Building with Coffee Stirrersand 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)

Coffee Stirrers and Fuzzy SticksCoffee 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.

Coffee Stirrers and Fuzzy SticksAdvantagesCheap (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.

DisadvantagesIf the faces are not triangles, then the model is not rigid (can hot glue to

make it pretty rigid).Can’t see the faces.

Coffee Stirrers and Fuzzy SticksMore things you can buildPyramid – and label the three heights.Geodesic domes and spheres.

Coffee Stirrers and Fuzzy Sticks

Beautiful Property of the IcosahedronThe 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!

We will make the Icosahedron withthe Three Perpendicular Golden RectanglesNeed:

30 coffee stirrersFuzzy 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

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.

Coffee StirrersListing 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

24

Thru centers of opposite faces

60⁰, 120⁰ 20

Thru midpoints of opposite edges

180⁰ 15

Total 60

Identity transformation 1

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