Math 2.0

Geometry Definition

Uniform polytope

Builds on Convex polytopeRegular polygonFacetSymmetry groupTransitive actionVertex of a polytope

Definition

Convex uniform polytopes of dimension d≥2d \ge 2 are defined by induction on dd.

  1. A convex polygon is uniform if it is a regular polygon.
  2. For d≥3d \ge 3, a convex polytope P⊆ℝdP \subseteq \R^d of dimension dd is uniform if both of the following hold:

Remarks

Example: the triangular prism whose bases are equilateral triangles of side 11 and whose sides are unit squares is uniform. Its facets are regular polygons, hence uniform. The symmetries of the prism, namely rotations about its axis, the swap of the two bases, and reflections, carry any vertex to any other.

Non-example: the square pyramid with equilateral triangles and a unit square base has regular facets, but it is not uniform. Its apex lies on 44 edges and each base vertex on 33, so no symmetry maps the apex to a base vertex, and the first condition fails. Second non-example: a box [0,1]×[0,1]×[0,2][0,1] \times [0,1] \times [0,2] is vertex-transitive, but its rectangular facets are not regular polygons, so the second condition fails.

Every regular polytope is uniform. In dimension 33 the convex uniform polytopes are the five Platonic solids, the thirteen Archimedean solids, and the prisms and antiprisms over regular polygons. The definition extends to non-convex star polytopes and to uniform tilings, by allowing star polygon faces or unbounded arrangements.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Convex polytope, concept, no evidence levelConvex polytopeconceptFacet, concept, no evidence levelFacetconceptRegular polygon, concept, no evidence levelRegular polygonconceptSymmetry group, concept, no evidence levelSymmetry groupconceptTransitive action, concept, no evidence levelTransitive actionconceptVertex of a polytope, concept, no evidence levelVertex of a polytopeconceptUniform polytope, definition, no evidence levelUniform polytopedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Convex polytope concept
  2. Facet concept
  3. Regular polygon concept
  4. Symmetry group concept
  5. Transitive action concept
  6. Vertex of a polytope concept
  7. Uniform polytope definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Uniform_polytope
About this page
TagPK3RPS6EF4
URIm2://PK3RPS6EF4
Kinddefinition
WritingW0 stub
MSC51M20, 52B15
DefinesUniform polytope
UsesConvex polytope, Regular polygon, Facet, Symmetry group, Transitive action, Vertex of a polytope

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