Polytope operations

An operation is a kind of transformation that can be applied to a polytope. It can be viewed as a function whose input and output are both polytopes.

This is an incomplete list of operations.

Basic operations

Dual

The cube and octahedron are each others' duals.
This figure can also be viewed as a compound of cube and octahedron, or as a stellation of the cuboctahedron (with red faces extending from the square faces, and yellow faces extending from the triangular faces).

Main article: Dual polytope

The dual of a polytope has vertices corresponding to the original polytope's facets, edges corresponding to the original's ridges (in 3D, edges), faces corresponding to the original's peaks (in 3D, vertices), and so on, culminating in facets corresponding to the original's vertices.

The dual of the dual is the original polytope. A polytope may be its own dual.

Polar set

Main article: Polar set

The polar set is an operation on convex sets in n-dimensional real coordinate space. The polar set functions similarly to the dual, but has a definition specific to the domain of convex geometry, and it is not translation-invariant (and therefore requires an origin). Convex polytopes may produce other convex polytopes under the polar set operation, although they may be degenerate.

Compounding

Main article: Polytope compound

A compound of polytopes consists of multiple polytopes made to overlap with one another, often in a symmetrical way.

The compounds themselves are not typically regarded as polytopes, because they are not fully connected.

Blending

Main article: Blending

Blending can be considered a special case of compounding, where the component polytopes' elements are made to coincide. Duplicate coincident elements are removed to obtain the final result.

Blends can be classified as "inner" or "outer." Inner blends typically involve many copies of a polytope inside the convex hull of a larger polytope, usually creating uniform polytopes with many self-intersections. Outer blends, however, are more similar to the "augmentation" process found in the Johnson solids and other CRF polytopes, and loosely speaking involve gluing two polytopes together at a facet.

Products

Main article: Polytope product

Prism product

Main article: Prism product

Pyramid product (join)

Main article: Pyramid product

Direct sum (tegum product)

Main article: Tegum product

Comb product

Main article: Comb product

Operations associated with regular or uniform polyhedra

Truncation

Main article: Truncation

Truncation is the result of "chopping off" a vertex and replacing it with its associated vertex figure. While it is meaningful to speak of truncating only one vertex, in the study of uniform polytopes, "truncation" usually means applying this operation symmetrically to all vertices.

Rectification

Main article: Rectification

If a polytope has all its vertices truncated, and the truncations cut deep enough to cause all the original polytope's edges to vanish, it is said to be rectified.

Wythoffian operations

Main article: Wythoffian operation

Wythoffian operations can create all the polytopes representable by ringing different nodes of a polytope's Coxeter-Dynkin diagram, as well as some degenerate polytopes for branched, looped, or non-integer-weighted diagrams. These operations are typically done to regular polytopes to produce Wythoffian uniform polytopes, although they may be done to uniform polytopes and produce degenerates (as seen in some of sidtid's other ringings, for example).

Alternation

An alternated octagonal prism, resulting in a square antiprism.

Main article: Alternation

Alternation basically removes every other vertex of a polytope with even-sided faces (note that this only holds true for convex finite polytopes). New edges are then added between remaining vertices that shared an adjacent removed vertex.

Holosnub

Main article: Holosnub

Holosnubbing is a variant of alternation that, in some cases, leads to new uniform polytopes.

Petrie dual

A Petrial cube. Note the different colors on the edges, corresponding to the skew faces.

Main article: Petrial

In 3D, one can make specific paths along the edges of polyhedra that form skew polygons. Several of these skew polygons together can constitute a polyhedron that looks like the original, but has skew faces. Doing this is known as the "Petrie dual" or "Petrial" operation.

Like the dual, the petrial of the petrial is the original polyhedron.

This operation may generalize to other dimensions.

Conjugation

Main article: Conjugate

Mathematically, replacing a polytope's vertex coordinates with their actual numerical conjugates can produce the polytope's conjugate.

Visually, this usually means switching polygons with "star" or "non-star" versions of themselves; pentagons with pentagrams, octagrams with octagons, and so on, producing a polytope with the same element counts but that may look radically different. Triangles and squares are unaffected. Polytopes are not turned into compound forms (such as the hexagram or the {8/2} compound of two squares).

More than two polytopes may be present in a field of conjugates. 4D m-n duoprisms for odd (>3) values of m or n typically have many conjugates, and the uniform polyhedra snid, gosid, gisid, and girsid are all conjugates of one another.

Stellation

A small stellated dodecahedron. The pentagonal faces of the dodecahedron are extended into pentagrams. This is viewed as distinct from a heightened pan-augmented dodecahedron. This figure can also be viewed as a faceting of the icosahedron.

Main article: Stellation

Generally, stellation is an operation that extends the facets of a polytope until they once again intersect one another. Its name comes from the star-like results it produces.

Stellation maintains the facet counts of the original polytope, even if each facet is composed of disjoint pieces (as occurs in some of the 59 stellations of the icosahedron).

Some simple stellations may be mistaken for augmentations of pyramids onto the faces of a polytope.

It can help to specify the dimension of the elements being affected by this operation. Thus, the following names were proposed:

Extending...	Operation is called	Result is called
Edges (of 2D+ figure)	Stellation	"Stellated"
Faces (of 3D+ figure)	Greatening	"Great"
Cells (of 4D+ figure)	Aggrandisement	"Grand"

Faceting

Many uniforms and compounds, or even more complex figures, can be produced by faceting. Shown here is a compound of five tetrahedra, which is a faceting of the dodecahedron. Note that the intersections between created elements are not vertices.

Main article: Faceting

A faceting of a polytope is simply any other polytope with vertices in the exact same locations. Faceting is typically discussed in the context of polytopes defined by symmetry, as many uniform polytopes are facetings of other uniform polytopes.

Blending (skew polytopes)

Main article: Blending (skew polytopes)

Other operations

Chamfering

A chamfered cube. Like an expanded cube (or small rhombicuboctahedron), it can be viewed as a cut along the cube's edges or as a dragging-apart of the cube's faces.

Main article: Chamfering

Chamfering is similar to the Wythoffian operation of expansion. In 3D, it adds rectangular-symmetric hexagons at the old edges instead of rectangles or squares, and in a general dimension, it adds bifrustrums at the old ridges based on the shape of the ridge (a dyad for polyhedra, a regular polygon for polychora, and so on). Facets equivalent to the original vertex figure are not produced (as they are in expansion), but a number of new vertices equal to the amount of vertices of the original are.

The result of chamfering also looks similar to adding short, wide frustrums to the faces of the original polyhedron, as long as one ignores the edges that bisect the hexagons this way.

Since chamfering does not produce uniform polytopes, it is not as widely used as similar Wythoffian operations, although it can produce Goldberg polyhedra and even near-miss CRF polytopes.

Goldberg polyhedron

A Goldberg polyhedron created by repeated chamfering of the cube.

Main article: Goldberg polyhedron

A Goldberg polyhedron is like a repeated chamfering of a polyhedron (usually but not necessarily a Platonic solid), separating the original faces by a great distance and filling in the gaps with many rectangular-symmetric hexagons. They and their duals resemble geodesic domes or insects' compound eyes.

Some Goldberg polyhedra are chiral and cannot be obtained simply by repeated chamfering, though.

Like chamfering, the concept of Goldberg polytopes may be generalized to higher dimensions by using distorted bifrustrums instead of hexagons.

Zonohedrification

Main article: Zonotope § Zonohedrification

The zonohedrification operation generates a new polytope based on certain vectors, which are in turn based on the position of the vertices of the original polytope relative to its center.

The operation is not widely used, likely because it can be confusing due to its several steps and non-bijectivity (the zonohedrified tetrahedron and zonohedrified cube are both rhombic dodecahedra).

Vertex figure

Main article: Vertex figure

The act of finding the vertex figure can be thought of as an operation, because it takes in a polytope ("What is the vertex figure of X?") and returns one ("The vertex figure is Y"), although they are not of the same dimension.

Since a non-vertex-transitive polytope will have multiple vertex figures (except in such cases as the pseudorhombicuboctahedron), the "Vertex figure" function would have multiple outputs for a single input, and thus cannot be considered a proper function.

Vertex figures can provide crude visualizations for higher-dimensional polytopes, although this will help few but an expert.