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)
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.
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.
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.
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!!
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.
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.
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.
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 :)
+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.
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.
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.)
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.
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.
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.
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°.
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.
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.
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
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.
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!
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?
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.
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.
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)
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?
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.
3:10 Actually, Australian 50c coins are dodecagons with flat faces around the edge, which work in vending machines here. (Perhaps having so many sides they're close enough to constant width for the machine to deal with).
I was thinking about this. Would these shapes be less material than a sphere of the same radius? If so there would be use for them to make cheaper products like bearings or anything that rolls without an axle.
Yes the Reuleaux triangle takes up less space then a cirlce. They are used in the real world for certain things. I know they are used in submarines and such, and also being able to drill holes besides circle shaped holes can be useful. I didn't watch the whole video so I don't know if he said it, but you can drill a square hole with one of these.
you would not want to use these for bearings because the center of mass is not in the symmetrical center so it would vibrate/shake at medium to high speeds
2:178:17 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.
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.
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.
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!
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.
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.
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?
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. :)
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.
Huh. I'm American, but the UK 20p coin is my favorite coin. And now I understand why it is that way. I have an Australian 50¢ coin. It's got 12 sides, I think, but they're all flat. What's going on there?
Our 50 cent coin is a regular dodecagon, that is, 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°. Vertex means a corner or a point where lines meet (plural: vertices.)
TheTurnUpTurnip I'd think it would bounce off at an angle. Since a perfect sphere is the only thing that's always perfectly linear with the ground, no matter which way you turn it. (if you get what i mean)
As long as you hit it squarely on either side, you should be just fine. It would behave more and more volatile the futher you approach any of the points, almost to the point of bouncing a football (american) on it's points.
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.
Surely solids of constant width could be used in place of regular ball bearings to reduce wear as well as not rolling away. The point of contact would be constantly changing as opposed to staying the same, as with ball bearings.
It would increase wear of the bearings though. Constant width shapes have edges. An edge will wear much faster than a sphere, and when the edge wears it is no longer constant width. By using spheres you have more wiggle room in terms of wear.
Is there a 4 dimensional version of both the sphere and this object described? Is there actually any sense to a two dimensional plane on a 3 dimensional object in a 4 dimensional verse?
Keeled, I guess so. The length of the sides of triangle will give you the radius, therefore giving you the circumcumference. And as it is 3x60 degree arcs, divide the circumference by 2. Although I could be totally wrong as I haven't done circle theorems in years.
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.
Colorado School of Mines (Colorado, USA) Has the Reuleaux Triangle as the main shape in their emblem. Each club fair gives out 5 of the tetrahedrons to incoming freshmen.
3D shapes already are shapes of 4D width because in the time they exist in the same state, throughout that time they still have the same constant width and constant time
you showed how to build some shapes of constant width in 2d, are there some that cannot be constructed that way? likewise, in 3d, you showed us a solid of constant width that is not of rotation, are there others? in other words, can you characterize shapes (or solids) of constant width?
In the Meisner Pyramid, why are three edges sharp and three rounded? Is there a way to make the shape completely symetrical by applying a little less rounding to all six edges?
I wonder, is there a way to round off the Meissner Tetrahedron so it's fully tetrahedrally symmetric again? - instead of rounding off three edges, round them all off but just slightly? Or is that mathematically impossible? Also, how about other platonic solids? Does this only work with tetrahedrons?
I'm Australian. Our fifty cent coins have twelve sides and they don't seem to be rounded; a regular dodecagon. How would our coin machines work? All our other coins are circular and smaller in diameter.
My guess would be that it would turn to the position were it has the smallest diameter when inserted in a coin machine. Also, the diameter vary by such a small amount that it may not make much of a difference anyways.
MIght the mazda rotary engine use the Reuleaux triangle for combustion chambers? Thank you for disabusing me of a long standing confusion. Here is a nice vid of the rotary engion design and function Rotary Engine .
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)
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.
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.
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.
Well said.
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!!
I understand that some manhole covers use these shapes, because they are constant width, the covers won't fall down the manhole.
"A 3 dimensional equivalent of a circle is a..."
"A sphere!"
Good job Brady!
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.
+pogogo51 Me too!
"I build a diorama - well I stuck a lego man to a ruler" Hahaha, ok you have my attention ;P
8:53 - "And the answer is…"
From his tone, I was so sure he was gonna say "…you can't."
Why do you keep popping up
What?
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.
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.
Why not?
Money. :Ü™
So not just a theory a game theory
and also it's really hard to look for videos about game theory without running into *That* channel
I haven't read it for years, but Martin Gardener's (?) books were great
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 :)
I already knew about shapes of constant width. What I learned today is that there are "lathe enthusiasts."
+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.
+altaroffire56 Lathes are awesome!! Yeah!! Check out +Clickspring
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.
Why wouldn't there be? There's 3D-printer enthusiasts, for instance...
How can anyone NOT be fascinated by lathes??
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.
It's so weird to see Steve Mould in numberphile! Great job, love to have you in this channel.
Wow, this video was epic. I could feel the epicness of his journey to get his constant width solids.
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.)
Awesome video! One remark: it's not a square hole that you can drill with a Reuleaux bit, it will be squircle-shaped.
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.
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.
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.
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°.
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.
Do this until you reach a perceivable level
It is people like the guys from Numberphile who make UA-cam a site worth visiting.
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.
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.
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.
Here in Australia, one of our coins doesn't have constant width, yet it still works with vending machines. Spoooky.
Isn't only non-constant-width coin bigger than all the others on it's shortest width? Or am I looking at obsolete coins?
I'm so glad you guys have a video on the Reuleaux Triangle! May sound weird, but it's my favorite shape!
And of course the reason these aren't used as bearings is they would wear out much faster than spheres.
That, and it takes more energy for them to roll than a sphere of the same mass.
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
Never though about that. Made my hour.
***** 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.
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.
The three rounded edges need explaining! Also what's the word for a 3d compass?
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!
"wankle engine" *giggity*
How much thrust does it provide?
I don`t know, but I just read up on it. It`s pretty cool. Has some pretty intersting insights.
@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.
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?
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.
Steve's channel is great, highly recommend it.
You are the nerdiest guys ever.
Love you guys
This is one of the coolest numberphile vids yet.
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"
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!
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!
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.
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
Wait,what about that Wankel engine? Why didn't you elaborate on that?
Other than that, one of the most fun episodes IMHO
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)
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?
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
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.
3:10 Actually, Australian 50c coins are dodecagons with flat faces around the edge, which work in vending machines here. (Perhaps having so many sides they're close enough to constant width for the machine to deal with).
Yeah that was my first thought, too. Completely goes against what was said in the video.
This is very counter-intuitive. Just looking at it you don't think it will work and then it works. Great video :D
I got really excited when he mentioned rotary engines...
More of this guy please!
I was thinking about this. Would these shapes be less material than a sphere of the same radius? If so there would be use for them to make cheaper products like bearings or anything that rolls without an axle.
Yes the Reuleaux triangle takes up less space then a cirlce. They are used in the real world for certain things. I know they are used in submarines and such, and also being able to drill holes besides circle shaped holes can be useful. I didn't watch the whole video so I don't know if he said it, but you can drill a square hole with one of these.
The machine process probably costs more than making spheres. Otherwise I don't see why they won't just use these in bearings.
you would not want to use these for bearings because the center of mass is not in the symmetrical center so it would vibrate/shake at medium to high speeds
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.
I'm 40 and I've learnt something new! Thanks! :)
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.
Hey, saw Steve present this almost exactly the same yesterday at the Maths Inspiration presentation in Birmingham!
"How big is your ruler"
So is a circle actually a reuleaux square?
+Nathan Williams
You could say that with any regular polygon with an even number of sides.
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.
It's technically a reuleaux octagon
A circle is a regular polygon of n-sides. So it would be a reuleux regular polygon of n-sides
Zapii112 doesn’t a circle have an infinite amount of sides? Or zero sides?
There's always that one crazy lathe enthusiast who just breaks all the rules... lol
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!
Just watched your presentation in Reading. Really nice
love the video. Metal Solids of Constant Width could be cast, then milled to a more precise finish.
Can't wait for those 4D shapes of constant width. Maths Gear promises them within a year or two!
lpreams and how exactly do you suppose they would represent 4 dimensions in the real world?!
Ross Girven it was a joke, I believe based on an actual post on Maths Gear's website
i see, my bad
Can you guys do a video on Euler's identity? You always explain things amazingly and in easy to understand terms.
i want to see that bike in action.
Steve is my second favorite
Saw this at the Maths Inspiration Lectures in Reading!! Such a nice guy! Great presentation, good day out :)
The hero with the lathe, made this video what it is
Who better to have some solids of constant width commissioned, than Steve Mould
this badass guy, steve mould, bring him more often!!!
Does every Platonic solid have a possible “Meissner” solid of constant width? I’d LOVE to see this from a dodecahedron!!
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.
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.
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?
Great Explanation, thanks
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. :)
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.
Just saw James doing a lecture in Manchester! :D Absolutely amazing! :)
Oh how I've missed seeing Steve on your videos! Have him do more videos! :)
Huh.
I'm American, but the UK 20p coin is my favorite coin. And now I understand why it is that way.
I have an Australian 50¢ coin. It's got 12 sides, I think, but they're all flat. What's going on there?
Probably it's enough sides to make it at like a circle for most intents and purposes
Our 50 cent coin is a regular dodecagon, that is, 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°. Vertex means a corner or a point where lines meet (plural: vertices.)
what if you rotate the triangle around another constant-width triangle instead of a circle? would the result still be a solid of constant width?
Steve is such an awesome guy!
"reauleaux" is simply pronounced "rolo" ! :)
But it's Reuleaux
Brady is my favorite!
So say one were to make one of the 3d shapes from the vid into a basketball. Would it bounce straight up or off at an angle?
TheTurnUpTurnip I'd think it would bounce off at an angle. Since a perfect sphere is the only thing that's always perfectly linear with the ground, no matter which way you turn it. (if you get what i mean)
As long as you hit it squarely on either side, you should be just fine. It would behave more and more volatile the futher you approach any of the points, almost to the point of bouncing a football (american) on it's points.
This guy makes maths fun and interesting
And you, Sir, are crazy for not earning affiliate commissions from maths gear!
Really liked this video! Loved the demonstrations.
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.
I love your shapes of constant width!
Surely solids of constant width could be used in place of regular ball bearings to reduce wear as well as not rolling away. The point of contact would be constantly changing as opposed to staying the same, as with ball bearings.
It would increase wear of the bearings though. Constant width shapes have edges. An edge will wear much faster than a sphere, and when the edge wears it is no longer constant width. By using spheres you have more wiggle room in terms of wear.
madogmdp
Good point, didn't think this all the way through.
Another problem is that the center of mass of the object has to move around, which will mean huge vibrations at any speed higher than hand cranked.
I really like Steve. Brady should do more videos with him! :)
How does the volume of these shapes compare with equivalent-width spheres?
Is there a 4 dimensional version of both the sphere and this object described? Is there actually any sense to a two dimensional plane on a 3 dimensional object in a 4 dimensional verse?
Yup. It doesn't have a name, but Matt Parker's things to make and do in the fourth dimension mentions it and matt calls it the Robert-Oudet body.
Very interesting but how could you just blow by the drill that makes square holes? Where can I see this amazing drill?
Keeled, I guess so. The length of the sides of triangle will give you the radius, therefore giving you the circumcumference. And as it is 3x60 degree arcs, divide the circumference by 2. Although I could be totally wrong as I haven't done circle theorems in years.
I had to have these - check out the dice as well.
Best nerd holiday gifts ever.
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.
Colorado School of Mines (Colorado, USA) Has the Reuleaux Triangle as the main shape in their emblem. Each club fair gives out 5 of the tetrahedrons to incoming freshmen.
"Shapes of constant width" sounds like some deep philosophical poem that's been turned into an art house movie! :-)
next step...FOUR-dimensional shapes of constant width!
3D shapes already are shapes of 4D width because in the time they exist in the same state, throughout that time they still have the same constant width and constant time
you showed how to build some shapes of constant width in 2d, are there some that cannot be constructed that way? likewise, in 3d, you showed us a solid of constant width that is not of rotation, are there others? in other words, can you characterize shapes (or solids) of constant width?
In the Meisner Pyramid, why are three edges sharp and three rounded? Is there a way to make the shape completely symetrical by applying a little less rounding to all six edges?
Saw the video for How Wide Is Your Circle. Had no idea what I was looking at until now
I wonder, is there a way to round off the Meissner Tetrahedron so it's fully tetrahedrally symmetric again? - instead of rounding off three edges, round them all off but just slightly? Or is that mathematically impossible?
Also, how about other platonic solids? Does this only work with tetrahedrons?
exactly what i want to know...
I'm Australian. Our fifty cent coins have twelve sides and they don't seem to be rounded; a regular dodecagon. How would our coin machines work? All our other coins are circular and smaller in diameter.
My guess would be that it would turn to the position were it has the smallest diameter when inserted in a coin machine. Also, the diameter vary by such a small amount that it may not make much of a difference anyways.
MIght the mazda rotary engine use the Reuleaux triangle for combustion chambers? Thank you for disabusing me of a long standing confusion. Here is a nice vid of the rotary engion design and function Rotary Engine .