Alternation (geometry)

Alternation (geometry)

In geometry, an alternation (also called "partial truncation") is an operation on a polyhedron or tiling that fully truncates alternate vertices. Only even-sided polyhedra can be alternated, for example the zonohedra. Every "2n"-sided face becomes "n"-sided. Square faces disappear into new edges.

An "alternation" of a regular polyhedron or tiling is sometimes labeled by the regular form, prefixed by an h, standing for "half". For example h{4,3} is an alternated cube (creating a tetrahedron), and h{4,4} is an alternated square tiling (still a square tiling).


A "snub" is a related operation. It is an "alternation" applied to an omnitruncated regular polyhedron. An omnitruncated regular polyhedron or tiling always has even-sided faces and so can always be alternated.

For instance the "snub cube" is created in two steps. First it is omnitruncated, creating the great rhombicuboctahedron. Secondly that polyhedron is alternated into a snub cube. You can see from the picture on the right that there are two ways to alternate the vertices, and they are mirror images of each other, creating two chiral forms.

Another example is the uniform antiprisms. A uniform "n"-gonal antiprism can be constructed as an alternation of a "2n"-gonal prism, and the snub of an "n"-edge hosohedron. In the case of prisms both alternated forms are identical.

Non-uniform zonohedra can also be alternated. For instance, the Rhombic triacontahedron can be snubbed into either an icosahedron or a dodecahedron depending on which vertices are removed.


Platonic solid generators

Three forms: regular --> omnitruncated --> snub.

The Coxeter-Dynkin diagrams are given as well. The omnitruncation actives all of the mirrors (ringed). The alternation is shown as rings with "holes".

Higher dimensions

This "alternation" operation applies to higher dimensional polytopes and honeycombs as well, however in general most forms won't have uniform solution. The voids created by the deleted vertices will not in general create uniform facets.

* Honeycombs
*# An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb.
*# An alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb.
* Polychora
*# An alternated truncated 24-cell is the snub 24-cell.
* A hypercube can always be alternated into a uniform demihypercube.
*# Cube --> Tetrahedron (regular)
*# "Tesseract" (8-cell) --> 16-cell (regular)
*# Penteract --> demipenteract (semiregular)
*# Hexeract --> demihexeract (uniform)
*# ...

See also

* Other operators on uniform polytopes:
** Truncation (geometry)
** Rectification (geometry)
** Omnitruncation (geometry)
** Cantellation (geometry)
** Runcination (geometry)
* Conway polyhedral notation


* Coxeter, H.S.M. "Regular Polytopes", (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (pp.154-156 8.6 Partial truncation, or alternation)

External links

*GlossaryForHyperspace | anchor=Alternation | title=Alternation

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Alternation — may refer to:*Alternation (card game) *Alternation (linguistics), a variation in the phonological form of a morpheme *Diathesis alternation *Alternation (algorithms), see Alternating Turing machine *R/N alternation, see Rhotacism * AlterNation, a …   Wikipedia

  • Truncation (geometry) — In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. Truncation in regular polyhedra and tilings When the term applies to truncating platonic solids or regular… …   Wikipedia

  • Greek arithmetic, geometry and harmonics: Thales to Plato — Ian Mueller INTRODUCTION: PROCLUS’ HISTORY OF GEOMETRY In a famous passage in Book VII of the Republic starting at Socrates proposes to inquire about the studies (mathēmata) needed to train the young people who will become leaders of the ideal… …   History of philosophy

  • Symbole de Schläfli — En mathématiques, le symbole de Schläfli est une notation de la forme {p,q,r, …} qui permet de définir les polyèdres réguliers et les tessellations. Cette notation donne un résumé de certaines propriétés importantes d un polytope régulier… …   Wikipédia en Français

  • List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

  • Western architecture — Introduction       history of Western architecture from prehistoric Mediterranean cultures to the present.       The history of Western architecture is marked by a series of new solutions to structural problems. During the period from the… …   Universalium

  • Rhythm — For other uses, see Rhythm (disambiguation). Rhythm, a sequence in time repeated, featured in dance: an early moving picture demonstrates the waltz …   Wikipedia

  • Europe, history of — Introduction       history of European peoples and cultures from prehistoric times to the present. Europe is a more ambiguous term than most geographic expressions. Its etymology is doubtful, as is the physical extent of the area it designates.… …   Universalium

  • HEBREW LANGUAGE — This entry is arranged according to the following scheme: pre biblical biblical the dead sea scrolls mishnaic medieval modern period A detailed table of contents precedes each section. PRE BIBLICAL nature of the evidence the sources phonology… …   Encyclopedia of Judaism

  • Parity of zero — Zero objects, divided into two equal groups Zero is an even number. In other words, its parity the quality of an integer being even or odd is even. Zero fits the definition of even number : it is an integer multiple of 2, namely 0 × 2. As a… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.