Professor Stewart's Hoard of Mathematical Treasures Read Online Free PDF

a polyhedron, and flexible, it is not a flexible polyhedron. Remember, the shapes of its faces are not permitted to change. They start out flat, so they have to stay flat, which means they can’t bend. Not even the tiniest bit. But when you play a concertina, and the flexible bit opens up, the faces do bend. Very slightly.

    Two positions of a concertina.
    Imagine the concertina partially closed, like the left-hand picture, and then opened, like the right-hand one. We’re viewing it from the side here. If the faces don’t bend or otherwise distort, the line AB can’t change length. Now, the sides AC and BD actually slope away from us, and we’re seeing them sideways on, but, even so, because those lengths don’t change in three dimensions, the points C and D in the right-hand picture have to be further apart than they are in the left-hand one. But this contradicts lengths being unchanged. Therefore the faces must change shape. In practice, the material that hinges them together can stretch a bit, which is why a concertina works.

The Bellows Conjecture
    Whenever mathematicians make a discovery, they try to push their luck by asking further questions. So when flexible polyhedra were discovered, mathematicians soon realised that there might be another reason why concertinas don’t satisfy the
mathematical definition. So they did some experiments, making a small hole in a cardboard flexible polyhedron, filling it with smoke, flexing it, and seeing if the smoke puffed out.
    It didn’t. If you’d done that with a concertina, or bellows, and compressed it, you’d have seen a puff.
    Then they did some careful calculations to confirm the experiment, turning it into genuine mathematics. These showed that when you flex one of the known flexible polyhedra, its volume doesn’t change. Dennis Sullivan conjectured that the same goes for all flexible polyhedra, and in 1997 Robert Connelly, Idzhad Sabitov and Anke Walz proved he was right.

    It doesn’t work for polygons.
    Before sketching what they did, let me put the ideas into context. The corresponding theorem in two dimensions is false. If you take a rectangle and flex it to form a parallelogram, the area gets smaller. So there must be some special feature of three-dimensional space that makes a mathematical bellows impossible. Connelly’s group suspected it might relate to a formula for the area of a triangle, credited to Heron of Alexandria (see note on page 282). 6 This formula involves a square root, but it can be rearranged to give a polynomial equation relating the area of the triangle to its three sides. That is, the terms in the equation are powers of the variables, multiplied by numbers.
    Sabitov wondered whether there might be a similar equation for any polyhedron, relating its volume to the lengths of its sides. This seemed highly unlikely: if there was one, how come the great mathematicians of the past had missed it?
    Nevertheless, suppose this unlikely formula does exist. Then the bellows conjecture follows immediately. As the polyhedron flexes, the lengths of its sides don’t change - so the formula stays
exactly the same. Now, a polynomial equation may have many solutions, but the volume clearly changes continuously as the polyhedron flexes. The only way to change from one solution of the equation to a different one is to make a jump, and that’s not continuous. Therefore the volume cannot change.
    All very well, but does such a formula exist? There is one case where it definitely does: a classical formula for the volume of a tetrahedron in terms of its sides. Now, any polyhedron can be built up from tetrahedra, so the volume of the polyhedron is the sum of the volumes of its tetrahedral pieces.
    However, that’s not good enough. The resulting formula involves all the edges of all the pieces, many of which are ‘diagonal’ lines that cut across from one corner of the polyhedron to another. These are not edges of the polyhedron, and, for all we know, their