Why are the formulas for the sphere so weird? (major upgrade of Archimedes' greatest discoveries)

0:05 Welcome to an extra-extra-extra-special Mathologer video.

0:09 Why extra-extra-extra-special, three extras?

0:11 Well, first extra, it's the 100th Mathologer video.

0:15 There, the thumbnails for the first 99 videos.

0:19 It's been quite a journey:) Second extra: it's my first crossover episode.

0:25 For the first time, I'm collaborating with another channel.

0:28 Mathematician and 3d printing artist Henry Segerman just

0:31 published a video that complements this Mathologer video.

0:34 I'll have more to say about Henry and his wonderful

0:37 claw contraption down there in the last part of this video.

0:41 Now, most importantly, third extra,

0:43 in today's video we'll make a little bit of mathematical history.

0:48 I'll tell you about a couple of brand new,

0:51 game changing discoveries about the good old sphere

0:54 that so far only a handful of mathematicians know about.

0:58 Yep, you've been invited to opening night,

1:00 a mathematical premier:) Have a look at this.

1:04 You're all familiar with this kind of baggage carousel, right?

1:08 Really ingenious how it flows around corners don't you think?

1:11 What makes this work is that the crescent moon shaped pieces the belt

1:15 consists of have circular sides that can seamlessly rotate one against another.

1:21 Let's steal this idea.

1:23 First create a circle-shaped conveyor belt.

1:25 And then have this conveyor belt perform a figure 8 dance like this.

1:30 Very pretty isn't it?

1:32 On close inspection what is happening here

1:35 is that the circle gets turned inside out.

1:38 Have another look.

1:40 Right, points that originally are close to the centre of the circle

1:44 end up on the outside of the new circle, and vice versa.

1:48 Very nice, but so what?

1:50 Is there any more to this?

1:53 Well, let's put a hemisphere on top of this circle,

1:59 Whoa, that animation really turned out very nice, don't you think?

2:02 Okay, now in the same way as before, let's turn the hemisphere inside out.

2:09 What on Earth is that?

2:12 What's that inside-out-shape?

2:14 Well, on close inspection it turns out to be approximately

2:19 a cylinder with a cone carved out from the top.

2:22 In fact, by using thinner and thinner moons,

2:25 the new shape will get arbitrarily close to a true cylinder minus a true cone.

2:32 There, thinner moons.

2:34 Even thinner.

2:35 Infinitely thin.

2:37 BUT, turning the hemi-sphere inside out like this doesn't affect the volume.

2:42 This means that the volume of the hemisphere is

2:45 equal to that of the cylinder minus the cone.

2:49 Now, finding the volume of the sphere that's a tricky one, right?

2:54 One of Archimedes' great achievements.

2:56 BUT with our baggage carousel trick,

2:58 all we have to do to find this tricky volume is to subtract

3:02 the volume of the cone from the volume of the cylinder, and then double.

3:07 And that's all easy.

3:08 The volume of the cylinder is just base times height.

3:11 AND, with the height being the same as the radius R of the sphere,

3:15 the volume of the cylinder pans out to be pi R^3.

3:18 And pointy things always have one third

3:20 the volume of cylinder things, also pretty easy.

3:24 So we subtract 1/3 pi r^3.

3:28 Then doubling to go from the hemisphere

3:31 to the sphere gives the volume formula of the sphere.

3:35 Tada.

3:35 4/3rds pi r^3.

3:36 From baggage carousel to the tricky volume formula

3:39 of the sphere pretty much at a glance.

3:42 How great is that?

3:44 Definitely made my day when a couple of weeks ago my colleague Andrew Kepert,

3:48 a mathematician at the University of Newcastle here in Australia pitched

3:52 this and a couple of other ingenious sphere related ideas to me.

3:56 And here we are, a Mathologer video featuring Andrew's ideas:) What I'll do

4:01 next is flesh out and milk this baggage carousel idea for all it's worth.

4:06 After that it's on to Archimedes' claw,

4:09 Andrew's second great idea and possibly the climax of this video.

4:13 That's also where 3d printing master Henry Segerman makes his appearance.

4:18 Anyway, prepare for a life changing experience as far

4:21 as all the spheres in your life are concerned:) Enjoy!

5:08 The inverted hemisphere is a cylinder minus a cone.

5:11 Really?

5:11 Well, the animation that I just showed you was very convincing but, of course,

5:16 to be absolutely sure that this really works,

5:19 we need to conduct a proper mathematical check that the shape

5:22 on the right is really a cylinder minus a cone,

5:25 with both shapes of just the right dimensions.

5:28 Okay, is there anything that can go wrong with the cylinder?

5:32 Well, let's have another look at how exactly the hemisphere turns inside out.

5:37 There, the orange arrow points at the highest

5:40 point of the hemi-sphere and the green arrow marks the point around which we are

5:46 figure-8-ting as we turn the hemisphere inside out.

5:48 Focus on what happens around those two special points.

5:53 Ready?

5:53 Go:) Okay, pretty obvious,

5:56 the mantle of the inside-out-shape is that of a cylinder

5:59 of the same radius and height as the hemisphere,

6:02 and the lowest point of the inside-out shape is right in the middle.

6:07 All convinced?

6:08 Great:) So, nothing can go wrong in term of the cylinder.

6:12 But what about the carved out bit?

6:15 Is that really a cone?

6:16 Basically what I am worrying about here is

6:19 that those mantle lines that appear to be straight,

6:21 as in a cone, may actually be a tiny bit curved.

6:26 I'll just quickly sketch one way of proving that looks are really not

6:30 deceiving:) The keen among you can then fill in the details in the comments.

6:34 Okay, for easier tracking of distances,

6:36 let's also display what the vertical cross-sections of our shapes appear to be.

6:42 There.

6:43 Now, cut everything in sight with a horizontal plane at height h.

6:50 Then it is clear that the red cross-section on the left will be

6:53 a circle and that the red cross section on the right will be a ring.

6:57 And it's also easy to see that the circle

6:59 and the ring have exactly the same area.

7:02 Is that clear?

7:03 Right?

7:03 During the inside-out dance, the circle cross section on the left continuously

7:07 turns into the ring cross section on the right,

7:10 and so the circle and the ring must have same area:) Now,

7:14 using this equality, you can then calculate the inner radius of the ring.

7:19 For this hire a couple of middle school kids and put them to work.

7:22 All they have to do is to write down the formulas for the areas

7:26 of the red circle and the red ring and then solve for this inner radius.

7:29 They've done stuff like this in school a million times.

7:33 Piece of cake.

7:34 Eventually they'll report back to you that the inner

7:38 radius exactly equals the height h at which we cut.

7:42 That's a linear relationship.

7:44 And that means that those mantle lines are really straight.

7:51 And so the missing part in the middle is really, truly a cone.

7:56 All good?

7:56 Everybody happy?

7:58 Great!

7:58 Proof complete.

7:59 As I said, just a sketch and as long as you got the gist of what is

8:03 going on here that's good enough to be

8:06 able to appreciate what I am about to say.

8:09 Okay, while Andrew's method of turning the hemisphere into the cylinder

8:13 minus cone shape on the right is definitely brand new,

8:16 and brilliant, the basic idea is not new.

8:20 400 years ago the mathematician Bonaventura Cavalieri

8:25 was the first to realise that the cylinder

8:27 minus cone configuration can be used to derive the volume formula of the sphere.

8:33 Of course, like Archimedes, Cavalieri was not familiar with air travel

8:38 and fancy baggage carousels and so it's

8:40 not surprising that his line of reasoning

8:43 was a bit different from ours:) In particular,

8:46 Cavalieri doesn't motivate why he is comparing

8:48 the two shapes over there and starts by showing,

8:51 from scratch, the crucial property of the two

8:54 shapes that the baggage carousel gives us for free.

8:57 What he does is he starts by showing that the circle and ring cross-sections

9:02 of the two shapes have the same area no matter the height at which we cut.

9:07 Right, we got that equality for free.

9:10 He then finishes his proof using Cavalieri's principle,

9:14 an ingenious idea named after him.

9:17 Cavalieri argues that BECAUSE there are cuts of equal areas at all heights,

9:22 the two shapes must also have the same volume.

9:26 Equal area cuts at all heights implies same volume.

9:31 Makes sense:) Cavalieri considering those two

9:34 shapes was probably inspired by Archimedes,

9:37 the original discoverer of the volume and area formulas of the sphere.

9:42 I won't go into the details of Archimedes' proof.

9:45 But let me at least give you a glimpse of his setup.

9:49 There is also a sphere/cylinder/cone combo at play

9:52 but it's all a bit more complicated.

9:55 If you are interested in more details about

9:57 Archimedes' proof there is a link in the description

10:00 of this video to an old Mathsmasters article by Marty

10:03 and me in which we rave about this proof.

10:07 Anyway, I think Andrew's new way of naturally turning the hemisphere

10:10 into the cylinder minus the cone shape

10:12 is an important upgrade of Cavalieri's discovery,

10:15 which in turn was an upgrade of Archimedes' original proof.

10:19 As I said, we are really making mathematical history today,

10:23 mathematical history that's been in the making for thousands of years.

10:27 You saw it first on Mathologer.

10:29 Pretty amazing, isn't it?

10:31 Okay, before I move on to the next main topic,

10:34 let me just mention one more beautiful insight that arises from Andrew's dance.

10:38 Here we go.

10:39 If we replace the hemisphere by a paraboloid something extra special happens.

10:44 What's a paraboloid.

10:45 Well, that's the solid we get when we spin the graph

10:49 of our good old friend y= x squared around the y-axis.

10:53 Well, to be nitpicky, that's NOT the whole paraboloid but rather part

10:58 of the paraboloid below a certain height and turned upside down.

11:03 Whatever, let's call this bit a paraboloid anyway.

11:06 Now here is what the corresponding inside-out shape looks like.

11:09 And here is the amazing bit.

11:11 The inside-out-shape turns out to be

11:13 the surrounding cylinder minus the same paraboloid.

11:17 Challenge for you and your army of middle school minions:

11:22 Work out the details in the comments.

11:24 Anyway, how great is that?

11:26 And so what do we do next?

11:28 Well, solve for the volume of the paraboloid of course:) The volume

11:34 of a paraboloid is equal to half the volume of the surrounding cylinder.

11:39 Archimedes was also the first to discover this surprising formula.

11:42 Did I ever mention that Archimedes is my no 1 mathematical hero,

11:47 with Euler being a close second?

11:50 Actually, at this point I just have to tell

11:53 you about yet another one of Archimedes' ingenious insights.

11:56 So much great stuff to point out here:)

11:59 For this insight let's spool back to this earlier slide.

12:03 The volume of the hemisphere is 2/3rds pi r cubed.

12:09 Oookay.

12:09 Now, pi r cubed by itself what's that?

12:13 Well, pi r cubed remember we had this before,

12:16 that's just the volume of the cylinder:)

12:18 Let's shuffle this formula around a bit.

12:22 There.

12:22 And there.

12:23 Hmm, okay.

12:24 But now its also clear that the hemisphere fits snugly inside the cylinder.

12:29 And so what our identity says is that the ratio of the volume

12:33 of the surrounding cylinder to the volume of the hemisphere is equal to 3 to 2.

12:38 And of course the same is true if we double everything in sight.

12:42 The ratio of the volume of the surrounding

12:45 cylinder to the volume of the sphere is 3:2.

12:49 How nice it that:) Archimedes was so

12:51 impressed by this simple relationship that he

12:53 had a version of this cylinder/sphere diagram

12:55 over there engraved on his tomb stone.

12:58 In fact, he noticed something else which makes all

13:01 this doubly amazing:) It turns out that the ratio

13:04 of the surface area of the cylinder

13:06 and the surface area of the sphere is ALSO 3:2.

13:10 So both the volumes and surface areas are in the ratio of 3:2.

13:15 Did you know that?

13:16 Faantastic:) Anyway,

13:17 this is our cue to now have a closer look at the surface area of the sphere.

13:30 Okay, so we've mastered the volume formula of the sphere.

13:32 What about the surface area?

13:34 Let me just go for it and power through a simple

13:37 and beautiful argument that gets us there without breaking a sweat.

13:40 I'll start with a quick warm-up exercise that quite

13:43 a few of you will already be familiar with.

13:46 Let's have a look at a disk of radius r.

13:49 We can think of this disk as being

13:51 made up of infinitely many concentric circles.

13:53 Here I've just highlighted a few of these circles.

13:56 Let's cut the circles open at the top and then straighten them out.

14:01 In this way the disk unfolds into a triangle of height r.

14:08 What's its base?

14:10 Well the circumference of the disk of course.

14:13 That's where the base comes from.

14:15 Right?

14:15 And so the area of the circle is equal

14:18 to the area of the triangle which is 1/2 base times height,

14:21 1/2 circumference times R.

14:23 Nice argument, don't you think?

14:25 What's extra nice is that in this way we've discovered a simple relationship

14:31 between the area and the circumference of the circle without any pi in sight.

14:36 Now, of course, once we know this relationship then

14:39 also knowing the formula for either one of these quantities,

14:43 immediately gives us the other formula.

14:45 For example, let's say we know the formula for the circumference 2 pi r.

14:49 Then plugging in we immediately get the area formula.

14:53 Hands up, who's seen this proof before?

14:57 Hm, as expected quite a few of you:) Okay.

15:02 The nested circles that we started with are reminiscent of an onion

15:08 and the argument that I just showed you is often called the onion proof.

15:13 Dad joke alert: Did that proof make you want to cry?

15:17 Yes, I know, that's a bad one, but I couldn't resist:) Now,

15:21 what even those of you in the know may not

15:24 know(:) is that the same argument also works in 3d.

15:27 Let me show you.

15:28 Replace the layered disk by a layered ball.

15:31 There layered.

15:32 Like an onion.

15:33 Ok that's it for onions, I promise.

15:37 Then we can unfold all layers into disks of the same area like this.

15:45 And so the ball unfolds into a circular cone of height r.

15:50 And, of course, the base area is equal to the surface area of the ball.

15:54 And so the volume of the ball is equal to the volume of the cone,

15:59 1/3rd base area times height, and so 1/3 surface area times r.

16:04 As in 2d, in this 3d onion proof pi is nowhere to be seen.

16:09 Cute.

16:10 And now, since we already know the volume formula of the sphere,

16:13 we can figure out the formula for the surface area.

16:19 Tada again:) And so at this point we've proved

16:26 both the volume and area formulas of the sphere.

16:31 And now you also know why these formulas look so

16:34 similar 4/3rds at the top and 4 at the bottom.

16:37 That was not hard, was it?

16:39 And Archimedes definitely looks pretty happy over there and has probably

16:42 pushed the like and subscribe buttons by now:) What about you?

16:46 Actually, although what I just showed you was a pretty quick

16:49 way to derive the the area formula from the volume formula,

16:51 there is an even quicker way.

16:53 If you are a calculus demon, at some point of your career you may have noticed

16:58 that the surface area formula is just the derivative of the volume formula.

17:03 Wait what?

17:04 Yes, quick check.

17:06 The derivative of r cubed is 3 r squared.

17:11 Okay simplify.

17:12 The derivative of the volume formula is the area formula.

17:16 Magic:) You may think that this is a coincidence

17:19 until you try the same for the circle, the 2d counterpart of our 3d sphere.

17:25 Turns out the derivative of the area formula

17:28 for the circle is the formula for the circumference.

17:31 There.

17:31 pi r squared and 2 pi r, the formulas for the circle.

17:35 And, of course, the derivative of pi r squared is indeed 2 pi r.

17:38 Magic again:) Really easy challenge for the calculus demons among you,

17:44 even those of you who are still in high-school:

17:47 Can you provide an onion powered calculus explanation for this derivative magic?

17:52 If so, dazzle the rest of us by leaving

17:54 your thoughts in the comments:) Go on give it a try.

17:58 Otherwise, I'll also link to a brief

18:01 explanation in the description of this video.

18:04 Another challenge.

18:05 Check that Archimedes' 3:2 really also works

18:08 for the surface areas of the cylinder and the sphere.

18:12 Okay back to our formulas.

18:14 Which of the two formulas do you like

18:16 better from a purely aesthetic point of view?

18:19 I think pretty much everybody would go for the area formula 4 pi r squared.

18:24 No fraction in sight and so cleaner, prettier, more beautiful.

18:29 Alright, we are now ready for the second main attraction of this video,

18:33 Andrew's ingenious new way of seeing at a glance

18:35 why the area formula is what it is.

18:44 Does the pi r squared in this formula ring a bell?

18:47 Of course!

18:47 pi r squared is just the area formula for a circle of radius r like,

18:51 for example, an equatorial circle cross-section of the sphere or its shadow.

18:57 In other words, the surface area of the sphere

19:00 is exactly 4 times the area of its shadow circle.

19:03 A couple of years ago 3blue1brown set out to find a nice

19:06 visual way of seeing at a glance why this is true.

19:10 Today I'll show you the ultimate one

19:12 glance way of understanding this relationship from scratch,

19:15 something that I think is even nicer

19:17 than what was shown in that 3Blue1brown video.

19:20 This is the result of another one of Andrew's brainstorms.

19:24 Andrew started by observing that there is a better way to write

19:29 the area formula if you're hunting for a one glance proof.

19:32 There pi times 4 r squared.

19:35 Since 4 is also 2 squared we can pull out the 2s in the exponents.

19:40 And what that means is that the area of the sphere

19:44 is equal to the area of the circle of double the radius.

19:50 And now, that this is true can indeed be seen at a glance

19:53 using an ingeniously weird unfolding

19:55 of the sphere's surface into this larger circle.

19:57 Here is what we do.

19:59 We start with the infinitely many meridians of longitude.

20:02 Here are a few of them.

20:04 Okay so every meridian is a semi-circle that joins

20:07 the north and the south poles of our sphere.

20:10 And, except for the two poles,

20:11 every point of the sphere is contained in exactly one of these meridians.

20:15 Let's focus on one of the meridians, this one here.

20:19 There is a radius of the pink circle right below this meridian.

20:23 There it is.

20:24 We now swivel the meridian 90 degree clockwise around this radius.

20:31 Swivel.

20:32 Fits perfectly.

20:33 So let's do the same with all the other infinitely many meridians.

20:38 I'll just illustrate with the few meridians highlighted over there.

20:42 Okay, after we folded down all

20:45 the meridians they completely fill the pink circle.

20:49 To really get a feel for this unfolding

20:51 let's turn the sphere into an Earth globe.

20:54 Ready?

20:54 There unfold.

20:56 Each individual meridian folds down nicely,

20:59 the meridians all snugly fit together and fill the big circle when folded down,

21:07 without overlapping.

21:08 Wonderful.

21:09 So, there, that's how Andrew's unfolding morphs the surface

21:11 of the sphere into the circle of twice the radius.

21:14 All clear?

21:15 Now, Andrew's simple transformation happens

21:17 to have an incredible, magical property.

21:20 Obviously, the transformation distorts the sphere like crazy,

21:23 but there is one thing it does NOT distort: area!

21:28 To use the jargon, Andrew's magical

21:30 transformation turns out to be area-preserving.

21:33 This means that any region on the sphere will be transformed

21:36 into a distorted region somewhere on the large circle of exactly the same area.

21:42 For example, have a look at Australia on the globe.

21:45 Then the area of Australia on the globe is exactly

21:47 the same as the area of its image on the circle.

21:51 Amazing, isn't it?

21:52 And the same is true for any other region

21:56 on the sphere and its image on the circle.

22:00 In particular, this means that the area of the entire

22:03 sphere is the same as the area of the circle.

22:08 Tada.

22:09 Proof complete.

22:10 Well, not so fast.

22:12 Definitely, everything I said is true.

22:14 And you know me, right?

22:15 Would I lie to you?

22:17 But of course that's not the way things work here in MathologerLand.

22:21 I have to at least sketch a proof.

22:24 First, to see how things could potentially go wrong,

22:28 let's do the same sort of unfolding for a cylinder of height h.

22:33 There this cylinder is of height H.

22:36 Now fold down all the mantle lines.

22:38 This means that the unfolded cylinder mantle is this ring.

22:41 And, so: Are the surface areas of the ring and cylinder mantle the same?

22:45 What do you think?

22:47 The answer is: Definitely not always.

22:49 That's easy to see.

22:50 Just think about it.

22:51 Keeping the height of the cylinder fixed, we can make its area as close to zero

22:56 as we wish by making the cylinder thinner and thinner.

22:59 On the other hand, the ring area will never get smaller than the area

23:04 of the circle whose radius is equal to the height of the cylinder.

23:08 There, basically that's a circle of radius H.

23:11 And so the two areas are definitely not the same.

23:15 Hmm.

23:15 So, why is the sphere so magical?

23:17 Why does unfolding the sphere not distort the area?

23:20 And can it be easily seen why this is the case?

23:24 Well, remember Henry's claw contraption,

23:26 which I showed you at the very beginning?

23:29 Henry's gizmo is a 3d printed version of Archimedes' claw,

23:34 another one of Andrew's ingenious inventions.

23:36 Have a look.

23:37 So, here we have the whole surface area of the sphere covered by special

23:42 3d crescent moons that are thickened up versions of the meridians from before.

23:46 Let's fold down those 3d moons together with the meridians.

23:51 As we fold, the moons keep touching without a gap

23:55 and end up forming a thickened up version of our target disk.

24:00 There again.

24:01 No gaps, completely smooth, magic.

24:03 And then, as we increase the number of moons,

24:06 we are getting closer and closer to the real thing:

24:12 there you have it, the surface of the sphere rigidly turning into the disk,

24:20 without any distortion.

24:22 Tada, NOW the proof's complete, right?

24:25 Well, again, not so fast:) Take my word for it,

24:29 we are getting closer to the true explanation of this area-preserving property.

24:33 However, I am fairly sure that what most of you are thinking here

24:38 is that because the moons approximate

24:40 the sphere and the circle better and better,

24:42 the total surface area of the bits of the moons facing out will

24:46 get closer and closer to the surface area of the sphere and the circle.

24:50 Which then somehow completes the proof.

24:53 Right, seems at least plausible?

24:55 However, that's not quite it.

24:57 Have a close look at the moons.

24:59 Arranged in sphere mode and circle mode we are

25:02 really looking at the moons from completely different perspectives.

25:06 And so, although there are never any gaps between the moons as they move,

25:12 how can we be sure that these different

25:15 perspectives really translate into the same surface areas.

25:18 Not obvious, right?

25:19 Okay, let's have a closer look at where those moons actually

25:24 come from and what is happening here in terms of areas.

25:27 There, the sphere and the circle before and after folding down the meridians.

25:32 The meridians divide the blue and the pink into the same number of regions.

25:37 And so, to show that the blue and the pink have the same area we

25:42 just have to show that a blue region has the same area as a pink region.

25:47 Okay, so to be able to compare these two regions,

25:51 let's fold down the blue region and align it with the pink one like this.

25:56 Okay, now watch this.

25:57 You just witnessed a little miracle.

25:59 Did you notice how the travelling sphere always exactly

26:03 touches the sides of the blue and the pink regions.

26:11 Let's watch that again There one 3d moon served.

26:18 Now let's get rid of the sphere and let's

26:21 focus on the growing moon surface by itself.

26:25 What you are looking at here is a circular

26:31 cross-section of the travelling sphere and the pink

26:36 and blue cross-sections are both roughly diameters

26:39 of this circle and so approximately of equal length.

26:43 And the same is true all along the 3d moon.

26:47 And now we are in a very familiar situation, where two shapes are the same,

26:52 cross section by cross section, at corresponding levels.

26:56 Remember Cavalieri?

26:56 And, as usual, we can conclude that we

26:59 are dealing with surfaces of the same area.

27:02 Well, since corresponding cross-sections are not exactly

27:05 the same in the case of large

27:09 moons the surface areas of the two regions will not be exactly the same.

27:13 However, as we push the number of moons to infinity,

27:16 in the limit we'll have equality which is what we've been chasing:) Okay,

27:21 so the sphere and the large circle have the same area.

27:25 Very, very pretty.

27:26 Also, looking a little closer,

27:28 we can similarly prove that Andrew's transformation is always area-preserving,

27:33 that any country on the sphere transforms to a country

27:37 on the circle of the exact same area.

27:40 To make this argument rigorous still requires quite a bit of detailed fiddling.

27:45 But what you see in front of you really nicely captures the core of the proof.

27:51 And that's definitely good enough for now.

28:01 Okay and that's pretty much it.

28:03 But before I sign off for today I just wanted to thank Andrew for inspiring

28:07 me to make this video and for creating

28:09 most of the fancy animations in this video.

28:12 And this is a Mathologer command: you must check out Andrew's playlist

28:16 of relevant animations linked in from the comments.

28:19 Also thank you to Henry for creating the 3d printed version of the claw.

28:24 Another command: make sure to head over

28:27 to Henry's channel for his video about the claw, plus free 3d printing files.

28:32 Then you can print your own copy of Archimedes' claw.

28:36 As usual, there will be more thank-you-s

28:39 at the very end of this video:) I should mention

28:42 that in terms of Archimedes' claw I've really just

28:45 touched upon a few of the secrets it holds.

28:48 There is definitely enough material for a part 2.

28:51 Just to whet your appetite,

28:53 there is the beautiful explanation and important implications of the fact

28:58 that as the claw opens all its moons will always touch without leaving any gaps.

29:04 Then there is the important connection to Heinrich

29:07 Lambert's famous area-preserving map of the sphere.

29:11 Remember polymath Heinrich Lambert?

29:13 We keep running into him.

29:14 He was the first to prove that pi is an irrational numbers,

29:17 he was one of the inventors of the hyperbolic trig functions.

29:20 Then there are all of Lambert's amazing maps, and so on.

29:23 Anyway, Archimedes' claw, there is a lot more to be said and to be discovered,

29:28 but let's call it a day.

29:30 I'll finish with one more of Andrew's

29:33 stunning animations illustrating a completely different

29:35 moon powered way to turn the surface of the sphere into the circle.

29:40 Enjoy.