These and other activities included in our book, Math Dance with Dr. Schaffer and Mr. Stern, now available from our Book Shop
(http://www.mathdance.org/store.html)
String Polyhedra
How to make 3-dimensional shapes out of a loop of string.

You will need:
- Four people.
- A loop of string at least 12 feet in circumference to make an octahedron with an edge length of 1 foot.

In this sequence four people make a triangle that turns into an tetrahedron, then an octahedron, then a cube. In the first five steps three people all do the same thing, moving symmetrically. Only in the last step is the fourth person needed. This figure is much easier to do than to read about, so don't be intimidated by the instructions.

We prefer thick nylon string, because it is easier to see and stronger than thin twine. For a smooth join, you can melt the ends together with a candle or lighter rather than tie them together.


1. Big triangle. Three people stand in a circle facing each other. Make a big horizontal triangle by hooking the string from underneath with your right thumb. Your thumb should be inside the triangle, and your fingers outside the triangle. Pull the string taut so the triangle has straight edges, all the same length.


2. Left hand ring. All three people take your left hand and form a ring with your thumb and forefinger that gathers both strings, right next to your right thumb. Slide your left hand halfway toward the center, shrinking the triangle to about half its size.


3. Tetrahedron. Leave your left hand where it is and swing your right thumb up and towards the center, so that all three thumbs join in the middle above the triangle. Be sure to keep the string taut at all times. This makes a tetrahedron. Note that three of the edges have doubled strings, whereas in the construction described in Scientific American only two of the edges were doubled.


4. Hook right index finger. Wiggle your right index finger so you all pay attention to it. Now hook your right index finger around one (not both!) of the strings on the thumb of the person to your right. Touch your right thumb and index finger so they make a ring, similar to the ring made with your left thumb and forefinger. Be sure the string can slide freely through your fingers; don't pinch it tight.


5. Octahedron. Pull your right hands slightly apart, opening a triangle between them. Slide your hands around until all the lengths of string are even, and you will notice that you have made an octahedron. Can you count how many triangles this shape has?


6. Cube. Notice that there are two triangles parallel to the floor: one on top and one on the bottom. To make a cube from the octahedron, the fourth person reaches one hand above the top triangle and the other hand below the bottom triangle. With the top hand, gather the midpoints of the three sides of the top triangle to a point. With the bottom hand, gather the midpoints of the three sides of the bottom triangle to a point.
     Everyone even out the lengths of string and you will see that you have made a cube, oriented so it is balanced on one corner. If you look at this figure from directly above, you will see that you have made the shape of the original SGI logo, which dance company member Scott Kim designed.

Variations. Where can you go after the cube? What if everyone drops their left hand and holds on with their right?
     Do steps 1-5 with two people instead of three and you will make a tetrahedron instead of an octahedron. This is essentially the same as the construction shown in the Scientific American article.
     Do steps 1-5 with five people instead of three and you will make what is called a pentagonal diprism, with a pentagon on top, a pentagon on the bottom, and ten triangles around the side that point alternately up and down. Make the top and bottom pentagons small and the triangles tall and skinny, then five more people can pinch strings together to turn this into a regular dodecahedron. Can you see how?
     Can you make a regular icosahedron (twenty triangles) out of a single loop of string? Hint: you'll need at least 6 people (12 hands), and 6 doubled edges (of 30 total).
     What else can you make? What about a cuboctahedron? What if you use more than one loop of string?

For more information on string polyhedra, email Scott Kim with the subject line "Math Dance Book". Or if you invent your own string polyhedra let us know and we may be able to include them in the book. Of course sending us a picture will help us understand your construction.
These and other activities included in our book, Math Dance with Dr. Schaffer and Mr. Stern, now available from our Book Shop
(http://www.mathdance.org/store.html)