The shares a projection can keep

A projection's share is how much of everything it keeps: for a q × q matrix, the number of directions kept divided by q. In the box on the left, θ runs across and the share runs up, and every line is mθ + n, with m and n integers; the lines drawn are at most the families with m from −6 to 6, those from −3 to 3 coloured by m, in the same colours as the gaps of the butterfly. The coral line marks the chosen θ, and the points on it are the values of mθ + n from 0 up to, but not including, 1, with m going as far as the slider allows; past 6 there are more points than lines drawn. These heights, together with 1, are the shares a projection can keep in this noncommutative algebra, and the number line on the right sets them out, the gaps between them coloured by length; whatever θ and m are, there are at most three gap lengths. With θ at 0.4 and |m| at 3 or more, the coral line holds only five points, 0, 0.2, 0.4, 0.6 and 0.8, however large |m| gets, and the corner reads “only 5 values”: at θ = 2/5 the algebra behaves like 5 × 5 matrices, and the shares can only be k/5. Try 0.35, which is 7/20: once |m| reaches 10 there are only 20 values and the points start to coincide. Most other values on the slider have denominators in the hundreds or thousands, and even at |m| = 60 nothing coincides. Choose “One over the golden ratio” and θ is irrational: no two points ever coincide, and the larger |m|, the denser they get, the smallest gap shrinking in steps. A share can be an irrational number between 0 and 1; the one for m = −1 is 1 − θ ≈ 0.382.