When it comes to the study and appreciation of polyhedra, there is nothing as instructive and satisfying as holding an actual model in the hands. A great deal of understanding about the symmetries of the icosahedron can be had by simply spinning one on your finger tips!

The first and most important step for producing high quality models is in the construction of accurate templates. By using a template pieces can be made quickly, and yet each piece will be highly uniform.

On stiff cardboard draw an equilateral triangle with a side of, say, 2 inches. Make your drawing as accurate as possible, and mark the vertices by pushing a pin through them. Draw a border of about 2/10 inch around the triangle and then cut it out (the border will become tabs that you will glue together to join two pieces).

Now use your template to draw several triangles on good quality art paper, and then cut these out. Stack 3 or 4 triangles with your template on top, and push a needle through the stack at each of the vertices.

Using a straight edge and needle, score the paper triangles between the holes. Finally, cut off the corners, and fold the resulting tabs down along the scored lines.

Glue the tabs of four triangles together and you have a tetrahedron, your first polyhedron model! (I assure you, if you go through the process outlined above just once it will all become clear - just do it!)

Ok, having completed the above exercise, you are ready to construct templates for the square and pentagon, making sure that they each have the exact same edge length as your triangle. From these three shapes you can construct a large number of highly interesting polyhedra, such as the cuboctahedron, snub cube, and rhombicosidodecahedron!

• When I cut off the corners to make the tabs I often find it necessary to make two cuts, so that when the tabs are folded down and joined they do not touch each other inside the model. All Triangles require this.

• If the angle between adjacent faces is small, the fold angle of the tabs is large. In this case I will sometimes cut off one of the tabs and glue the remaining tab to the back of the joining face.

  This technique will produce a more pleasing join in some cases. I used to always use a single tab to join two faces, but the edge where the tab was cut off was not always straight, and sometimes the edges did not line up as well as when two tabs are used.

• I have found tweezers or needle-nose pliers to be very helpful for reaching inside a model to squeeze two tabs together. It gets pretty tight near the end, when you are closing a model!

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  Materials and References

  The board from the back of a pad of paper is great for making templates out of. Poster or art board also works well.

  For my scoring tool I use one of those needles with a wooden handle found in disection kits. I also use a large needle pressed into a wine cork.

  You definitely want to use color in your models, so find a good quality art paper. Basic contruction paper is not to be used, as it tends to be soft and fades badly. I use Strathmore Colored Art Paper, available in individual sheets as well as in packs.

  I use Elmer's School Glue, as it does paper well and comes off of everything with water. I also spray my models with a clear acrylic to seal the paper from moisture and dirt.

  The best book (IMHO) is Polyhedron Models by Magnus J. Wenninger, Cambridge University Press. You may also enjoy Shapes, Space, and Symmetry by Alan Holden, Columbia University Press.

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  There is a set of polyhedral models which I constructed on display in the Department of Mathematics at the California State University Chico. These models are dedicated to the memory of my father, Thomas O. Gettys, and to that of my friend Michael J. Dixon.

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  Email: Tom Gettys (polypages@comcast.net)

  Page created : June 20, 1995