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“Platonic Solids, Archimedean Solids, and Geodesic Spheres”

“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.

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“Platonic Solids, Archimedean Solids, and Geodesic Spheres”

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

  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 http://faculty.wiu.edu/JR-Olsen/wiu/B3D/Archimedean/front.html

  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: • Vertices from Faces: • Euler’s formula:

  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 http://faculty.wiu.edu/JR-Olsen/wiu/B3D/Archimedean/front.html

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

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