The Platonic Solids

A polyhedron is said to be regular if

All the faces are the same regular polygon

The same number of faces meet at each vertex

The polyhedron is convex.

Theorem: There are only 5 regular polyhedra

Proof:

The faces can be equilateral triangles, squares or regular pentagons.

If the faces are equilateral triangles, 3 could meet at a vertex, 4 could meet at a vertex or 5 could meet at a vertex. Therefore, there are 3 regular polyhedra with triangular faces.

If the faces are squares, there is only one possibility: 3 meet at a vertex. Therefore, there is only one regular polyhedron with square faces.

If the faces are regular pentagons, there is only one possibility: 3 meet at a vertex. Therefore, there is only one regular polyhedron with pentagonal faces.

There can be no regular polyhedron with hexagonal faces.

Therefore, there are only 5 regular or Platonic Solids.

The dual of a polyhedron with V vertices and F faces is a polyhedron with F vertices and V faces.

Constructing the Platonics and Their Duals Using ZomeTools and SketchUp

  1. Introduction to SketchUp
  1. Build a Cube using the Push-Pull Tool
  1. Inscribe in the Cube an Octahedron
  1. Build an Icosahedron using ZomeTools
  1. Build an Icosahedron using SketchUp
  1. Build a dodecahedron using ZomeTools
  1. Build a dodecahedron using SketchUp

Fill in the table below

Name / Schafli / V / E / F / TOA / Dihedral / Dual
Tetrahedron
Octahedron
Hexahedron
Dodecahedron
Icosahedron

Nets for the Platonic Solids

Duals

Weisstein, Eric W. "Platonic Solid." From MathWorld--A Wolfram Web Resource.

Equations for the Platonic Solid

Euler’s Theorem: In 3 space, V –E + F = 2

If a Platonic solid has faces with n edges and p faces meet at a vertex then

nF = 2E

pV = 2E

V – E + F = 2

Polyhedron / Equations
Tetrahedron
Octahedron
Hexahedron
Dodecahedron
Icosahedron

Finding Dihedral Angles

Symmetry Operations

Tetrahedron

Hexahedron/Octahedron

Dodecahedron/Icosahedron