In
geometry, the alternated hypercube honeycomb (or demicubic honeycomb) is a dimensional infinite series of
honeycombs, based on the
hypercube honeycomb with an
alternation operation. 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 ...
h representing the regular form with half the vertices removed and containing the symmetry of
Coxeter group for n ≥ 4. A lower symmetry form
can be created by removing another mirror on an order-4
peak.
[Regular and semi-regular polytopes III, p.318-319]
The alternated hypercube facets become
demihypercubes, and the deleted vertices create new
orthoplex facets. The
vertex figure for honeycombs of this family are
rectified orthoplexes.
These are also named as hδ
n for an (n-1)-dimensional 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 ''h'' prefix: h=; h=, h=
*# 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 ...
, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'',
ath. Zeit. 200 (1988) 3-45
{{Honeycombs
Honeycombs (geometry)
Polytopes