1. Beauty, Form and Function: An Exploration of Symmetry
Asset No. 20
PART II
Plane (2D) and Space (3D) Symmetry
Lecture II-6
The Platonic Solids

2. Objectives By the end of this lecture, you will be able to:
see that tessellation in three dimensions creates polyhedra
label polyhedra according to Schäfli notation
recognize the 5 Platonic solids
describe 1 Archimedean solid

3. Polyhedra are closed figures with polygonal faces.
Convex polyhedra have all dihedral angles (angles between faces) less than 180o when viewed from the outside.
Regular polyhedra have all vertices related by symmetry and all faces congruent.
A cube (43) is a regular polyhedron with:
8 vertices
12 edges
6 faces 1 4 2
3 squares (4-gons)
around each vertex 5 3 6

4. The Tetrahedron and Octahedron - Platonic Solids1 2 3 4
A tetrahedron (33) has:
4 vertices
6 edges
4 faces 1 2 3 4 5 6 7 8
A octahedron (34) has:
6 vertices
12 edges
8 faces
The five convex regular polyhedrons - tetrahedron, octahedron, cube, icosahedron, dodecahedron - known collectively as Platonic Solids.

5. The Icosahedron and Dodecahedron
A dodecahedton (53) has:
20 vertices
30 edges
12 faces 1 2 4 6 3 5 7 8 9 10 11 1 12 2 3 4 5 8 10 6 7 9 11 12 14 13
An icosahedron (35) has:
12 vertices
30 edges
20 faces 15 16 17 18 19 20

6. Properties of Platonic Solids33 34 43 53 35
All the Platonic Solids have vertices on the surface of a sphere and are constructed with a single type of polygon - triangle, square, pentagon.
Singular
Plural
tetrahedron* tetrahedra
octahedron octahedra
cube (hexahedron) cubes (hexahedra)
icosahedron icosahedra
dodecahedron dodecahedra
Greek
tetra 4
octa 8
hexa 6
icosa 20
dodeca 12
Triangle
Square
Pentagon

7. Cuboctahedron - An Archimedean Solid
Polyhedra with equivalent (i.e. symmetry related) vertices but more than one kind of regular polygonal face are the semi-regular or Archimedean Solids.
An important Archimedean solid in crystallography is the cuboctahedron. 1 2 3 4 5 7 10 6 9
A cuboctahedron ( ) has:
12 vertices
24 edges
14 faces 8 11 12 13 14
The Euler Rule of Platonic and Archimedean Solids states that V-E+F = 2 which relates the number of vertices (V), edges (E) and faces (F).

8. Summary
There are 5 regular polyhedra known as the Platonic solids - tetrahedron, octahedron, cube, icosahedron, dodecahedron
Regular polyhedra are composed of one type of polygon and every vertex is identical
Semi-regular polyhedra are composed to two or more polygons with every vertex regular
The cuboctahedron is an Archimedean solid