Conformal

Prologue

1. 0:05A function bends a sheet of graph paper: straight lines curve; cells swell, shrink and turn, each in its own way.
2. 0:12Yet the lines still cross at right angles. Divide the grid more finely, and the cells become more nearly square.
3. 0:19This film is about functions that preserve angles.

INumbers in the plane

1. 0:25Add a vertical axis: one step above zero is i.
2. 0:29The complex number 3 + 2i is a point: three right, two up. It's also an arrow, with length and direction.
3. 0:37Add a complex number, and every point slides the same distance in the same direction: a translation.
4. 0:43Multiply 1 by i: a quarter-turn anticlockwise takes it to i. Again, to −1: that is i · i = −1. The whole plane makes the same quarter-turn.
5. 0:54Multiply by any nonzero complex number, and the plane turns by its angle and scales by its length; squares stay square. Multiply by another: lengths multiply, angles add.
6. 1:06Multiplying by 0 is different: the whole plane collapses to a point.

IIPainting a function

1. 1:14A real function can be graphed as a curve. A complex function's inputs and outputs need two coordinates each: four dimensions in all.
2. 1:23On the output plane, a colour wheel shows only direction; from ring to ring, length grows by the same factor. Rings and rays cut the plane into nearly square cells.
3. 1:33Each input point takes its output's colour. With z − 1, the wheel's centre moves to 1, where z − 1 = 0.
4. 1:41Where cells crowd, a step turns the value faster or scales it by a bigger factor.
5. 1:46Switch to z³ − 1: the colours gather at three points where the value is 0, the zeros. Loop tightly around one, anticlockwise: each colour appears once, in the wheel's order.
6. 1:59Divide by z³ + 1. At three points the denominator is 0, the numerator isn't: these are poles. Loop the same way: the colours run backwards. Near a pole, the value grows without bound.
7. 2:12On the input plane, rings and rays meet at right angles, but for a few points. At a zero, rays meet only because it maps to the wheel's centre.

IIIPreserving angles

1. 2:24What kind of function preserves these right angles? A complex-differentiable one: take a small step h, and the value changes by roughly h times a complex number, the same one in every direction.
2. 2:37That number is the derivative. Where it isn't 0, zoom in close, and you see only turning and scaling.
3. 2:44Turning and scaling preserve angles: their size, and which way you turn from arrow 1 to arrow 2. Wherever the derivative isn't 0, the map preserves angles.
4. 2:53Zoom in on any smooth map and its lines straighten, but the squares may be stretched or sheared. Complex differentiability demands one multiplier for all directions, which is far stricter.
5. 3:05So in the colour plots, wherever the derivative isn't 0, small enough cells are nearly square.
6. 3:11The exception is where the derivative is 0. At 0, z² doubles every direction's angle: the multiplier vanishes, and higher powers of h take over.
7. 3:21Such a point is called a critical point. In the last chapter, one such point will shape the whole picture.

IVThe point at infinity

1. 3:31Under 1/z, everything far away, in every direction, lands near 0. So both planes gain one more point: ∞.
2. 3:40Put a light at a sphere's top: every other point casts a shadow on the plane. This is stereographic projection. The top itself is ∞; add that point, and the plane becomes the sphere.
3. 3:53Stereographic projection preserves angles: small squares cast nearly square shadows, and circles become circles, or lines through the light. Angles at ∞ are measured at the top.
4. 4:05Move the sphere around, keeping the light at its top, and the shadows move by a Möbius transformation; every Möbius transformation arises this way.
5. 4:14Turn the sphere upside down: 0 and ∞ trade places, and lines can bend into circles. Project up, move, project down: every step preserves angles, even at ∞.
6. 4:26To map the whole sphere onto itself, one to one and preserving angles everywhere, only Möbius transformations will do. z² won't: 2 and −2 both go to 4.

VAn infinite plane in a disc

1. 4:41In the plane, a regular hexagon's angles are 120°, so three fit exactly around a point; an octagon's are 135°, and three won't fit.
2. 4:51In the hyperbolic plane, three fit: octagons with 120° angles.
3. 4:56Map the whole plane into a disc: the octagons are all one size, drawn smaller toward the rim. The rim is infinitely far away: walk off in different directions, and you head for different points on it.
4. 5:09The map preserves angles but not lengths. Hyperbolic straight lines, the shortest paths, are drawn as arcs meeting the rim at right angles, or as diameters.
5. 5:19The triangle's sides are mirrors. Reflecting in a circle is called inversion; it swaps inside and outside. This one meets the rim at right angles, so the disc maps to itself.
6. 5:30Angles keep their size, but the image is flipped. A second mirror flips it back: only the pair makes a Möbius transformation. Reflection after reflection fills the disc.
7. 5:41Reflect once in each of these two mirrors: the result is a translation, also a Möbius transformation taking the disc to itself. The octagons keep their true size and shape.
8. 5:52Motions preserve angles, and the map does too: so the motions that don't flip are exactly these transformations.

VICircles in circles

1. 6:03Inside a big circle, three equal circles: all four touch one another. Each gap holds exactly one circle touching all three sides; the new circles leave new gaps, and the filling never ends.
2. 6:15Hollow out every circle inside the big one, and what remains is the Apollonian gasket.
3. 6:21Four circular mirrors each pass through three points of contact, crossing the circles at right angles, so each of those circles maps to itself.
4. 6:29Reflected, the big circle lands in the middle gap as the new circle, still touching them: touching is an angle of zero, and inversion maps circles to circles, preserving angle sizes. Every combination of mirrors builds the gasket.

VIIOne piece or dust?

1. 6:47Apply a function again and again: the points visited form an orbit. Under z², points inside the unit circle fall toward 0, points outside fly off to infinity, points on it stay on it.
2. 7:00Next, add a constant, c. Some points never escape; others escape to infinity. The boundary between them is the Julia set; as c moves, the circle twists into a crinkled curve. Brightness shows only how fast they escape.
3. 7:15Move c right, past 1/4, and the Julia set breaks into dust: no two points are joined, yet every point has others arbitrarily close. Either one piece, or all dust.
4. 7:28One piece or dust? Watch the orbit of 0: if it never escapes, the Julia set is in one piece; if it escapes, dust. Zero is special only because the derivative is 0 there: it is the only critical point in the plane.
5. 7:42Points escaping equally fast form level curves. The first crossing, a figure eight, can only be at 0, where angles aren't preserved: the right angle doubles and opens flat, into a smooth curve through c.
6. 7:55If 0 escapes, the curves pinch in two, splitting again and again at points that eventually land on 0. If not, they stay whole.
7. 8:04A new picture: each point is a value of c. Blacken each c for which 0 never escapes: that is the Mandelbrot set, a catalogue of the Julia sets. With c in the set, the Julia set is one piece; outside, dust.
8. 8:19The Mandelbrot set is itself in one piece. The proof uses another angle-preserving map: it takes every circle around a disc, one to one, to a level curve around the set.
9. 8:30No circle pinches in two, so no level curve does; as they close in, the set must be one piece.

Coda

1. 8:37In every one of these pictures there is a map that preserves angles: a complex-differentiable function, in each small patch, only turns and scales; a mirror flips as well.
2. 8:48Yet a whole picture can hinge on a single point: the one where angles aren't preserved.
3. 8:53Preserving angles has a name: conformal.