Tuesday, May 5, 2015

Five Intersecting Tetrahedra

Five Intersecting Tetrahedra by Thomas Hull.  Click for a larger version.

Today I bring one of the most inspiring origami models ever created.  It's Five Intersecting Tetrahedra, and the instructions are online.  It consists of five tetrahedra which interlock in a symmetrical way.  Thomas Hull says the following:
People's first reaction, when being shown the object, is usually to stop and stare at it for a few hours in fascination. Try it!
Don't you think that's odd?  How do we make a symmetrical figure with five tetrahedra?  Five, of all numbers?  A tetrahedron has four faces, four vertices, and six edges.  So where does the five come  in?

If you count all the vertices in the model, there are 5 * 4 = 20.  Now that's a more reasonable number, and it's the same as the number of vertices in a dodecahedron.  Indeed, the vertices of this model are arranged the same way as they are for a dodecahedron.  But still, it's strange that you can divide up the 20 vertices in this particular way.

The Five Intersecting Tetrahedra also has some profound things to say about group theory.

For one, this model single-handedly proves that the A5 group is isomorphic to the group of dodecahedral rotations.  As you can see in the photo, the tetrahedra have five different colors.  Each of the "faces" of the dodecahedron looks like a five pointed star with five colors.  If you look at the way that the colors of the star are ordered, you find that every even permutation of colors is realized somewhere in the model.  So rotating the dodecahedron is equivalent to making an even permutation.

Incidentally, the A5 group is really important in Galois theory.  Certain properties of the group prevent there from being any general algebraic solution to quintic equations!

And here's another thing.  In group theory, there's the idea of a quotient group.  For instance, say that we start with the group of all 3D rotations (the SO(3) group).  Also consider the smaller group of all 3D rotations that leave a regular tetrahedron unchanged (the Td group).  The group of all unique rotations of a tetrahedron is the quotient group SO(3)/Td.

The Five Intersecting Tetrahedra basically consists of five points in the SO(3)/Td group.  I have a hard time wrapping my head around this group, but apparently it is possible to select five points in the group such that they obey some sort of symmetry.

This is an inspiring way to look at modular origami shapes.  Most shapes merely consist of points arranged symmetrically in the SO(3) group, but we can also consider quotient groups.  Upon reflection, I realize that the planar models really exist within RP3 (which is the quotient group SO(3)/Z2), which is what makes those models cool.

Anyway, yeah, maths.