Comb
A drop of liquid crystal pressed thin between two slides, under a microscope with its polarizers crossed. The dark brushes are where the molecules lie along a polarizer; the points where brushes meet are defects, places where the molecules cannot agree. The film was quenched a moment ago and is still sorting itself out. Drag across it and your stroke combs the molecules into line, and the film has to undo the damage; time runs slower while your hand is in it. The polarizers turn on their own; the brackets turn them by hand.
what the film can say about itself
A nematic is a liquid whose molecules are rods. They flow like oil, but they agree, locally, on a direction, and the direction has no arrow: a rod turned end for end is the same rod. That small fact is why the points in this picture come in halves. Walk once round a defect where two brushes meet and the molecules have turned by half a revolution, which for a rod is enough to come back to itself. A whole revolution would cost four times the energy, so whole defects split into halves and you will not find one here after the first few seconds.
The film is solved, not drawn. Every cell of a disk carries the order parameter of Landau and de Gennes, an amount of order and a direction folded into one tensor, and each step relaxes it a little down the slope of its own free energy: order costs less than disorder below the transition, and neighbours that disagree cost elastic energy. The film starts quenched, with fresh order pointing every way at once, and it coarsens. Defects of opposite sign find each other and vanish; the ones that are left move apart as far as the rim lets them. At the rim the molecules are held tangent, and a tangent field around a circle carries a charge of exactly one, so the film can never be rid of its last two half defects. Left alone, it ends with those two on a diameter, each about two thirds of the way from the centre to the rim, where the theory of image charges puts them: the fourth root of one fifth of the radius.
The light is the other half of the physics. Between crossed polarizers a birefringent film passes light only where its molecules lie diagonal to the polarizers; where they lie along either one, nothing gets through, and that is a brush. The brightness at each pixel is the Jones calculus for a half-wave film at full order, sin² of twice the angle between the molecules and the polarizer, times sin² of half the retardation, which falls to nothing in a defect's core where the order itself goes to zero. Turn the polarizers and the brushes turn round their points at twice the rate, with the polarizers around a positive defect and against them around a negative one. That is how a microscopist reads the sign of a defect without touching the slide.
The picture is monochrome because the lamp is. One wavelength, the yellow line of sodium, is what this page assumes; in white light the same film would show the colours of its thickness. Nothing in the render was chosen to be gray. It is gray because the gallery's rule and the physics of one lamp happen to agree.
Your stroke is a field. Inside a small brush around the pointer a term is added to the equation that pulls the molecules toward the direction you are moving, the same term by which a voltage switches a pixel in a display. The stroke leaves an aligned band, and where the band meets the rest of the film the mismatch breeds new pairs of defects, which the film then has to annihilate again. The link regrows the quench from its seed to the step you copied, exactly, because the solver only adds, subtracts, multiplies, divides and takes square roots. It does not regrow your combing. That leaves with the print, and with nobody else.
What this is not. One elastic constant, where a real nematic has three. No flow: the liquid in a real cell moves when its molecules turn, and here it does not. Zero temperature after the quench, a film so thin it has no thickness, anchoring so strong the rim never yields. A real slide under a real microscope is all of those things at once, and the picture it shows is this one with more life in it.
Eight checks and five guesses were written down before the solver existed, and run blind. The energy never rose in four thousand steps. The charge summed over the film was one at every sample and zero on a periodic test lattice. The same seed gave the same bytes, and a kept film's link regrew it in a fresh page to the hash. Two checks failed as written and the fault was the examiner's: one compared the stepped model with the continuous relaxation rate, which the step itself makes 2.107 and not 2, and one asked for a relative error where the test field is zero. Both are amended on the record and neither changed the solver. Of the five guesses, two held blind: the brushes turn at twice the polarizers' rate with the sign of the defect, measured on twelve of them to half a degree, and no whole defect survives the first hundred time units. Three were wrong as registered. The guess about where the last pair settles was graded before the pair had settled, because the test read positions at half a cell and a pair creeping at a fiftieth of a cell per thousand steps looked still; run to thirty thousand time units on three seeds, the two half defects sit at 0.666 to 0.670 of the radius, on a diameter to half a degree, against the fourth root of one fifth, 0.669. The coarsening exponent came out at 0.68 on a small lattice with too few defects to fit, and 0.79 on a lattice four times larger. A placed pair annihilated six times slower at twice the distance where the guess allowed at most 4.8; at twice that distance again the ratio fell to 5.4, on its way toward four with the logarithm the theory adds. All of it stays as graded.
Keeping a film saves it at 2,400 pixels with its seed line printed beneath it. The line regrows the quench; the combing was yours.
lattice 360 across · disk radius 174 cells · 95,115 cells of film · coherence length 2 cells