Many commercial products are available to model polyhedra, such as Zometools, Stella 4d, Polydrons, Astro-Logix, and magnetic spheres which can be assembled into polyhedral shapes, sometimes with brightly-colored struts for the edges of the polyhedron. The first three tools, I can recommend without reservation (and I simply haven’t tried Astro-Logix, yet), but there is a problem with using rare-earth “ball magnets” to model polyhedra: the magnets don’t last long, for, while their magnetic fields are powerful, the neodymium-iron-boron alloy used to make these magnets is not durable, and such spherical magnets break easily.
For this reason, I decided to try a variation of the “ball magnet” idea, and instead obtained some (non-magnetic) steel balls, along with small, cylindrical rare-earth magnets to go between them, thus serving as polyhedral edges, while the steel balls serve as polyhedral vertices. With the steel balls keeping these cylindrical magnets separated (rather than smashing into each other), the magnets are more durable, and the steel balls, of course, do not have a durability problem. Here’s what I was able to produce when I attempted to make a set of Platonic solids, using this method:
The icosahedron, cube, octahedron, and tetrahedron shown above were easy to make, but attempting to construct a dodecahedron from these materials was an exercise in frustration. Forming one pentagon of this type is easy, but pentagons of this type lack the rigidity of triangles, or even the lesser rigidity of squares, and I was never able to get twelve such pentagons formed into a dodecahedron without the whole thing collapsing into a big ferromagnetic glob, which isn’t what I wanted at all.
Every polyhedron-modeling system has advantages and disadvantages, and the weakness of this particular system was made apparent by my failed attempt to construct a dodecahedron. I next tried adding triangles to pentagons, hoping the rigidity of the triangles would stabilize the pentagons, and allow me to construct an icosidodecahedron, the Archimedean solid which combines the twenty triangles of an icosahedron with the twelve pentagons of a dodecahedron. This method of combining triangles with pentagons did work, and I was able to construct an icosidodecahedron.
A major advantage of this medium for polyhedral modeling is that it is incredibly economical, compared to most specialized-purpose polyhedron-building tools. The materials are readily available on eBay. Non-magnetized steel balls are much less expensive than their magnetic counterparts; also, small cylindrical magnets are inexpensive as well, especially in large quantities. These will not be the last polyhedra I build using these materials — but they are suited for certain polyhedra, more so than others. With this system, the more equilateral triangles a given polyhedron has as faces, the better, for the rigidity of triangles adds to the overall stability of triangle-containing polyhedral models.