Mathematical Models

Way back when I was at school, I came across a fascinating little book called Mathematical Models, by Cundy and Rollett.  It had instructions on how to build various mathematical objects, such as stellated polyhedra.  I liked the book so much I actually bought my own copy, new, for £2.95, which was a lot of money back then!

I made a few of the simpler models, but never got much further than the dodecahedron.  I certainly never got as far as making any of the fiddly stellated ones.

So today, when I came across a posting in Google+ about Anselm Levskaya's website polyHédronisme, I was taken right back to those days.  Playing with this interactive web-based systems is much easier than fiddling with card, glue, and scissors, though.  Type in a few commands, and a zoomable, rotatable polyhedron appears!

I've spent my afternoon playing around on this site, and reading up on Conway polyhedron notation that is used to define shapes, and now I can say I have at last "made" some of these polyhedra.

The small stellated dodecahedron is made by raising a pentagonal-based pyramid on every face of a regular dodecahedron. If the pentagons making up the dodecahedron have side length \(1\), then the height of each pyramid should be* \[ \frac{\sqrt{4\sqrt{5}-1}}{2} \approx 1.41\] The Conway notation command in polyHédronisme that achieves this is \(k(5,1.41)D\), which means: start with a dodecahedron \(D\), then raise a pyramid of height \(1.41\) on each \(5\)-sided face.

The great dodecahedron is made by making a pyramidal dimple in every face of a regular icosahedron. If the triangles making up the icosahedron have side length \(1\), then the depth of each pyramid should be* \[ \sqrt{\frac{1}{2}- \frac{\sqrt{5}}{6}} \approx 0.36\] The Conway notation command in polyHédronisme that achieves this is \(k(3,-0.36)I\), which means: start with an icosahedron \(I\), then indent a pyramid of height \(0.36\) on each \(3\)-sided face.

The great stellated dodecahedron is made by raising a pyramid on every face of a regular icosahedron. If the triangles making up the icosahedron have side length \(1\), then the height of each pyramid should be* \[ \sqrt{\frac{7+3\sqrt{5}}{6}} \approx1.51\] The Conway notation command in polyHédronisme that achieves this is \(k(3,1.51)I\), which means: start with an icosahedron \(I\), then raise a pyramid of height \(1.51\) on each \(3\)-sided face.

So much for standard polyhedra.  But polyHédronisme doesn't stop there.  I had great fun playing about with the notation language, producing weird and wonderful forms:

(i)  \(k(20,1)bk(3,2)I\)    (ii)  \(k(12,1)k(10,2)bk(5,1)D\)    (iii)  \(k(20,-0.3)k(6,0.3)bk(3,-0.3)I\)
(iv)  \(k(24,-0.5)k(6,0.2)k(20,-1)bk(3,-0.1)k(5,1)D\)

The results of play can only really be appreciated on the site itself, rotating the polyhedra, and getting a real feel for their shapes, with all their dips and bumps.  A marvellous site.

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* The book Polyhedron Models has helpful stellation diagrams that allow these heights to be calculated, with a little trigonometry.