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 alternated hypercube honeycomb (or demicubic 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 ...

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 Peak or The Peak may refer to: Basic meanings Geology * Mountain peak ** Pyramidal peak, a mountaintop that has been sculpted by erosion to form a point Mathematics * Peak hour or rush hour, in traffic congestion * Peak (geometry), an (''n''-3)-di ...

.Regular and semi-regular polytopes III, p.318-319 The alternated hypercube facets become

demihypercube In geometry, demihypercubes (also called ''n-demicubes'', ''n-hemicubes'', and ''half measure polytopes'') are a class of ''n''-polytopes constructed from alternation of an ''n''-hypercube, labeled as ''hγn'' for being ''half'' of the hype ...

s, and the deleted vertices create new orthoplex facets. The

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

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