In geometry, an alternation or ''partial truncation'', is an operation on a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all o ...

, tiling, or higher dimensional polytope that removes alternate vertices.Coxeter, Regular polytopes, pp. 154–156 8.6 Partial truncation, or alternation Coxeter labels an ''alternation'' by a prefixed ''h'', standing for ''hemi'' or ''half''. Because alternation reduces all polygon faces to half as many sides, it can only be applied to polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is usually reduced to a single edge. More generally any vertex-uniform polyhedron or tiling with a vertex configuration consisting of all even-numbered elements can be ''alternated''. For example, the alternation of a vertex figure with ''2a.2b.2c'' is ''a.3.b.3.c.3'' where the three is the number of elements in this vertex figure. A special case is square faces whose order divides in half into degenerate digons. So for example, the cube ''4.4.4'' is alternated as ''2.3.2.3.2.3'' which is reduced to 3.3.3, being the

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...

, and all the 6 edges of the tetrahedra can also be seen as the degenerate faces of the original cube.

Snub

A snub (in Coxeter's terminology) can be seen as an alternation of a truncated regular or truncated quasiregular polyhedron. In general a polyhedron can be snubbed if its truncation has only even-sided faces. All truncated rectified polyhedra can be snubbed, not just from regular polyhedra. The snub square antiprism is an example of a general snub, and can be represented by ss, with the square antiprism, s.

Alternated polytopes

This ''alternation'' operation applies to higher-dimensional polytopes and honeycombs as well, but in general most of the results of this operation will not be uniform. The voids created by the deleted vertices will not in general create uniform facets, and there are typically not enough degrees of freedom to allow an appropriate rescaling of the new edges. Exceptions do exist, however, such as the derivation of the

snub 24-cell In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular face ...

from the

truncated 24-cell In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell. There are two degrees of truncations, including a bitruncation. Truncated 24-cell The truncated 24- ...

. Examples: * Honeycombs *# An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb. *# An alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb. * 4-polytope *# An alternated

is the

. * 4-honeycombs: *# An alternated truncated 24-cell honeycomb is the snub 24-cell honeycomb. * A

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...

can always be alternated into a uniform demihypercube. *#

Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

→

Tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...

(regular) *#* → *# ''Tesseract'' (

8-cell In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...

) → 16-cell (regular) *#* → *#

Penteract In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tessera ...

→

demipenteract In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' ( penteract) with alternated vertices removed. It was discovered by Thorold Gosset. Since it was the only semiregular ...

(semiregular) *#

Hexeract In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It can ...

→ demihexeract (uniform) *# ...

Altered polyhedra

Coxeter also used the operator ''a'', which contains both halves, so retains the original symmetry. For even-sided regular polyhedra, a represents a

compound polyhedron In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connect ...

with two opposite copies of h. For odd-sided, greater than 3, regular polyhedra a, becomes a star polyhedron. Norman Johnson extended the use of the altered operator ''a'', ''b'' for blended, and ''c'' for converted, as , , and respectively. The compound polyhedron known as the

stellated octahedron The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depic ...

can be represented by a (an altered

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

), and ,

. The star polyhedron known as the small ditrigonal icosidodecahedron can be represented by a (an altered dodecahedron), and ,

. Here all the pentagons have been alternated into pentagrams, and triangles have been inserted to take up the resulting free edges. The star polyhedron known as the great ditrigonal icosidodecahedron can be represented by a (an altered great stellated dodecahedron), and ,

. Here all the pentagrams have been alternated back into pentagons, and triangles have been inserted to take up the resulting free edges.

Alternate truncations

A similar operation can truncate alternate vertices, rather than just removing them. Below is a set of polyhedra that can be generated from the

Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan so ...

s. These have two types of vertices which can be alternately truncated. Truncating the "higher order" vertices and both vertex types produce these forms:

References

* Coxeter, H.S.M. '' Regular Polytopes'', (3rd edition, 1973), Dover edition, * Norman Johnson ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 * * Richard Klitzing, ''Snubs, alternated facetings, and Stott-Coxeter-Dynkin diagrams'', Symmetry: Culture and Science, Vol. 21, No.4, 329-344, (2010

External links