Trying to be Edgy

Maths and making stuff! I spent some time this weekend folding up the Dodecahedron pictured below. Instructions on how to build your own origami dodecahedron come from MathCraftNZ, great site with some high quality and fun math related builds.

dodecahedron

A Dodecahedron is a member in the platonic solid family. A platonic solid is a 3D shape where each face is the same regular polygon and the same number of polygons meet at each vertex. A dodecahedron has 12 faces, 20 vertices and 30 edges. The 12 faces are in the shape of a pentagon, and 3 pentagons meet at each vertex.

There are only five platonic solids:

  1. Tetrahedron
  2. Cube
  3. Octahedron
  4. Dodecahedron
  5. Icosahedron

A proof of the fact that there can only be five platonic solids is offered by Euler’s Formula (noting this is a different formula than Euler’s Formula for Complex Numbers).

Euler’s Formula states:
\(F + V - E = 2\)
\(F\) is the Number of Faces
\(V\) is the Number of Vertices
\(E\) is the Number of Edges

An easy example is the cube. For the cube:
\(F = 6, V = 8, E = 12\)
\(F + V – E = 2\)
\(6 + 8 – 12 = 2\)

Let’s say that we wanted to build a platonic solid out of a polygon with \(N\) number of edges. Then \(NF = 2E\), because in the fully assembled 3D shape, every edge is shared by two faces, meaning each edge is counted twice. \(R\) number of edges are going to meet at each of the vertices as we build up this shape, giving, \(RV = 2E\).

exploded_cube

Exploded Cube

If we put the two equations together, rearrange and substitute back into Euler’s Formula:

\(NF = 2E\)
\(F = \dfrac{2E}{N}\)
\(RV = 2E\)
\(V = \dfrac{2E}{R}\)
\(F + V - E = 2\)

\(\dfrac{2E}{N} + \dfrac{2E}{R} {–E} = 2\)
\(\dfrac{1}{N} + \dfrac{1}{R} {-\dfrac{1}{2}} = \dfrac{1}{E}\)

Looking at the last term, we note that the number of Edges can’t be zero, which yields the equation that proves there are only five platonic solids.

\(\dfrac{1}{N} + \dfrac{1}{R} - \dfrac{1}{2} > 0\)
\(\dfrac{1}{N} + \dfrac{1}{R} > \dfrac{1}{2}\)

Plugging in any positive integer values for \(N\) and \(R\) shows that only five combinations of \(N\) and \(R\) are possible! Those combinations of \(N\) (Number of Edges) and \(R\) (Number of Edges that meet at each of the vertices), are the combinations for the Tetrahedron, Cube, Octahedron, Dodecahedron, and Icosahedron.

No coffee required to make these regular solids! Thanks for reading! Feel free to share pictures of your own Dodecahedron builds, comments or questions.