My cover picture for Astronomical Calendar 2015
The simplest of all straight-edged solids is the tetrahedron. A solid can’t have fewer faces, yet its mere four have endless aspects and connections. One of the minor remarks I cut out was that “If the tetrahedron is flattened, its four faces become an equilateral triangle stellated; or, put the other way, you make a paper tetrahedron by drawing an equilateral triangle and stellating it.”
I mentioned that hidden in the tetrahedron is circularity (the cylinder that, pinched in two places, becomes a tetrahedral milk carton) but didn’t have space to illustrate it –
– also that hidden in the tetrahedron is a squareness. It’s all triangles, but if you cut it into two equal parts, the two new surfaces are squares. I would have liked to show the paper model I once made of this, but must have thrown it away. But this is what it looked like:
If you were sawing a wooden tetrahedron, you would start by making marks at the halfway points along four of the six edges. Each of the two new faces made by the cut is a square. And each of the two new solids has five faces: the square, two isosceles triangles, and two isosceles trapezoids. Quite un-simple chunks! The fun of making the model is that you toss these two chunks, with their total of ten faces, on the table and ask your friends to fit them together into something with only four faces. Yes, you now easily see how to do this, but that’s only because you’ve just been reading out tetrahedrons.
Alastair McBeath told me he had been wondering about the possibility of paper polyhedral star globes, and asked whether I had seen any. Yes: James Weightman in 2013 sent me this:
It is the all-sky map from my Astronomical Calendar, cut into a large number of polygons so that, folded and glued together by their tabs, they become the faces of a polyhedron, which is an approximation to the celestial sphere.
It would be a celestial sphere backwards, that is, with the constellations correctly mapped as seen from the outside but mirror images of themselves as seen from the center; as in ancient celestial spheres such as the one carried by the sculpture called the Farnese Atlas; and as distinct from modern transparent celestial globes into which you peer, to see the constellations the right way around on the far side.
The polyhedron James chose was the truncated icosahedron. An icosahedron has 20 (Greek eikosi) triangular faces, 5 of which meet at each of 12 vertices.
Therefore, truncating – slicing off – a vertex produces a pentagonal face. Doing the same to all the vertices produces 12 such pentagons, and by chopping off the corners of the 20 original triangles reduces them to hexagons. So the result is 32 faces in all. The more faces, the closer the approximation to a sphere, which could be considered a polyhedron with an infinite number of faces.
James used a “Photoshop add-on” called Flexify2. It probably can make all the polyhedrons I laboriously figured out how to calculate, and more, but I’m glad I got as far as I did in understanding them.
A further challenge, and I don’t know whether it would be solved by trigonometry or sheer dextrous fingers, would be to make a celestial sphere by taping together 89 pieces, of very varied size and irregular shapes, for the 88 constellations (including the two separated parts of Serpens). This would be like building a house, or a vault, out of – well, I can’t think of any object in nature that’s shaped like the constellation Draco.