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

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

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. 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. 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. 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. 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