Slide-glide cyclides

0:01 Hi, I'm Henry Segerman and this is slide-glide cyclides.

0:06 So it looks like a sphere made out of eight bananas.

0:10 Each banana is attached to the base

0:11 with an axle that it can rotate around like this.

0:15 And there are also bevel gears at the end of the banana,

0:18 that mesh with a crown gear that goes all around the base,

0:23 and it's connected to this part of the base down here.

0:26 So if I twist the base down here,

0:29 the crown gear is going to mesh with the bevel gears on the bananas,

0:33 and each banana will rotate down.

0:36 Okay, well, let's see it.

0:40 So the sphere opens out into a disk in a very satisfying way.

0:46 This design was discovered by Andrew Kepert in the process

0:49 of working on a video with Burkard Polster, also known as Mathologer.

0:55 Their video is about ways to see why the area of a sphere is 4 pi r squared,

0:59 four times the area of the disk of the same radius as the sphere.

1:03 And you can see something of the idea in this mechanism.

1:07 So if the radius of the sphere is r say, so this distance here,

1:13 then when you open it out into a disk, here's the radius of the sphere.

1:19 So the diameter of the sphere is here.

1:21 That's the radius of the disk.

1:23 So the area of the disk is 2 times r squared, times pi,

1:28 which works out to 4 pi r squared,

1:30 which is the formula for the area of the sphere.

1:33 And you can immediately see that the same amount of stuff makes up

1:37 the sphere as makes up the disk because it's all just the same bananas.

1:43 As the number of bananas goes up,

1:45 they get thinner and the shapes at either end of the movement

1:48 get closer to being the actual sphere and the actual disk.

1:52 As usual with this kind of visual proof though there are

1:55 some details to worry about to show that things actually converge.

2:00 For more on comparing areas of spheres and discs

2:03 with constructions like this, check out Mathologer's video,

2:08 link in the description.

2:14 So anyway, Andrew came up with the design,

2:15 then nerd-sniped me into making this physical geared version.

2:20 But why does it work?

2:21 How come the bananas slide against each other

2:23 the whole way through their motion, staying in contact?

2:28 The technical name for the banana shape is a cyclide.

2:31 One way to define cyclides is that they are the surfaces that you

2:34 can make as an envelope swept out by spheres in two different ways,

2:39 both from the inside and the outside.

2:42 This particular cyclide can be thought of as a kind

2:45 of reflection of a simpler shape, a cone.

2:47 If you reflect it in an ordinary planar mirror,

2:49 then you get back the same cyclide.

2:51 But if you instead reflect in the correct shape

2:53 or rather invert in the correct sphere, you get a cone.

2:58 Eight cones like this can slide together in a kind of irising motion.

3:01 If we move the cones up and down around the sphere as they iris,

3:05 then the inverted cyclides give us back the original motion

3:09 Inversion in the sphere turns the sliding motion on the cones

3:12 into rotation of the cyclides and it turns the lines

3:15 of contact on the cones into arcs of contact between the cyclides.

3:23 If you want to print out one of these yourself,

3:25 I've uploaded the files- link in the description,

3:28 and to finish up, here are some questions

3:30 that came up as we were thinking about this.

3:32 First, is there a way to get the surfaces

3:35 of the cyclides themselves to do the gearing?

3:36 I added the standard bevel gears and the crown gear

3:39 to get the cyclides to move in the right way.

3:42 But even without the gears,

3:43 they push each other around because of the arcs of contact.

3:47 Here I've removed the crown gear so you

3:49 can see how well they move without the gearing.

3:51 And the answer is...

3:53 well...

3:55 yeah, here we go.

3:56 Once it's near the sphere position, it's not bad.

3:59 I can just move one of them and it moves all of them,

4:01 but when it gets down towards the disc,

4:03 they sort of don't really have much to do with each other.

4:06 So yeah, here the gearing is really necessary.

4:10 So can we design the surfaces to ripple in some way so

4:14 that they grip on to each other and better enforce the motion?

4:18 Following on from that, I'm interested in mechanisms where

4:21 pairs of parts are in contact throughout the movement,

4:24 but they aren't just rotating or translating

4:25 or doing a screw motion relative to each other.

4:28 Gears are the classic example of this.

4:30 But these cyclides seem to be a completely new example.

4:34 What other ways are there of making mechanisms

4:36 with continuous contact between parts that aren't just rotation,

4:40 translation, or screw motion?

4:42 All right.

4:43 This is slide-glide cyclides.

4:47 Thanks for watching.