Question

Find pictures of the five Platonic solids. Imagine you are an ant that likes to walk along the edges of solids. For which of these solids is it possible to walk along each edge exactly once, ending at your starting point? Give justification, and the name(s) of the solid(s) solids where such a walk is possible.

Answer 1

Five Platonic Solids are as follows.

1. Tetrahedron (4 Faces) (Equilateral Triangles)

2. Cube (6 faces) (Squares)

3. Octahedron (8 Faces) (Equilateral Triangles)

4. Dodecahedron (12 Faces) (Pentagons)

5. Icosahedron (20 Faces) (Equilateral Triangles)

Following are the pictures of the five platonic solids with their 1-Skeleton (Polyhedral graphs)

To find whether is it possible to walk along each edge exactly once, ending at your starting point, we will use concept of Euler circuit.

Euler Trail:

A trail that traverse every edge of G is called an Euler trail of G.

Euler Circuit:

An Euler Circuit of G is a closed Euler Trail of a graph G.

Theorem:

A non-trivial connected graph G is Eulerian if and olny if every vertex of G has an even degree.

i.e. A non trivial connected graph G contains an Euler Circuit if and only if every vertex of G has an even degree.

Thus to check whether it is possible to walk along each edge exactly once we have to check that every vertex in a graph has an even degree.

Out of the 5 platonic solids only polyhedral graph of Octahedron satisfies above theorem. (degree of every vertex is 4)

Thus Octahedron is a platonic solid where such walk is possible.