Crystal Movement

If we plot the vertices of the octahedron as nodes of the graph, we will acquire $6$ nodes, each with a degree of $4$ , and the Eulerian path can be drawn as shown below:

In theory, if all nodes have even degrees (4 connections to other nodes in this case), the graph will be an Eulerian cycle : the path can be drawn without repeating the edges and returning to the starting point.

It's because each corner has 4 sides coming off it, which is an even number. It doesn't work with a cube because a cube has 3 sides coming off each corner so the lines end at the corners.

For me, it is a logic... we can color each edges by rotating your screen and we can color it downward. Lol

If we mark the vertexes from the given figure then we will see; Vertex a has 4 degrees, Vertex b has 4 degrees, Vertex c has 4 degrees, Vertex d has 4 degrees, Vertex e has 4 degrees, Vertex f has 4 degrees.

As all of the vertexes has even no of degrees, then according to Euler's law it has both Eulerian path & circuit. So, we can color the edges of the octahedron above without lifting the pencil nor coloring the same edge more than once