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“Platonic Solids, Archimedean Solids, and Geodesic Spheres”
Jim OlsenWestern Illinois University
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