Last week I was at a conference on mathematical biology in Durham. I will get to the biology in a bit, but first the mathematics, in particular geometry. You can make precisely five (not four or six) different regular solid shapes using only regular polygons, and where at every corner of the shape the same number of these polygons meet. The cube is probably the most familiar one. This is made from six squares (a square is a four-sided polygon), and at each corner, three of these squares meet. There are exactly four other shapes, one of which is the icosahedron shown to the left. The icosahedron is made of triangles (three-sided polygons), not squares, and here at each corner five of these triangles meet. The icosahedron has 20 of these triangles.
I find it somehow continually surprising, that there are precisely five and only five such shapes possible. It just seems a bit weird to me. Even more weirdly, we have known that you can only make five of these solid shapes since the time of the ancient Greek philosopher Plato (born about 427 BC). These five shapes are the most symmetric, and so many people think, the most beautiful. They are called the Five Platonic Solids, because they are mentioned in works by Plato.
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