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# Coffee stirrers (Beauty of Three Dimensional Polyhedra Workshop)

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This is the portion of my workshop on Coffee Stirrers and Fuzzy Sticks (given at MathFest Washington DC 8/7/2015).

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### Coffee stirrers (Beauty of Three Dimensional Polyhedra Workshop)

1. 1. Building with Coffee Stirrers and Fuzzy Sticks with a few more properties of the Icosahedron Jim Olsen, Western Illinois University JR-Olsen@wiu.edu Homepage: http://faculty.wiu.edu/JR-Olsen/wiu/ Main webpage: http://wp.me/P6mrPm-E Main Prezi: http://bit.ly/1Uh8ePp Coffee Stirrers and Fuzzy Sticks (#5 of 5)
2. 2. Coffee Stirrers and Fuzzy Sticks Coffee Stirrers (straws with extra reinforcement) are used for the edges of the polyhedron. Fuzzy Sticks (pipe cleaners/chenille stems) are twisted together to make the vertices.
3. 3. Coffee Stirrers and Fuzzy Sticks Advantages Cheap (at WalMart) Versatile (can make anything) Easy to assemble. Colored Fuzzy Sticks make the vertices more obvious. Can see through the polyhedron. Can be taken apart (and put in a bag) for future use. (or) Can be made permanent, with hot glue at the vertices. Disadvantages If the faces are not triangles, then the model is not rigid (can hot glue to make it pretty rigid). Can’t see the faces.
4. 4. Coffee Stirrers and Fuzzy SticksMore things you can build Pyramid – and label the three heights. Geodesic domes and spheres.
5. 5. Coffee Stirrers and Fuzzy Sticks Beautiful Property of the Icosahedron The 12 vertices of the Icosahedron are the vertices of 3 rectangles, which are mutually perpendicular and are Golden Rectangles. And the proof is also beautiful!
6. 6. We will make the Icosahedron with the Three Perpendicular Golden Rectangles Need:  30 coffee stirrers Fuzzy sticks to make 12 degree-5 vertices (joints) Make these first. Assembly:  Slide in the 3 Perpendicular Golden Rectangles. At each vertex insert a fuzzy stick joint. Make Triangles  Coffee Stirrers
7. 7. A Few more Properties of the Icosahedron ◦There are 120 transformations of the icosahedron onto itself. ◦60 of the transformations can be done physically (and 60 are reflections). ◦The 60 physical transformations are isomorphic to A5 (even permutations of 5 items) ◦The 60 physical transformations are all rotations. Let’s count them.
8. 8. Coffee Stirrers Listing of the 60 ‘Physical’ Transformations (Rotations) of the Icosahedron Axis of Rotation Angles of Rotation (non Identity) Calculation Total Number of Rotations Thru opposite vertices 72⁰, 144⁰, 216⁰, 288⁰ 12 2 ∙ 4 24 Thru centers of opposite faces 60⁰, 120⁰ 20 2 ∙ 2 20 Thru midpoints of opposite edges 180⁰ 30 2 ∙ 1 15 Total 60 Identity transformation 1
9. 9. Coffee Stirrers and Fuzzy Sticks (Page with links) Jim Olsen, Western Illinois University JR-Olsen@wiu.edu Homepage: http://faculty.wiu.edu/JR-Olsen/wiu/ Main webpage: http://wp.me/P6mrPm-E Main Prezi: http://bit.ly/1Uh8ePp