Maths Moments II

Shapes and the nature of space
(a) A collection of triangles, squares, pentagons, hexagons, septagons and octagons. How can you put them together to cover the floor? If all the shapes are the same then only triangles, squares or hexagons will work. But what if you mix them up?
(b) What 3 dimensional shapes can you build with the flat shapes? How many are there if the faces are all the same? Plato proved there are five shapes. Their symmetry makes them perfect as dice. Plato associated the shapes with fire – tetrahedron; earth – cube; air – octahedron; water – icosahedron; and finally the shape of the universe corresponded to the dodecahedron. If you mix shapes then Archimedes proved there are another 13 shapes you can make.
(c) What are the shapes of the universe? What is the shape of the library? If the library only has one floor, then it is like a flat universe. The library could extend without limit in all directions, the hexagons just tiling an infinite flat expanse. But more interestingly it might fold up on itself. It could be like the surface of a sphere or the surface of a torus. But how could the librarian tell which shape he lives in, stuck as he is on the surface of this universe? On a sphere if you draw a closed path, a loop, on the surface, then it is possible to continuously morph the loop until it vanishes to a point. On a torus there are loops that can’t be shrunk like this. These loops are like closed journeys on these surfaces. The way the hexagons are arranged and the doors going from one to another, it looks like the floors of the library are actually like torus shaped universes. As you travel through the hexagons you find yourself returning to the original hexagon from which you started. But what are the other possible shapes that you can use as plans for wrapping up the floors of the library? Poincare proved at the beginning of the twentieth century that any library can be morphed until it either looks like a sphere, or a torus with one hole, or a torus with two holes and so on. Perhaps you could see the librarian going through the proof of this projected up onto the walls.
(d) But the library is many layered, it is a 3 dimensional universe where you can go from one floor to the next…a third dimension. So how can this universe be wrapped up? Now we are having to wrap up a three dimensional universe in 4 dimensions. If we head up to the higher layers of the library we suddenly find ourselves returning to the layer we started at then the library is wrapped up into a higher dimensional bagel or torus. But what other shapes are possible? This is what Perelman answered in his solution to the Poincare conjecture, possible one of the greatest achievements in mathematics in the last century.
(e) There is also the interesting issue of whether the library is a Euclidean geometry or possible a non-Euclidean geometry. In non-Euclidean geometry, triangles have angles that don’t add up to 180 degrees. Consider triangles drawn on the surface of a sphere. See this link for an interesting article: plus article
(f) And what about fractals? Could the library reflect in some way some fractal characteristics?