Editing Regular polyhedron (section)

== Characteristics ==
===Equivalent properties===

The property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition:

*The vertices of a convex regular polyhedron all lie on a [[sphere]].
*All the [[dihedral angle]]s of the polyhedron are equal
*All the [[vertex figure]]s of the polyhedron are [[regular polygon]]s.
*All the [[solid angle]]s of the polyhedron are congruent.<ref>{{cite book
  | last = Cromwell | first = Peter R. | title = Polyhedra | title-link = Polyhedra (book) | publisher = Cambridge University Press | year = 1997 | page = 77 | isbn = 0-521-66405-5 }}</ref>

===Concentric spheres===

A convex regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre:
* An [[insphere]], tangent to all faces.
* An intersphere or [[midsphere]], tangent to all edges.
* A [[circumsphere]], tangent to all vertices.

===Symmetry===

The regular polyhedra are the most [[symmetry|symmetrical]] of all the polyhedra. They lie in just three [[symmetry group]]s, which are named after the Platonic solids:
*Tetrahedral
*Octahedral (or cubic)
*Icosahedral (or dodecahedral)

Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry.

===Euler characteristic===

The five Platonic solids have an [[Euler characteristic]] of 2. This simply reflects that the surface is a topological 2-sphere, and so is also true, for example, of
any polyhedron which is star-shaped with respect to some interior point.

===Interior points===

The sum of the distances from any point in the interior of a regular polyhedron to the sides is independent of the location of the point (this is an extension of [[Viviani's theorem]].) However, the converse does not hold, not even for [[tetrahedra]].<ref>Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", ''[[The College Mathematics Journal]]'' 37(5), 2006, pp. 390–391.</ref>