A 3-dimensional golden star
Here Greg Egan has drawn a dodecahedron with 5 tetrahedra in it. This picture is 'left-handed': if you look at where the 5 tetrahedra meet, you'll see they swirl counterclockwise as you go out! If you view this thing in a mirror you'll get a right-handed version.
Putting them together, you get a dodecahedron with 10 tetrahedra in it. You can see it here:http://math.ucr.edu/home/baez/mathematical/dodecahedron_with_10_tetrahedra.gif
The two kinds of tetrahedra are colored yellow and cyan. Regions belonging to both are colored magenta. It's pretty - but it's hard to see the tetrahedra, because they overlap a lot!
A cube has 8 corners. If you take every other corner of the cube, you get the 4 corners of a tetrahedron. But you can do this in 2 ways. If you choose both, you get a cube with 2 tetrahedra in it:http://math.ucr.edu/home/baez/mathematical/cube_with_2_tetrahedra.gif
This picture is from Frederick Goodman's book Algebra: Abstract and Concrete
All this is just the start of a much more elaborate and beautiful story which also involves the golden ratio, the quaternions, and 4-dimensional shapes like the 4-simplex, which has 5 tetrahedral faces, and the 600-cell, which has 600 tetrahedral faces! You can read it here:http://blogs.ams.org/visualinsight/2015/05/01/twin-dodecahedra/
I learned some of this story from Adrian Ocneanu at Penn State University. Greg Egan and I figured out the rest... or most of
the rest. There's an unproven conjecture here, which needs to be true to make the whole story work. Can you prove it?Puzzle:
If you take a regular 4-simplex whose vertices are unit quaternions, with the first equal to 1, can you prove the other 4 vertices generate a free group on 4 elements?
Hmm, I see that this puzzle has been solved by +Ian Agol
and someone else on Mathoverflow:http://mathoverflow.net/questions/204464/do-unit-quaternions-at-vertices-of-a-regular-4-simplex-one-being-1-generate-a
I don't understand the solution yet, because I don't know what a 'Bass-Serre tree' is... but I'll try to learn about this. Math is infinite, there's always more to learn. #geometry #4d