One row of the butterfly

Press somewhere on the butterfly: the height picks a row, snapping to the nearest fraction p/q whose denominator is no larger than the slider allows, and that fraction is the flux θ through each small square of the lattice; the left–right position picks an energy E. The chosen row is drawn thick, and the corner counts its bands: always q. When q is even, the two middle bands touch at E = 0, so only q − 2 gaps are open, not q − 1. The gaps are coloured by an integer m, as in the fourth movement of Plate II; only m from −3 to 3 gets a colour, and the rest are drawn dark grey. If r bands lie below a gap, the states there make up r/q of the total; write r/q as mθ + n, with m and n integers, and take the m no larger than q/2 in size. Move E into a gap and the bottom line gives its m: with the electrons filled up to there, the Hall conductance is m times e²/h. Move it into a band and it tells you which band. The whole butterfly behind is four hundred and ninety-one rows, one for every fraction with denominator up to 40; the slider only decides which rows you can pick, and at 40 you can pick any of them.