This is one of the elements I used in Spiral Run (http://www.furaffinity.net/view/32345003/)

I made this cool object that looks pretty neat on its own and can be thrown into anything!

It started with a subdivided icosphere (hence, psychosphere.) I made four shape keys (morph targets) for each type of spikiness, then I used the "select similar" function to select vertices depending on how many other verts they were connected to and scaled those ones out from the center.
I made a shape key like this for the points with 3, 5, and 6 connections (the ones with 4 connections looked kinda bad so I didn't use it)
Then I just generated a sine wave controlling the values of each key and offset them by 1/3

The one on the right has an additional "smooth" modifier that fades in and out over the course of the animation.

Oh, and they're both spinning on their X and Y axes.