Composing the first frame…
The Order Beneath · No.19 · Live plates
Noncommutative geometry
Plates on noncommutative geometry, each one computed in the browser as you watch: the irrational winding, a straight line of irrational slope on a torus that never closes, whose lines no continuous function can tell apart and whose space is replaced by two operations that commute only up to a turn, UV=e2πiθVU; the Hofstadter butterfly, an electron hopping on a lattice in a magnetic field whose two hops obey the same relation, its energies computed row by row for every flux p/q, a small butterfly inside one band and the gaps coloured by the Hall conductance, an integer in units of e²/h; a projection of trace θ, built by Rieffel from two functions on the circle, squared along one orbit without changing, the blocks the plate cuts out of it having eigenvalues 0 and 1 only, the ones a Fibonacci fraction of the sites; distance without paths, Connes's formula giving back the length of an arc, the transport distance between two distributions and a distance between two points with nothing between them, and the dimension of a torus read from how its spectrum grows; and primes at a temperature, the Bost–Connes system whose energy levels are log n, its partition function the Riemann zeta function and its modes the primes, its phase transition at β = 1, and the values its equilibrium states give the roots of unity, which move from 0 out towards the circle as the system cools. The corner keeps the numbers that describe what is drawn and checks them against the picture.
Plate I · 4 movements · 1:00 loop
Glue the opposite edges of a square and it becomes a torus; a straight line on the square becomes a line wound around it, once around the big circle each time it crosses the square, and round the tube as often as its slope says. The plate draws the line twice, flat in the square on the left and wound on the torus on the right. If the slope is a fraction p/q the line closes after q turns. If it is irrational it never closes, and comes as close as you like to every point. Then a continuous function that takes one value on each line must take the same value everywhere: the usual way of describing a space, by the functions on it, sees the space of these lines as a single point. Alain Connes's answer is to describe it by operators instead: here a shift by θ and a multiplication by e2πix, which no longer commute. The corner keeps the numbers that describe the line and checks them against the picture.
This volume's pictures, in your hands: change a parameter and watch the picture follow.
Plate IA slope just offMove the slope just a little: does the line still close?Open →
Plate IIOne row of the butterflyWith a flux of p/q, how many bands are there?Open →
Plate IIIThe shares a projection can keepChange θ: which shares can a projection keep?Open →
Plate IVThe distance between two distributionsHow far apart are two distributions on a circle?Open →
Plate VRoots of unity as the system coolsAs the system cools, where do the equilibrium state's values at the roots of unity go?Open →Places in the world where this volume's mathematics turns up. Each card links to a related Studio.
Lissajous figuresFeed one sine wave to each axis of an oscilloscope and the spot draws a Lissajous figure. When the two frequencies are in a ratio of whole numbers, the figure closes after some turns, the longer the larger the reduced denominator; when the ratio is irrational, it never quite closes, and in time it all but fills the box.Open in the Studio →
The quantum Hall effectIn 1980 Klaus von Klitzing, working in a strong magnetic field at low temperature, measured a Hall conductance that was exactly a whole number times e²/h, and won the 1985 Nobel Prize in Physics for it. In 2013 several groups, measuring the conductance of graphene laid almost in line on boron nitride in strong magnetic fields, found the signature of Hofstadter's butterfly.Open in the Studio →
The earth mover's distanceTo compare the colours of two images, think of them as the same amount of earth heaped in two different shapes. Move one heap into the other, multiply each bit of earth by how far it travels, and add it all up; the total for the thriftiest way of moving it is the earth mover's distance. Machine learning often uses it to compare two sets of data too.Open in the Studio →
A magnet coolingCool iron below its Curie temperature and, within each small domain, the atomic magnets line up in a direction of their own choosing, with no field applied: spontaneous magnetization. Cooling turns one possible equilibrium into several, and settling on one of them breaks the symmetry; the number-theoretic Bost–Connes system does the same as it cools.Open in the Studio →A compact Hausdorff space can be recovered from the algebra of continuous complex functions on it, and that algebra is commutative. Some spaces, like the space of lines in the first plate, have too few functions to be recovered this way. Noncommutative geometry keeps an algebra anyway, one that no longer commutes, and studies it as if it were the functions on a space. For the irrational winding that algebra is the noncommutative torus, generated by U and V with UV=e2πiθVU.
Plates I to III are tied together by the same integers. The traces of the projections in the noncommutative torus take exactly the values mθ + n between 0 and 1, with m and n integers (Pimsner and Voiculescu, Rieffel); plate III runs the rotation of plate I backwards, which leaves these values unchanged, builds the projection whose trace is θ, and draws a sample of the rest on a line. For a gap of the butterfly at flux α, the fraction of states below it is mα + n, the trace of the matching spectral projection, so the gaps of plate II are labelled by the K-theory of the algebra of plate I.
Nothing on screen is read from a precomputed table. The edges of the bands at θ = p/q are where D(E) = 4 and D(E) = −4, the eigenvalues of a periodic and an antiperiodic q × q matrix. Each is found by counting: factor the matrix minus E, count the negative pivots, and that is how many eigenvalues lie below E; bisection then pins each one down to the precision of the arithmetic, however close it is to its neighbour, with no scan over energies to miss a narrow gap. The 491 rows of the full butterfly take about a tenth of a second on a laptop; the rows of the zoomed window are computed between frames while the earlier movements play, and kept once computed.