“Platonic Solids, Archimedean Solids, and Geodesic Spheres”

Jim OlsenWestern 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 Solidshttp://faculty.wiu.edu/JR-Olsen/wiu/B3D/Archimedean/front.html

• 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).

Platonic & Archimedean Solids

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: • Euler’s formula:

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 webpagehttp://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