Treebeard's Stumper

Tricky Tetrahedra
Here's a tricky visualization stumper. Imagine a perfect tetrahedron. That's a triangular pyramid where every side including the base is an identical triangle. Now imagine lots of tetrahedra and start to color their sides. If you have four colors to work with, then how many different pyramids can you paint so that a duplicate cannot be found by rotating or turning the pyramids? For help, there's a nifty way to make a tetrahedron from a small envelope at http://tremor.nmt.edu/tetra.html. How about coloring a cube with six colors?
Four different ways of visualizing a tetrahedron: as a wire frame, or an animation (by Rüdiger Appel), or a colored model, or a paper plan ready to fold up. How many unique colorings are possible, with just four colors to work with, if every face has a single color?
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