## Summer Research 2005-- Gillian Brunet

### Summary of Research:

This summer I used computer software to design three-dimensional models of all five Platonic solids in Leonardo da Vinci’s solid edge form. I used the computer to capture renderings and learned to operate the three-dimensional printer (also known as a rapid prototyping machine) to create physical models of all five shapes. I did most of my work in Alias Studio 12, but towards the end of the summer I also used Mathematica to construct the models in a different way.

The five Platonic solids are the cube, the dodecahedron, the icosahedron, the octahedron, and the tetrahedron. The different shapes provided different challenges. The dodecahedron has twelve faces, each of which is a regular pentagon. As it was my first shape, much of the difficulty was in deciding how to build the shape. I ended up rotating five rectangular prisms and fitting them together into a pentagonal face. I then duplicated this shape and rotated the copy so that the faces fit at the correct dihedral angle. Next came the cube, which was without complications. The tetrahedron (composed of four faces, each of which is an equilateral triangle) proved more challenging. The angles of the tetrahedron are all acute, which means that the faces stick out through each other if they are not beveled correctly. It was a long process of trial and error, but eventually it worked out. The icosahedron was complicated because of the sheer number of rotations involved. It has twenty faces (again equilateral triangles), each of which must meet at an angle of approximately 138.19°. The rotations become more difficult to compute when the faces are further removed from the original face. Like the tetrahedron, the octahedron had to be beveled to keep edges from sticking out through each other.

Models of Platonic Solids, from left to right: cube, octahedron, icosahedron, and tetraehdron. All were created using Alias Studio 12.

One of the reasons we built the Platonic solids was to build a replica of Kepler’s model of the solar system. This was Diana’s project, but she used my models of the Platonic solids. In order to get the shapes thin enough for the punch bowl, Professor O’Rourke and I worked to create the Platonic solids using Mathematica. With these model designs it is possible to change the thickness of the edges by changing only a single variable in one equation instead of starting over from the beginning (as you would have to do in Alias).

Left: A computerized rendering of the dodecahedron. Center: Image of tetrahedron as designed with Mathematica. Right: Kepler’s model of the solar system.