1. “Platonic Solids, Archimedean Solids, and Geodesic Spheres”
Jim Olsen
Western Illinois University

2. 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

3. 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.

4. The Icosahedron

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

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

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

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

9. Find data for the truncated octahedron

10. 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

11. 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

12. Geodesic Spheres and Domes
Go right to the website – Pictures!