“Platonic Solids, Archimedean Solids, and Geodesic Spheres” Jim Olsen Western Illinois University JR-Olsen@wiu.edu

Platonic ~ Archimedean Plato (423 BC –347 BC) Aristotle (384 BC – 322 BC) Euclid (325 and 265 BC) Archimedes (287 BC –212 BC) *all dates are approximate Main website for Archimedean Solids http://faculty.wiu.edu/JR-Olsen/wiu/B3D/Archimedean/front.html

Platonic & Archimedean Solids There are 5 Platonic Solids There are 13 Archimedean Solids For all 18: Each face is regular (= sides and = angles). Therefore, every edge is the same length. Every vertex "is the same." They are highly symmetric (no prisms allowed). The only difference: For the Platonics, only ONE shape is allowed for the faces. For the Achimedeans, more than one shape is used.

The Icosahedron

V, E, and F (Euler’s Formula: V – E + F = 2) Two useful and easy-to-use counting methods for counting edges and vertices.

Formulas Edges from Faces: 2𝐸=𝑁𝐹 𝐸= 𝑁𝐹 2 Vertices from Faces: 𝐾𝑉=𝑁𝐹 𝑉= 𝑁𝐹 𝐾 Euler’s formula: 𝑉−𝐸+𝐹=2

One Goal: Find the V, E, and F for this:

Truncate, Expand, Snubify - http://mathsci.kaist.ac.kr/~drake/tes.html

Find data for the truncated octahedron

How many V, E, and F and Great Circles in the Icosidodecahedron? Note: Each edge of the Icosidodecahedron is the same! Systematic counting Thinking multiplicatively

Interesting/Amazing fact Pugh (1976, p. 25) points out the Archimedean solids are all capable of being circumscribed by a regular tetrahedron so that four of their faces lie on the faces of that tetrahedron. Archimedean Solids webpage http://faculty.wiu.edu/JR-Olsen/wiu/B3D/Archimedean/front.html

Geodesic Spheres and Domes Go right to the website – Pictures! http://faculty.wiu.edu/JR-Olsen/wiu/tea/geodesics/front.htm