Ratio of vertices to edges is always 2:3 I noticed that the ratio of the vertices to edges on shapes with more than one vertice is always 2:3. Is there any equation or mathematical proof that backs this up?


For example, a cube has 8 vertices and 12 edges, and the ratio of vertices to edges is 8:12, which simplifies to 2:3.

Assuming your observation is about polyhedra, it is only true if the skeleton of the polyhedron is a $3$-regular graph, or alternatively if three edges meet at every vertex. So the octahedron doesn't fit, having $6$ vertices and $12$ edges.
The $2:3$ ratio for polyhedra with three edges meeting at every vertex is an easy consequence of the hand-shaking lemma.
