symmetry

Transititvity of the pentagonal dodecahedron

The following grasshopper definition is meant to explain the concept of transitivity with the pentagonal dodecahedron (mind that its upper face is missing). Below the video, a description of what is going on.

The first axis to emerge, in red, is a 5-fold axis of rotational symmetry. If we rotate the dodecahedron around this axis, the vertices, edges and faces will move to a position that is indistinguishable from the initial, provided that the angle of rotation is 1/5 of the complete turn, that is, 72º, 72º more and so on until we get to 360º.

The second axis to appear is a 3-fold axis of rotational symmetry. Again, we rotate the dodecahedron around this axis, and it moves into a position indistinguishable from the first at consecutives 1/3 of a complete turn: 120º, 120º more, until a third gives us the complete turn.

With each of the 2-fold axis of rotational symmetry, there are only twp positions in which the dodecahedron moves into a new position that is indistinguishable from the initial: at half a turn (that is, at 180º) and at 360º.

For the regular dodecahedron, the number of 5-fold axis of rotational symmetry is half the number of faces (6); the 3-fold axis is half the number of vertices (10); and the 2-fold axis, half the number of edges (15). That is, 31 axis of rotational symmetry in total: six 5-fold, 10 3-fold and 15 2-fold.

With the necessary adaptations, the same happens with any regular polyhedron, because they are all transitive in their faces, edges and vertices (they are flag-transitive, but I will not address the concept of flags here).