Euler's Gem Now

Ensuring 3D meshes are "manifold" (water-tight).

It leads to the concept of the Euler Characteristic , which helps mathematicians classify surfaces in higher dimensions. Euler's Gem

The "2" in the formula represents the "internal" connectivity and the "external" face that was removed. Ensuring 3D meshes are "manifold" (water-tight)

A common way to visualize the proof is by "flattening" a polyhedron: Euler's Gem

Remove one face of a polyhedron (like a cube) and stretch the remaining shell flat onto a plane.