“Platonic Solids, Archimedean Solids, and Geodesic Spheres”

Jim Olsen Western Illinois University [email protected]

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

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