath. Zeit. 200 (1988) 3-45See p31
*
{{Honeycombs
Honeycombs (geometry)
Polytopes