Math Circle Session 11: More Platonic Solids

Tuesday saw the next installment of our math circle. In the previous session, we had been investigating Platonic solids, and the aim of this session was to build up to a proof that there are exactly five such solids.

Since in the previous session the children had produced a lot of their own definitions, I produced a sheet summarising their definitions as well as standard mathematical definitions of some of the concepts we had been dealing with during the previous session. Below you can see the summary sheet of the children’s definitions, which was laid alongside the standard definitions. Names have been changed for privacy reasons.

Definition: Line-Shape

A line-shape is any shape made by a finite number of straight lines.

(Note the Thomas-Adrian Condition: The number of lines used by the shape must be finite, because an infinite number could lead to “curvy shapes”.)

Definition: Eve-Regular

An Eve-Regular line-shape is one whose sides all have the same length.

Definition: Line-Shape-Shape

A line-shape-shape is any 3D shape made by a finite number of line-shapes.

(Note the Christopher Convexity Test: Place a pencil on the object so it rests on at least two points; if you cannot create a space between the pencil and the object between these two points, the object is convex.)

The first part of the session was then taken up comparing these definitions to more standard ones. In particular, I pointed out that the standard name for Eve-Regular polygons is equilateral, and that polygons that are both equilateral and equiangular are referred to as regular. This approach seemed to work very well, especially with the younger children, who were very excited to see their names in print as part of a definition! Fortunately, I had been able to include the names of all children who had been present in the previous session, as all had been able to contribute.

After reminding the children of the {n,m} notation for the Schläfli symbol of a Platonic solid, I led them towards trying to systematically enumerate the solids. They were able to quickly tell me that the minimum values of n and m must be 3. We then worked our way through all possible Schläfli symbols until they found that m is bounded from above for each n, due to the need for the internal angles to add to a value strictly less than 360 degrees. By physically building each polyhedron as they went, they were able to show that every allowable m does indeed lead to a polyhedron, and we were able to construct all 5, showing that there were indeed only 5 of them in the process: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron.

In case they finished early, my plan was to have them compute the Euler characteristic for each Platonic solid, and then try to build other polyhedra to see whether the value held, however the proof of five solids took up the whole session. After two sessions building polyhedra with Polydron, some of the children may have reached the limits of their interest in the topic for now, so if we do explore Euler characteristics for polyhedra, it will be in a later session.