Building with Octahedra and Tetrahedra

I led a MoMath workshop for Math for America in which teachers and students created structures based on octahedra and tetrahedra.




I began by showing some magnetic polyhedral building blocks and explaining aspects of the "oct-tet" lattice (a.k.a. "FCC" lattice).




Then participants started playing with the blocks, to get a feel for the possible structures.




The blocks are fun to play with, so I wish I had lots more of them.  But this was just the warm-up.




The main activity involved a kit with nodes and struts, designed for making the oct-tet lattice.




These laser-cut parts snap together to make the nodes.




It is something of a puzzle to assemble them.




Once the nodes are assembled, they join with struts to make structures such as this tetrahedron.




And then, to make larger structures, you need to make more nodes.




Here's the start of a square pyramid, which grows into...




...an octahedron.




Twist ties can be used to hold the struts to the nodes.




Once you have the idea, you want to make larger and larger structures.




So, keep building!




The plan here is to surround an octahedron with eight tetrahedra.




It looks so cool that other groups make one also.




I explain that the structure is sometimes called the "stella octangula," Latin for "eight-pointed star".




Or you can build in a more free-form manner.
Here the base is a tetrahedron, so there is a 3-fold axis vertical.




In this structure, a 4-fold axis is vertical.




With some careful thought, ...




...and careful modifications, ...




The result is a stack of stella octangulas, which is the design of one of the towers in Escher's Planaria. 




We had some snacks, and then built this larger structure with an archway.