Math 2.0

Geometry Definition

Regular polytope

Builds on Convex polytopeFace of a convex setSymmetry groupTransitive action

Definition

Let P⊆ℝdP \subseteq \R^d be a convex polytope of dimension dd. A flag of PP is a sequence (F0,F1,…,Fd−1)(F_0, F_1, \dots, F_{d-1}) of faces of PP with F0⊂F1⊂⋯⊂Fd−1F_0 \subset F_1 \subset \cdots \subset F_{d-1} and dim⁡Fi=i\dim F_i = i for each ii. The symmetry group of PP is the group of isometries of ℝd\R^d that map PP onto itself. It acts on the set of flags of PP. The polytope PP is regular if this action is transitive. That is, for any two flags Φ\Phi and Ψ\Psi of PP there is a symmetry gg of PP with g(Φ)=Ψg(\Phi) = \Psi.

Remarks

Example: the cube [0,1]3[0,1]^3 has 88 vertices, each vertex lies on 33 edges, and each edge lies on 22 square faces, so it has 8⋅3⋅2=488 \cdot 3 \cdot 2 = 48 flags. Its symmetry group has order 4848. It can move any vertex to any vertex and then permute the three edges at a vertex freely, so it acts transitively on flags. So the cube is regular.

Non-example: the rectangle [0,2]×[0,1][0,2] \times [0,1] has 88 flags but only 44 symmetries. No symmetry maps a flag whose edge has length 22 to a flag whose edge has length 11, so the action is not transitive.

For a convex polytope, a symmetry that fixes one flag is the identity, so the action of the symmetry group on the flags of a regular polytope is simply transitive. In dimension 22 the regular polytopes are the regular polygons, and in dimension 33 they are the Platonic solids. The same flag condition defines regular star polytopes, which are not convex. It also defines regular abstract polytopes, where the symmetry group is replaced by the automorphism group of a partially ordered set.

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To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Convex polytope, concept, no evidence levelConvex polytopeconceptFace of a convex set, concept, no evidence levelFace of a convex setconceptSymmetry group, concept, no evidence levelSymmetry groupconceptTransitive action, concept, no evidence levelTransitive actionconceptRegular polytope, definition, no evidence levelRegular polytopedefinition
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  1. Convex polytope concept
  2. Face of a convex set concept
  3. Symmetry group concept
  4. Transitive action concept
  5. Regular polytope definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Regular_polytope
About this page
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Kinddefinition
WritingW0 stub
MSC51M20, 52B15
DefinesRegular polytope
UsesConvex polytope, Face of a convex set, Symmetry group, Transitive action

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