Shapes and Solids of Constant Width - Numberphile

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Steve Mould discusses shapes and solids of constant width, including the Reuleaux triangle and the UK's 50p coin.
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[@Japanthewoman]

"I build a diorama - well I stuck a lego man to a ruler" Hahaha, ok you have my attention ;P

[@s_oh]

I just ordered the 3d set from you guys and was trying to explain the concept to my wife and 1 year old. They weren't getting it. I was lying on the rug surrounded by my son's toys and was astonished to see a little Fisher Price tractor attachment with a bunch of little 2d 3-sided solid width shapes in it!!

[@thepip3599]

He's just like "Oh yeah, they can also be used to drill square holes" Why did he say it with such a bored expression. FREAKING DRILLING SQUARE HOLES!!! That it really cool and counter-intuitive! Why didn't he seem to care!? (I had already known that, but if I hadn't, it would have blown my mind)

[@annas7847]

I know, right? Especially that you do it with a TRIANGLE! You make a SQUARE hole with a freaking TRIANGLE. That's everything else than mundane! EDIT: inb4 anyone points it out, yeah, I know, it's a triangle of constant with with rounded edges, but when talking about intuitive vs. counter-intuitive things, this has still a triangular-ish shape, so at the first glance it doesn't make sense at all.

[@brokenwave6125]

Drilling square holes is actually pretty mundane. Its been done for longer than youve been alive. Just buy a 10 dollar morticing bit and drill all the square holes you want.

[@uruiamnot]

You can probably google a square hole drill. But I don't think I have to. I saw one on display at a Ripley's Believe it or Not! museum when I was a kid. Mind blown, unforgettable. They exist.

[@jasonfaulkner8644]

Well said.

[@NoriMori1992]

8:53 - "And the answer is…" From his tone, I was so sure he was gonna say "…you can't."

[@RSPikachuAlpha]

Why do you keep popping up

[@NoriMori1992]

What?

[@2Cerealbox]

I already knew about shapes of constant width. What I learned today is that there are "lathe enthusiasts."

[@altaroffire56]

+Ryan N Ever heard of hobby machinists? Plus it's actually an awesome tool, and an unsurprising source of enthusiasm. It's the one of the few tools that can make itself (you can make a lathe with a lathe) and in case of apocalypse you could probably rebuild civilization with lathes alone.

[@AnasHart]

+altaroffire56 Lathes are awesome!! Yeah!! Check out +Clickspring

[@InsideInterpreting]

As a conference interpreter, I can assure you that there is no interest so small or esoteric that it doesn't have its own international conference.

[@smorrow]

Why wouldn't there be? There's 3D-printer enthusiasts, for instance...

[@FarceeTheFire]

How can anyone NOT be fascinated by lathes??

[@AntonAdelson]

"wankle engine" *giggity*

[@blazebluebass]

How much thrust does it provide?

[@FlyingButterHorse]

I don`t know, but I just read up on it. It`s pretty cool. Has some pretty intersting insights.

[@AntonAdelson]

@Fremen I saw a video on it. The problem was that the chamber wore off at specific places and most of the engine had to be replaced.

[@RACH5188]

"A 3 dimensional equivalent of a circle is a..." "A sphere!" Good job Brady!

[@DavidFosterZen]

I understand that some manhole covers use these shapes, because they are constant width, the covers won't fall down the manhole.

[@pogogo51]

Aw, man. Halfway through the video I thought: "That's cool, but could you do the same thing by drawing a sphere through each of the faces?" and you guys totally delivered.

[@NoriMori1992]

+pogogo51 Me too!

[@unvergebeneid]

I think a show about non-euclidean geometries would be a perfect fit for a future Numberphile episode. Goes right to the heart of geometry and its history :)

[@cakeman58]

If you visit the Museum of Mathematics in NYC, they have a "life-sized" ride called Coaster Roller based on these. It's maybe 10' long and you sit on a plexiglass platform and pull yourself across a variety of these shapes using a rope.

[@xenomann442]

You guys should start doing videos on the different branches of math, and what they're all about! Like topology, game theory etc. I'm interested to learn more about them, since I will probably never take a course in any of the advanced maths.

[@annas7847]

Why not?

[@Darker7]

Money. :Ü™

[@danthiel8623]

So not just a theory a game theory

[@thatoneguy9582]

and also it's really hard to look for videos about game theory without running into *That* channel

[@nokian9005]

It's so weird to see Steve Mould in numberphile! Great job, love to have you in this channel.

[@Centigonos]

It is people like the guys from Numberphile who make KZbin a site worth visiting.

[@AuroraDashPteriforever]

Here in Australia, one of our coins doesn't have constant width, yet it still works with vending machines. Spoooky.

[@timsonins]

Isn't only non-constant-width coin bigger than all the others on it's shortest width? Or am I looking at obsolete coins?

[@umcarainteressante]

Wow, this video was epic. I could feel the epicness of his journey to get his constant width solids.

[@doorhanger9317]

The ball-bearing that doesn't roll idea is already used - in a way. Those pencils that are designed for carpenters and builders are Reuleaux triangles so that they don't roll off slightly slanted surfaces that are common when building things

[@Metusalem979]

Never though about that. Made my hour.

[@doorhanger9317]

***** well, pencils with enough straight edges to be round enough to use tend to roll quite easy, so i guess it makes more sense than a square pencil.

[@benjaminhackett8896]

Zeet, Reuleaux triangles do not roll under the influence of gravity if there is not a surface on top of them. When a Reuleaux triangle tries to roll downhill, it's center of gravity shifts upward, so it just ends up rocking back and forth (unless it's a really steep incline). For the same reason, Reuleaux triangles would make terrible ball bearings, because they would take momentum from a system due to rotational energy and their changing center of mass. When people say Reuleaux triangles roll, it's always as sandwiched between two surfaces, never alone on a surface or about an axle.

[@grandexandi]

Another interesting possibility allows you to get a smooth ride from any shape, you just have to adapt the ground, instead of rolling it on a straight surface, you roll the shapes on a surface that evens out the irregularities of the shape. For a lumpy road, a triangular 'wheel' may be more efficient that a circular one. Those bumps you rounded off of the reuleaux triangle, if you place them on the ground, bump up, you can use triangular wheels.

[@MusicThatILike234454]

Just looking at it - I think from an engineering perspective you could use these objects in gears to transform rotational motion into any arbitrary combination of rotation + translation. (By varying the number of sides and the relative lengths of each side to the others.)

[@TheCSJY]

I remember having this 'lecture' from him last year, and two other people at Leeds playhouse! Was a fun day and it helps when he's as energetic and passionate about it as he is.

[@RobMoerland]

As young boy I used to draw lots of Reuleux triangles when lessons were boring. Didn't know what they were, but found them to be intriguing.

[@andreipastushuk3234]

Awesome video! One remark: it's not a square hole that you can drill with a Reuleaux bit, it will be squircle-shaped.

[@Thesupremeone34]

I dunno. I've used one before and I gotta say, on a square about a half inch wide, the corner radius couldn't have been more than a 32nd. I'd call that pretty damn square, considering how fast it is and the low likelihood of breaking the bit, it is clearly a massive improvement over attempting to mill the hole with any of those microscopic 1/32 or 1/64 end mills.

[@alpratsibz]

I cannot stress how much I like your channels, Brady. This video is astonishing - and the interviewed math man very funny. Thanks a lot and keep it on!

[@Larwood.]

And of course the reason these aren't used as bearings is they would wear out much faster than spheres.

[@brokenwave6125]

That, and it takes more energy for them to roll than a sphere of the same mass.

[@HatimERrbibi]

You are the nerdiest guys ever. Love you guys

[@jebwatson]

There's always that one crazy lathe enthusiast who just breaks all the rules... lol

[@ishouldhavetried]

I'm so glad you guys have a video on the Reuleaux Triangle! May sound weird, but it's my favorite shape!

[@Omni315]

The three rounded edges need explaining! Also what's the word for a 3d compass?

[@turboblazer]

I love math for stuff like this. Objects that have almost no practical use in application but exist anyways, because math says they work. Great video.

[@mariasmith8840]

Really liked this video! Loved the demonstrations.

[@RincewindIsMyHero]

This is one of the coolest numberphile vids yet.

[@musikSkool]

To make a wheel out of this, put a constant-width shape with gear-teeth inside a wheel that is also constant-width. It wont remove all the up and down motion, but it will reduce it to half.

[@jack_elliott]

Do this until you reach a perceivable level

[@tomatoso27]

Before I even watch it, let me tell you how happy it makes me to see a new numberphile or sixtysymbols video on my stream: A completely absurd amount!

[@Mathertron]

Steve's channel is great, highly recommend it.

[@andrewwmitchell]

Hi Brady, I love watching your channel with my 11 yr old son. We were just watching this video and he mentioned Australia's 50 cent piece. This is a do-decagon with no rounded edges and it works in vending machines even though it is not of constant width. We thought you might like to know. Keep up the good work. Andrew.

[@Meijimack]

Well Andrew, your son was pretty smart then and must be very smart now! " A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational symmetry of order 12.: the internal angle at each vertex of a regular dodecagon is 150°.

[@ZimmervisionCZ]

There's so much enthusiasm from Steve in this video. Shapes are fun, more then in a Sesame Street type of capacity. It's good that people think about this. There could be some application of these where a proper sphere just simply would not do.

[@Aviator27J]

I was thinking the same thing he was: that ball bearings can be made in those shapes to prevent rolling when loose. Pretty neat things!

[@TomboFry]

Hey, saw Steve present this almost exactly the same yesterday at the Maths Inspiration presentation in Birmingham!

[@RambobolaProductions]

"How big is your ruler"

[@DJalex155]

Saw this at the Maths Inspiration Lectures in Reading!! Such a nice guy! Great presentation, good day out :)

[@CAMERONJTC97]

Just saw James doing a lecture in Manchester! :D Absolutely amazing! :)

[@adamhansraj2314]

Who better to have some solids of constant width commissioned, than Steve Mould

[@eugenejefferson1838]

Just watched your presentation in Reading. Really nice

[@awesomesauceaaron]

More of this guy please!

[@HunterJE]

Shapes of constant width are also used in tamper-proof nuts, which need a specially made socket to turn (since a conventional parallel-jaw wrench will of course smoothly rotate all the way around)

[@ketoabigail3306]

Oh how I've missed seeing Steve on your videos! Have him do more videos! :)

[@nathanwilliams4554]

So is a circle actually a reuleaux square?

[@Zapii112]

+Nathan Williams You could say that with any regular polygon with an even number of sides.

[@benjaminhackett8896]

A Reuleaux shape works by sweeping out an arc from a corner to an opposite edge (when it rolls, exactly one corner will always be touching the top or bottom surface). All the points on the opposite edge are a radius of the corresponding corner, so by definition of a radius the shape is constant width. This is possible only for polygons of odd number of sides. On a polygon with an even number sides, you would have to sweep from one corner to the two corners on either side of the opposite corner, or from one corner to the opposite corner and let the arc extend to either side, or sweep from the midpoint of an edge or the two opposite corners, et al. However you sweep the arcs, as long as they are spaced evenly (for a regular polygon), you'll end up getting a shape that looks like it could be of constant width, but the curves are too shallow and therefore have less distance between them than the corners. In order to fix this, you can bend the edges outward further until it's measurably a shape of constant width. However, the furthest distance in a polygon of even number of sides is between two opposite corners, and when you push the edges outward to make the width constant, you end up making it completely uniform so that the shape becomes a perfect circle, with the constant width becoming the diameter of the circle.

[@heather19515]

It's technically a reuleaux octagon

[@josefranco480]

A circle is a regular polygon of n-sides. So it would be a reuleux regular polygon of n-sides

[@colleen9493]

Zapii112 doesn’t a circle have an infinite amount of sides? Or zero sides?

[@olipolygon]

The Soviets made a film projector using a mechanism utilizing a Reuleaux triangle in a square, rotated on an axis near one of its corners. This was so that a frame would be shown for a moment before being moved by the mechanism to the next frame, wait, next frame, wait, next frame, wait... which made the film look smoother.

[@guillaumegeaymond503]

quite hard to write it phonetically in english actually. Reuleaux is: the french "r", the "u" of "mud", a regular "L" and finally an "O"

[@IllidanS4]

I love your shapes of constant width!

[@ItsSansom]

Saw the video for How Wide Is Your Circle. Had no idea what I was looking at until now

[@tinkerhelll]

I had to have these - check out the dice as well. Best nerd holiday gifts ever.

[@Mala2335]

Can you guys do a video on Euler's identity? You always explain things amazingly and in easy to understand terms.

[@domvito4728]

Incredible video. I love these :)

[@Ragmulokos]

this badass guy, steve mould, bring him more often!!!

[@markdouglas2464]

i want to see that bike in action.

[@iainhewitt]

The realeaux triangle was mentioned on QI some time ago as how to drill a square hole. Ross Noble was very excited as he had just been talking about a rolo toblereone combo (which sounds amazing in a Geordie accent - look it up) and he got points for accidentally stumbling on almost the right answer.

[@Swimmer7596]

I really like Steve. Brady should do more videos with him! :)

Wait,what about that Wankel engine? Why didn't you elaborate on that? Other than that, one of the most fun episodes IMHO

[@KheledB]

So just a thought/question...If a circle is a shape of constant width, and the ratio of the diameter to the circumference is equal to pi do these other shapes share a similar constant with the ratios of their diameter to the circumference? I.E. Could you use a similar formula to figure the circumference (or is it perimeter?) of a Reuleaux triangle?

[@sjholden]

Yes. And it's the same formula: pi*d, for all the curves of constant width. It's Barbier's theorem - it's pretty obvious for the equalateral triangle case - it's 3 arcs that are each 60 degrees of a circle with twice the diameter of the final shape (see the construction in this video). For the other polygon's the proof is more complicated, but googling the name will turn up the details.

[@greeninphysics]

You could offer metal shapes of solid width if you use a 3D printer to make them. Selective laser melting allows the creation of metal 3D shapes. Also, have you considered other shapes, like the square, or even a non-symmetric polygonal shape?

[@Darkassassin09]

i don't know about a non-symetric shape, but a square or any other shape with an even number of sides/points gives you a circle

[@Nugglashine]

I got really excited when he mentioned rotary engines...

[@MinuteMaths]

Great Explanation, thanks

[@superhunk1989]

Absolutely love this!

[@PeradonCentury]

Great stuff. Might even buy a couple of those things.

[@Car0linaPh03nix]

Can't wait for those 4D shapes of constant width. Maths Gear promises them within a year or two!

[@rossgirven5163]

lpreams and how exactly do you suppose they would represent 4 dimensions in the real world?!

[@Car0linaPh03nix]

Ross Girven it was a joke, I believe based on an actual post on Maths Gear's website

[@rossgirven5163]

i see, my bad

[@MrXanaduWorld]

I'm 40 and I've learnt something new! Thanks! :)

[@LeeLekCheong]

Steve is such an awesome guy!

[@Ndo01]

great vid!

[@Iskaldr]

I saw you at the Maths Inspirational Talks in the Palace Theatre!

[@simoncrabb]

And you, Sir, are crazy for not earning affiliate commissions from maths gear!

[@easye9186]

The hero with the lathe, made this video what it is

[@MG-cp8xk]

love the video. Metal Solids of Constant Width could be cast, then milled to a more precise finish.

[@mohammadalmasri1386]

Nice video!

[@TheRandyTown]

I am so getting these for Christmas.

[@tamarababala]

Actually Fantastic!!

[@TIMS3O]

Great video

[@QuixoticVerities]

I seem to remember an older video of someone using the reuleaux triangle as a shape for bearings for some reason, can't quite remember why.

[@pdcoates]

Hi i have been interested in curves and solids of constant width since the late great Martin Gardener alerted me to them, he gave as an illustration one of the reasons early submarine makers used wooden curved templates to check profiles. In my back yard i have found one in nature, a gum nut. At this time of the year my patio is covered with them and they roll under the feet like ball bearings, they are only 3 or 4 millimeters in diameter. I have a photo of them that i am trying to copy into this thread, but so far no luck. Anyway thanks for the great series.

[@pdcoates]

p.s. you are having far too much fun for a physicist and probably putting the profession in a bad light. When i went to school it was not seemly for serious scientists to behave in such a flippant manner.

[@scottaseigel5715]

Does every Platonic solid have a possible “Meissner” solid of constant width? I’d LOVE to see this from a dodecahedron!!

[@MrSuperXToast]

Brady is my favorite!

[@ronaldbeimel9575]

Although you may not be able to turn the Meissner Tetrahedron on a lathe, it is possible to make metal castings of it using the plastic ones as the pattern.

[@ArjanGroenenboom]

I love this kind of nerdy things! Give me more!

[@OmegaRainbow]

super interesting!!

[@JonathanMLM]

This is mind blowing.

[@tedchirvasiu]

Yes, finally!

[@derekbloom633]

I would think that while used as a ball-bearing, the edges and points would wear more easily than a spherical bearing. Of course wearing would also be affected by the material of which it is made, but would a spherical object be less prone to distortion than say the "corner" of one of these objects?

[@chrisofnottingham]

Having hoovered up a UK 50p piece that exactly fitted into the extension wand, I can confirm there was no orientation in which it became loose, despite the curvey sides.

[@uzimonkey]

I think my spirograph set had some of these. They made some interesting patterns.

[@SoeaOu]

great vid, very entertaining :)

[@tamarababala]

Very nice!

[@Merthalophor]

That is so cool! Have to get one of these :D

[@marcusrees5364]

Can someone explain why some of the tetrahedron's edges have to be rounded? Also, when making a solid of revolution, would that work around any axis going through the centre? Because if so that could get you some wacky looking shapes of constant width!

[@skilliyay]

In the video, the solid of constant width is produced by "sweeping" the Reuleaux triangle through 360 degrees. Can some explain what solid shape would result if at the same time as sweeping 360 degrees, the triangle is also rotated 360 degrees about its centre? A bit like a corkscrew sweep if that helps my explanation.

[@rossgirven5163]

skilliyay Take any shape and rotate it 360 deg on its plane and you create a circle. then sweep it to make 3d you got a sphere

[@lawrencedoliveiro9104]

2:17 8:17 Actually, no, the Wankel rotor doesn’t need to be a Reuleaux triangle. The curvature of the sides can be shaped to give a particular compression ratio. Remember, the rotor slides around the casing while spinning, it doesn’t roll, so the Reuleaux property is irrelevant. In fact, you can see the discrepancy between the curvatures of the Reuleaux and the actual rotor in that diagram at 2:17.

[@Ghost00117]

This is very counter-intuitive. Just looking at it you don't think it will work and then it works. Great video :D

[@frankharr9466]

This is the video that brought me into this world! Incidentally, yay yardstick! Hey, I just noticed, he never gave use the formula for the are or volume! I'm sure I could find it.

[@harrytodhunter5078]

This guy makes maths fun and interesting

[@bontrom8]

I had the equilateral triangle version pop up in a project where I was depending on the constant diameter to determine roundness. Took me by surprise.

[@Kevin_Eder]

I like that you can actually get the two wooden rulers upgrade.

[@general9064]

It's quite easy to explain why it happens so , just a simple property of a circle and it's tangent. the the distance from the tangent to the center ( vertex of the opp side) is always its radius or side of the triangle. :)