Uniform polytope

2D	3D
Truncated triangle or uniform hexagon, with Coxeter diagram.	Truncated octahedron,
4D	5D
Truncated 16-cell,	Truncated 5-orthoplex,

In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the same must also be true within each lower-dimensional face of the polytope. In two dimensions a stronger definition is used: only the regular polygons are considered as uniform, disallowing polygons that alternate between two different lengths of edges.
This is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures are allowed, which greatly expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs of Euclidean and hyperbolic space to be considered polytopes as well.

Operations

Nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram. Notable exceptions include the great dirhombicosidodecahedron in three dimensions and the grand antiprism in four dimensions.
Equivalently, the Wythoffian polytopes can be generated by applying basic operations to the regular polytopes in that dimension. This approach was first used by Johannes Kepler, and is the basis of the Conway polyhedron notation.

Rectification operators

Regular n-polytopes have n orders of rectification. The zeroth rectification is the original form. The -th rectification is the dual. A rectification reduces edges to vertices, a birectification reduces faces to vertices, a trirectification reduces cells to vertices, a quadirectification reduces 4-faces to vertices, a quintirectification reduced 5-faces to vertices, and so on.
An extended Schläfli symbol can be used for representing rectified forms, with a single subscript:

k-th rectification = t_k = kr.
Truncation operators

Truncation operations that can be applied to regular n-polytopes in any combination. The resulting Coxeter diagram has two ringed nodes, and the operation is named for the distance between them. Truncation cuts vertices, cantellation cuts edges, runcination cuts faces, sterication cut cells. Each higher operation also cuts lower ones too, so a cantellation also truncates vertices.

t_0,1 or t: Truncation - applied to polygons and higher. A truncation removes vertices, and inserts a new facet in place of each former vertex. Faces are truncated, doubling their edges.
:
* There are higher truncations also: bitruncation 't_1,2 or 2t, tritruncation t_2,3 or 3t, quadritruncation t_3,4 or 4t, quintitruncation t_4,5 or 5t, etc.
t_0,2 or rr: Cantellation - applied to polyhedra and higher. It can be seen as rectifying its rectification. A cantellation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically expanded copies of themselves.
:
* There are higher cantellations also: bicantellation t_1,3 or r2r, tricantellation t_2,4 or r3r, quadricantellation t_3,5 or r4r, etc.
* t_0,1,2 or tr: Cantitruncation - applied to polyhedra and higher. It can be seen as truncating its rectification. A cantitruncation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically expanded copies of themselves.
** There are higher cantellations also: bicantitruncation t_1,2,3 or t2r, tricantitruncation t_2,3,4 or t3r, quadricantitruncation t_3,4,5 or t4r, etc.
t_0,3: Runcination - applied to Uniform 4-polytope and higher. Runcination truncates vertices, edges, and faces, replacing them each with new facets. 4-faces are replaced by topologically expanded copies of themselves.
* There are higher runcinations also: biruncination t_1,4, triruncination t_2,5, etc.
t_0,4 or 2r2r: Sterication - applied to Uniform 5-polytopes and higher. Sterication truncates vertices, edges, faces, and cells, replacing each with new facets. 5-faces are replaced by topologically expanded copies of themselves.
* There are higher sterications also: bisterication t_1,5 or 2r3r, tristerication t_2,6 or 2r4r, etc.
* t_0,2,4 or 2t2r: Stericantellation - applied to Uniform 5-polytopes and higher.
** There are higher sterications also: bistericantellation t_1,3,5 or 2t3r, tristericantellation t_2,4,6 or 2t4r, etc.
t_0,5: Pentellation - applied to Uniform 6-polytopes and higher. Pentellation truncates vertices, edges, faces, cells, and 4-faces, replacing each with new facets. 6-faces are replaced by topologically expanded copies of themselves.
* There are also higher pentellations: bipentellation t_1,6, tripentellation t_2,7, etc.
t_0,6 or 3r3r: Hexication - applied to Uniform 7-polytopes and higher. Hexication truncates vertices, edges, faces, cells, 4-faces, and 5-faces, replacing each with new facets. 7-faces are replaced by topologically expanded copies of themselves.
* There are higher hexications also: bihexication: t_1,7 or 3r4r, trihexication: t_2,8 or 3r5r, etc.
* t_0,3,6 or 3t3r: Hexiruncinated - applied to Uniform 7-polytopes and higher.
** There are also higher hexiruncinations: bihexiruncinated: t_1,4,7 or 3t4r, trihexiruncinated: t_2,5,8 or 3t5r, etc.
t_0,7: Heptellation - applied to Uniform 8-polytopes and higher. Heptellation truncates vertices, edges, faces, cells, 4-faces, 5-faces, and 6-faces, replacing each with new facets. 8-faces are replaced by topologically expanded copies of themselves.
* There are higher heptellations also: biheptellation t_1,8, triheptellation t_2,9, etc.
t_0,8 or 4r4r: Octellation - applied to Uniform 9-polytopes and higher.
t_0,9: Ennecation' - applied to Uniform 10-polytopes and higher.

In addition combinations of truncations can be performed which also generate new uniform polytopes. For example, a runcitruncation is a runcination and truncation applied together.
If all truncations are applied at once, the operation can be more generally called an omnitruncation.

Alternation

One special operation, called alternation, removes alternate vertices from a polytope with only even-sided faces. An alternated omnitruncated polytope is called a snub.
The resulting polytopes always can be constructed, and are not generally reflective, and also do not in general have uniform polytope solutions.
The set of polytopes formed by alternating the hypercubes are known as demicubes. In three dimensions, this produces a tetrahedron; in four dimensions, this produces a 16-cell, or demitesseract.

Vertex figure

Uniform polytopes can be constructed from their vertex figure, the arrangement of edges, faces, cells, etc. around each vertex. Uniform polytopes represented by a Coxeter diagram, marking active mirrors by rings, have reflectional symmetry, and can be simply constructed by recursive reflections of the vertex figure.
A smaller number of nonreflectional uniform polytopes have a single vertex figure but are not repeated by simple reflections. Most of these can be represented with operations like alternation of other uniform polytopes.
Vertex figures for single-ringed Coxeter diagrams can be constructed from the diagram by removing the ringed node, and ringing neighboring nodes. Such vertex figures are themselves vertex-transitive.
Multiringed polytopes can be constructed by a slightly more complicated construction process, and their topology is not a uniform polytope. For example, the vertex figure of a truncated regular polytope is a pyramid. An omnitruncated polytope will always have an irregular simplex as its vertex figure.

Circumradius

Uniform polytopes have equal edge-lengths, and all vertices are an equal distance from the center, called the circumradius.
Uniform polytopes whose circumradius is equal to the edge length can be used as vertex figures for uniform honeycombs. For example, the regular hexagon divides into 6 equilateral triangles and is the vertex figure for the regular triangular tiling. Also the cuboctahedron divides into 8 regular tetrahedra and 6 square pyramids, and it is the vertex figure for the alternated cubic honeycomb.