Curvature

Prologue

1. 0:00Squeeze this ball however you like: its total curvature is always this number.

IHow much a curve bends

1. 0:09How do we measure how much a curve bends?
2. 0:11At every point on this ellipse, one circle hugs the curve most closely.
3. 0:16Where the curve bends sharply, the circle is small; where it bends gently, the circle is large.
4. 0:22Curvature is one over the circle's radius.
5. 0:25Keep every centre, and they trace a path.
6. 0:27At the ends of the long and short axes, the bend is sharpest or gentlest; the centre doubles back there, leaving a cusp. Four cusps make a star.
7. 0:37Draw curvature as spines: sharper bends, longer spines.
8. 0:41Following the arrow, left turns are plus, right turns minus.
9. 0:45At the waist, the spines shrink to zero, then flip across: the curvature turns negative.
10. 0:52Draw the perpendicular at every point. Where they crowd, a star lights up: four cusps on this egg.
11. 0:58Each marks a spot that bends more, or less, than its neighbours.
12. 1:02This triangle, six.
13. 1:05This square, eight.
14. 1:06Any closed curve, if it doesn't cross itself, has at least four.
15. 1:10Walk the curve: over equal lengths, a sharper bend turns you more.
16. 1:14One full turn: 2π.
17. 1:16The figure eight turns right, then left: they cancel.
18. 1:19Zero in all.
19. 1:21The one with an inner loop turns twice.
20. 1:24Around a closed curve: always a whole number of turns. The ball's total stays fixed too.

IIGaussian curvature

1. 1:32How much does a surface bend at a point?
2. 1:34Slice straight down: every direction cuts a curve, each with its own bend.
3. 1:39Bends one way are plus, the other minus; the extremes are the principal curvatures.
4. 1:44Outside, both match, like a ball; inside, they're opposite, like a saddle.
5. 1:49Multiply the two: that is the Gaussian curvature.
6. 1:52Positive is painted warm, negative cool.
7. 1:55Up here, one slice runs straight, so the product is zero.
8. 1:59So the warm outside is like a ball; the cool inside, like a saddle.
9. 2:04Draw a small loop on the surface. Take the outward direction at each point, and carry it to a sphere.
10. 2:10Their tips trace a second loop; divide its area by the first loop's.
11. 2:14As the loop shrinks, the ratio nears the Gaussian curvature.
12. 2:18On the saddle side, the loop on the sphere runs the other way, so the curvature is negative.
13. 2:23Flat paper has zero Gaussian curvature. Roll it into a cylinder, no stretching or squeezing: straight along the axis, so still zero.
14. 2:32To lay it on a sphere without wrinkles, lengths on the paper have to change.
15. 2:36Gauss proved it: this curvature depends only on lengths, measured within the surface. Without stretching or squeezing, no bending can change it.

IIISoap films

1. 2:47A soap film between two rings: why this shape?
2. 2:50To reduce its area, the film draws in at the waist, rather than forming a straight tube.
3. 2:55This shape is called a catenoid.
4. 2:59Pick any point on the film, and slice it two ways.
5. 3:03Along the axis, it bends like the blue circle, outside the film; around it, like the gold, towards the axis, just as much.
6. 3:10The two curvatures add to zero, at every point on the film. That's what holds it still.
7. 3:16Pull the rings slowly apart, and the waist grows thinner.
8. 3:20It has more area than two flat discs, yet it holds: small changes only add area.
9. 3:26The film snaps: beyond this gap, no catenoid spans the rings.
10. 3:31Now take a catenoid on its own, and cut it along one line.
11. 3:34Without stretching or squeezing, it twists into a spiral staircase: a helicoid.
12. 3:40As it twists back, lengths stay the same, and the two bends still cancel.
13. 3:44The catenoid and the helicoid: one surface, bent two ways.

IVGeodesics

1. 3:51On a curved surface, what does it mean to go straight?
2. 3:54Keep going forward, never turning left or right: the path you trace is a geodesic.
3. 3:59On a sphere, they are great circles: halfway round, they meet again on the far side.
4. 4:04The sphere bends inwards everywhere, and bends them back together.
5. 4:08Between two cities, the shortest path is always a geodesic: on a sphere, the shorter arc of a great circle.
6. 4:15Both sit at 45° north. Along that latitude, the route looks direct, but it's a detour; the great circle arcs north, nearly a fifth shorter.
7. 4:24Long flights often arc towards the pole.
8. 4:27Stretch the sphere three ways, each by a different amount.
9. 4:30From one point, straight paths go out, but no longer meet at a single point; where neighbours crowd, a four-cusped star lights up.
10. 4:39Jacobi predicted these four cusps, nearly two hundred years ago. The proof came in 2004.

VTotal curvature

1. 4:47Without leaving a surface, can you tell that it bends?
2. 4:50Draw a triangle of straight paths, and add up its angles.
3. 4:54On a plane they make π, 180 degrees; on a sphere, always more.
4. 4:59The extra is the curvature it encloses: on this sphere, its area.
5. 5:04Another way to measure: slide an arrow along the sphere, around the triangle, never letting it turn.
6. 5:09Along each straight side, its angle to the path never changes.
7. 5:16Back at the start, it's off by a right angle: exactly the curvature inside.
8. 5:21For each patch, multiply curvature by area, then add the results.
9. 5:27A sphere gives 4π. Squeeze in bumps and dents, and some parts become saddles.
10. 5:32Still 4π, the number from the start.
11. 5:35Why always 4π? Map the outward directions to this sphere: extra and reversed layers cancel, leaving exactly one.
12. 5:43And a doughnut?
13. 5:44The outer half is like a ball: its directions cover the sphere once. That's 4π.
14. 5:50The inner half is like a saddle: it covers the sphere again, backwards. Minus 4π.
15. 5:56Together, exactly zero. For any closed surface, the total depends on one thing: how many holes.

Coda

1. 6:03Curvature changes from point to point.
2. 6:06Added up, it keeps an order: round a closed curve, a whole number of turns;
3. 6:11a sphere, always 4π;
4. 6:13a doughnut, zero.
5. 6:14That order ignores the shape. It counts only turns and holes.