There is an Anathem wikia (inevitable I suppose and quite useful for my purpose) but unfortunately it doesn’t have a picture of the proof of Pythagoras’s theorem that the aliens put on the outside of their spaceship. So here it is:
[Note: the colours are mine; and I have reconstructed the picture from memory of the book and working out what it is trying to prove; hopefully I've got it right.I mean, I know I've got the proof-picture right; I'm less sure that this is exactly whats in the book. The line that looks orthogonal inside the larger square, is.]
The proof isn’t critical to the story (at least not up to p 500, which is where I’ve got to) and isn’t explained (ditto). The main character puzzles over it for a bit, then gives up. I haven’t seen it before, so thought I would try to puzzle it out myself. It turns out to be Euclids basic proof, so I think the book is being unreasonable by having the character fail to get it; he should recognise it instantly, because I think that (unlike me) they would have had a proper classical geometrical education.
How does it work? Not in the way I first though. I thought that double-counting all the triangle areas would do it, but only 2 pairs of triangles are similar. Lets colour them in:
The two red triangels are similar because they have sides of the same length (being erected on squares) and have a common angle. So you can rotate them on top of each other. Ditto the blue.
Next step is to draw some extra lines, which the book doesn’t do, because that would give the game away.
Then you realise that the red triangles from the previous pic are each of equal area to a right angled triangle (half-base-times-height, and so you can slide the apex along a line parallel to the base) as shown here, one of which is clearly half the area of the square, and the other of which is clearly half one of the rectangles that comprises the larger square. Ditto for the blue. So half of the large square is equal to the sum of half the sum of the smaller squares, WWWWW.
Wiki, of course, has far more of this. One interesting point is how this fails to work in non-euclidean geometry.
[Note that seed doesn't have a maths channel, and it clearly isn't tech or science, so I'll call it chatter]