Figure 1. Net of an icosahedron.
Ed Pegg Jr., March 29, 2004
When I was young, my babysitter Mark asked me and my younger brothers if there was any construction paper in the house. We had some sturdy white board behind a cabinet, so I brought that out for him, along with a compass, yardstick, pencil, glue, and Exacto knife. He proceeded to construct lots of equilateral triangles. At that point, I didn't know about simple compass constructions, so I was very impressed. Soon, he had a net of 20 equilateral triangles and tabs, and cut it out with the knife. Then he built a very sturdy icosahedron. I had never seen an actual icosahedron before -- certainly not in grade school. I had that icosahedron for years.
Figure 1. Net of an icosahedron.
Years earlier, Albrecht Durer pondered how to make polyhedral nets. At George Hart's page, you can read about Durer's polyhedral writings. At Mathworld, Eric Weisstein has made PDF files of polyhedral nets, suitable for printing and cutting. Programs Stella and Poly both allow creation of polyhedral nets.
The Zome system allows for extensive polyhedral construction. An expanded small rhombicosidodecahedron serves as the connecting sphere between triangular, rectangular, and pentagonal rods. An excellent book about Zome by George Hart and Henri Picciotto is available. One advantage to Zome over the magnetic sets below is that large constructions are stable.
Figure 2. Expanded 120-cell made with Zome.
Strange Attractors and Roger's Connection both use magnetized rods of various size and steel bearings for polyhedral constructions. Strange Attractors is relatively new. As of the the time of this writing, only the Starter kit is available. A more advanced kit will be available sometime in April 2004. Strange Attractors can build more polyhedra than Zome, but is unfortunately unstable in large constructions.
Figure 3. The Strange Attractors construction set.
Polymorf, Polydron, and Geofix all use polygonal shapes that snap together. The Polymorf site has a very extensive discussion of the importance of polyhedra.
Figure 4. A snub cube in the Polymorf construction set.
These are all great construction sets. Lately, I've been experimenting with neodymium-iron-boron supermagnetic spheres. Safety note - I use the smallest possible spheres. Back when I was a kid, I bought a large (dime-sized) samarium-cobalt supermagnet. Within 15 minutes of its arrival, I managed to shatter the magnet, and badly damaged my fingers. For magnets this strong, it is vital to play around with the tiny ones to get an appreciation of their extreme power. "Giant NIBs just want to smash themselves, and you. They're about as fun to play with as alligators." -- Dan's Data.
I got these magnets at wondermagnet.com. They have a strong emphasis on safety, and won't sell big magnets to novices. Their page on Magnetic Sculptures was an inspiration for this article. Other interesting photos are available at execpc.com's Magnetic Toys page and Dan's Data Amazing Magnet's Superball kit.
Here are three of the stable structures that can be made using supermagnetic spheres and nothing else.
Figure 5. The magnetic cube, icosahedron, and small rhombicuboctahedron.
With some rod magnets, it is possible to see where the poles of the spheres are pointing. Can anyone figure out the angles of the magnetic icosahedron? It's a minimization problem of some sort, but I'm not sure how to set it up.
Figure 6. Poles of the magnetic cube, icosahedron, and small rhombicuboctahedron.
The cube has a fourfold symmetry, and is somewhat shiftable. The magnetic icosahedron has a fixed three-fold symmetry, as does the small rhombicuboctahedron. Bigger polyhedra are possible, and the angles of stabilization become even more strained. I don't know how to determine whether a polyhedra is constructible with magnetic spheres. The regular dodecahedron, for one, seems unconstructible.
Figure 7. The magnetic octahedron and great rhombicuboctahedron.
Larger objects tended to collapse upon themselves in mid-construction. If you can build any polyhedra not mentioned here with supermagnetic spheres, I would like to hear about it.
References:
Richard Engel. Knowhere. http://www.polymorf.net/knowhere.htm.
George Hart. Durer's Polyhedra. http://www.georgehart.com/virtual-polyhedra/durer.html.
Eric Weisstein. Polyhedron Nets. http://mathworld.wolfram.com/topics/PolyhedronNets.html.
Comments are welcome. Please send comments to Ed Pegg Jr. at ed@mathpuzzle.com.
Ed Pegg Jr. is the webmaster for mathpuzzle.com. He works at Wolfram Research, Inc. as the administrator of the Mathematica Information Center.