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Coordinates and other statistics of the 5 Platonic Solids

They are the tetrahedron, cube (or hexahedron), octahedron, dodecahedron and icosahedron.

Their names come from the number of faces (hedron=face in Greek and its plural is hedra). tetra=4, hexa=6, octa=8, dodeca=12 and icosa=20.

Remember that in these pages Phi is 1·61803.. and phi is 1/Phi = Phi-1 = 0·61803... .

The "wire-frame" views are symmetrical plan views of the frame of the object with wire edges and the faces missing, dotted lines being edges that would be hidden by solid faces.

The Tetrahedron

Wireframe Views
tetrahedron views

4 vertices with coordinates:
(1, 1, 1), (1, -1, -1), (-1, 1, -1), (-1, -1, 1)
6 edges of length 2

2, mid point of edge to centre of solid = 1
4 triangular faces,
Surface area = 8

3, volume = 8/3,

The Cube or Hexahedron

Wireframe Views
cube views

8 vertices with coordinates (±1, ±1, ±1), i.e.:
(1, 1, 1), (1, 1, -1), (1, -1, 1), (1, -1, -1),
(-1, 1, 1), (-1, 1, -1), (-1, -1, 1), (-1, -1, -1)
12 sides of length 2, mid point of edge to centre of solid =

2
6 square faces each of area 4, surface area = 24, volume 8

The Octahedron

Wireframe Views
octahedron views

6 vertices with coordinates (±1, 0 0), (0, ±1, 0), (0, 0, ±1) i.e.:
(1, 0, 0), (-1, 0, 0), (0, 1, 0), (0, -1, 0), (0, 0, 1), (0, 0, -1)
12 sides of length

2, mid point of edge to centre of solid = 1/

2
8 triangular faces, surface area = 4

3, volume 4/3

The Dodecahedron

Wireframe Views
dodecahedron views

20 vertices with coordinates (0, ±phi, ±Phi),(±Phi, 0, ±phi),(±phi, ±Phi, 0), (±1, ±1, ±1), i.e.:
(0, phi, Phi), (0, phi, -Phi), (0, -phi, Phi), (0, -phi, -Phi),
(Phi, 0, phi), (Phi, 0, -phi), (-Phi, 0, phi), (-Phi, 0, -phi),
(phi, Phi, 0), (phi, -Phi, 0), (-phi, Phi, 0), (-phi, -Phi, 0),
(1, 1, 1), (1, 1, -1), (1, -1, 1), (1, -1, -1),
(-1, 1, 1), (-1, 1, -1), (-1, -1, 1), (-1, -1, -1)
30 sides of length 2 phi, mid point of edge to centre of solid = Phi, vertex to centre =

3 =

(phi² + Phi²)
12 pentagonal faces,
surface area = 60

(phi/

5) =

(360(5-

5)) = 15

(5 Phi + 10 ), volume = 8 + 4 Phi = 10 + 2

5

The Icosahedron

Wireframe Views
icosahedron views

12 vertices with coordinates (0, ±Phi, ±1), (±1, 0, ±1Phi), (±Phi, ±1, 0) i.e.:
(0, Phi, 1), (0, Phi, -1), (0, -Phi, 1), (0, -Phi, -1),
(1, 0, Phi), (1, 0, -Phi), (-1, 0, Phi), (-1, 0, -Phi),
(Phi, 1, 0), (Phi, -1, 0), (-Phi, 1, 0), (-Phi, -1, 0)
30 sides of length 2, mid point of edge to centre of solid = (1+Phi)/

3
20 triangular faces,
Surface area = 20

3, volume = 20( 1 + Phi)/3 = 20 Phi²/3 = 10 (3 +

5)/3

The Dual of a Solid

There are two more important relationships between the dodecahedron and the icosahedron. First, the mid-points of the faces of the dodecahedron define the points on an icosahedron and the mid-points of the faces of an icosahedron define a dodecahedron. The same is true of the cube and the octahedron. If we try it with a tetrahedron, we just get another tetrahedron. Each is called the dual of the other solid where the number of edges in each pair is the same, but the number of faces of one is the number of points of the other, and vice-versa.

Golden sections in the Dodecahedron, Icosahedron and Octahedron

If we join mid-points of the dodecahedron's faces, we can get three rectangles all at right angles to each other. What's more, they are Golden Rectangles since their edges are in the ratio 1 to Phi.
The same happens if we join the vertices of the icosahedron since it is the dual of the dodecahedron.