Platonic Solid is Uniform Polyhedron

Theorem

Let $P$ be a Platonic solid.

Then $P$ is also uniform.


Proof

Recall the definition of Uniform Polyhedron:

A uniform polyhedron is a polyhedron:

$(1): \quad$ which is isogonal
$(2): \quad$ whose faces are all regular polygons (but not necessarily all of the same type).


Recall the definition of Platonic Solid:

A platonic solid is a convex polyhedron:

$(1): \quad$ whose faces are congruent regular polygons
$(2): \quad$ each of whose vertices is the common vertex of the same number of faces.


From Platonic Solid is Isogonal, $P$ is isogonal.

That all the faces are regular polygons follows a fortiori.

Hence the result.

$\blacksquare$


Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): polyhedron: 1. (plural polyhedra)
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): polyhedron: 1. (plural polyhedra)
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.