Computation — page 5
The rectangles a square can be cut into
Cut a square into rectangles that all have the same shape — the same ratio of long side to short, in any sizes, lying either way. Two to one is easy. The golden ratio is impossible, however many pieces are used and however small. The ratio φ squared works with six. What decides it is not the ratio but its partner: the number obtained by changing the sign of the square root, which must be positive. Changing that sign is a map that keeps every sum and product and scrambles every size, and areas that must add up to one cannot all turn negative.
A code built from an icosahedron
Write down which corners of an icosahedron are not joined by an edge, put the table beside a diagonal of ones, and add the twelve rows together in every combination. The 4,096 strings of 24 bits that come out differ from each other in at least eight places, occur in only five weights, and — with one bit deleted — fill the space of 23-bit strings so exactly that every one of the 8,388,608 lies within three flips of exactly one of them. This is the Golay code, which carried Voyager's pictures home, and its words of weight eight cover every five of the twenty-four positions exactly once.