182 Are the Androids Dreaming Yet? We know from our childhood that infinity plus one is still infinity. Is there anything we can do to make infinity bigger? Perhaps multiplying infinity by infinity will do the trick. Infinity times infinity can be visualized as a square with edges of infinite length. We can show that this square is the same size as a one- dimensional infinity through a clever trick — the zigzag method. Mark the infinity square into a grid. Start in the corner square, go across, diagonally down, then across, diagonally up, and so on. I'll draw you a picture. We visit every square in our grid using a single line. We can then lay down our infinite zigzag line next to the infinite line of one of the edges. The lines are the same length as they are both infinitely long! So infinity, times infinity, can be matched to infinity, they are the same. Cantor thought this a very strange result and wrote to a fellow mathematician, Dedekind, “Je le vois, mais je ne le crois pas!”, “I see it, but I don’t believe it!” If you are struggling with this, don’t worry. We just jumped forward to quite a complex concept. Let’s take it more slowly. One way to get a better grip on infinity is through the stories of David Hilbert and the Infinity Hotel. Infinity for Dummies Hilbert’s Hotel is a mythical building with an infinite number of rooms. Other than this strange feature it is a regular hotel complete with minibar, dodgy TV, and slightly mad manager. The rooms are numbered sequential starting at one, then two, three, four, and so on. The hotel allows you to play a series of mathematical games to see how infinity behaves. Are there the same number of minibars as there are rooms? That's easy. I said every room has a minibar. We can use the matching technique to match minibars with rooms. Go to the first room. There is a number on the door and a minibar inside. The same goes for room 2 and 3 and this goes on forever. I’ve just proven two infinite things are the same -