Making Math Visible

Rhombic Triacontahedron Puzzle

The rhombic triacontahedron (RT) is an interesting and beautiful polyhedron that can be the basis for many mathematical explorations. Its close relationship to the dodecahedron and icosahedron lead to the discovery of many rich geometrical interconnections. It can also be used as a non-threatening way to introduce students to combinatorics. The RT is a shape that students will get excited about when they construct the pieces for a physical puzzle that they assemble themselves. Building a large cardboard version leaves the class with a beautiful object students will want to play with whenever they can, furthering their love of geometry.

Here are three workshops based on the RT. In the first, students will create a paper puzzle that assembles inside a clear plastic shell. The second workshop is a continuation in which some of the combinatorics is explained and students add a color aspect to the paper model, thereby increasing the difficulty of the puzzle. In the third workshop, a human-scale cardboard version is created, providing a puzzle that students can work on as a group.  To cut out the cardboard pieces accurately, you will need access to a laser-cutter.  If no laser-cutter is available, the colored paper version can be built instead, as it covers the same mathematical material.

Part I. Paper

Part II. Coloring and Combinatorics

Part III. Cardboard Colossus

The colored RT dissection was first described by Gerhard Kowalewski in his book Der Keplersche K├Ârper und andere Bauspiele, Koehlers, Leipzig, 1938, available in English translation as Construction Games with Kepler's Solid, tr. David Booth, Parker Courtney Press, 2001.  We wrote a paper summarizing these three workshops and some of the underlying mathematics, which you can download from here.  It appeared in the 2016 Bridges Conference on Mathematics and Art.

Note: We welcome feedback on this new workshop, especially variations which work well with your students.  One might start with the large cardboard version, because once students understand it they will be inspired to make their own paper model as accurate as possible.