A Variant of the Great Icosahedron


Faceted Dual

The great icosahedron has 20 faces which are interpenetrating equilateral triangles, most of which are hidden in the interior of that polyhedron. The non-hidden, and therefore visible, parts are called “facelets” — and there are 180 of them: 120 scalene, and 60 isosceles.

In this variant of the great icosahedron, the sixty isosceles facelets are simply missing, which changes the shape of the remaining 120, still-scalene facelets. The color scheme is one which gives each facelet a different color — except for coplanar or parallel facelets, which are the same color, making them easier to spot.

Software credit: see http://www.software3d.com/stella.php — with a free trial download available there.

My Polyhedral Nemesis: The Great Icosahedron


My Polyhedral Nemesis:  The Great Icosahedron

I used Stella 4d, a program you can find at http://www.software3d.com/stella.php, to make the rotating .gif file you see here. You can many such rotating pictures of other polyhedra elsewhere on this blog.

Older versions of this program would only create still images. In those days, I would also make actual physical models out of paper (usually posterboard or card stock). However, I’ve stopped doing that, now that I can make these rotating pictures.

There is one polyhedron for which I never could construct a physical model, although I tried on three separate occasions. It’s this one, the great icosahedron, discovered, to the best of my knowledge, by Johannes Kepler. Although it only has twenty faces (equilateral triangles), they interpenetrate — and each triangle has nine regions visible (called “facelets”), with the rest of each face hidden inside the polyhedron.

To create a physical model, 180 of these facelets must be individually cut out, and then glued or taped together, and there’s very little margin for error. On my three construction-attempts, I did make mistakes — but did not discover them until I had already built much more of the model. When making paper models, if errors are made, there is a certain point beyond which repair is impossible, or nearly so.

Although I never succeeded in making a physical model of the great icosahedron myself, and likely never will, I did once have a team of three students in a geometry class successfully build one. One of the students kept the model, and all three received “A” grades.