Jim Olsen Western Illinois University JROlsenwiuedu Platonic Archimedean Plato 423 BC 347 BC Aristotle 384 BC 322 BC Euclid 325 and 265 BC Archimedes 287 BC ID: 495066
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“Platonic Solids, Archimedean Solids, and Geodesic Spheres”
Jim Olsen
Western Illinois UniversityJR-Olsen@wiu.eduSlide2
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
approximateMain website for Archimedean Solidshttp://faculty.wiu.edu/JR-Olsen/wiu/B3D/Archimedean/front.htmlSlide3
There are 5 Platonic SolidsThere are 13 Archimedean SolidsFor 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.Slide4
The IcosahedronSlide5
V, E, and F(Euler’s Formula: V – E + F = 2)
Two useful and easy-to-use counting methods for counting edges and vertices
.Slide6
Formulas Edges from Faces:
Vertices from Faces:
Euler’s formula:
Slide7
One Goal: Find the V, E, and F for this:Slide8
Truncate, Expand, Snubify - http://mathsci.kaist.ac.kr/~drake/tes.htmlSlide9
Find data for the truncated octahedronSlide10
How many V, E, and F and Great Circles in the Icosidodecahedron?
Note:
Each edge
of the Icosidodecahedron is the same!Systematic countingThinking multiplicativelySlide11
Interesting/Amazing factPugh (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 Slide12
Geodesic Spheres and DomesGo right to the website – Pictures!
http://faculty.wiu.edu/JR-Olsen/wiu/tea/geodesics/front.htm