New Reuleaux Triangle Magic
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- Опубліковано 15 лют 2019
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Today's video is about plane shapes that, just like circles, have the same width in all possible directions. That non-circular shapes of constant width exist is very counterintuitive, and so are a lot of the gadgets and visual effects that are "powered" by these shapes: interested in going for a ride on non-circular wheels or drilling square holes anybody?
While the shapes themselves and some of the tricks they are capable of are quite well known to maths enthusiasts, the newly discovered constant width magic that today's video will culminates in will be new to pretty much everybody watching this video (even many of the experts :)
Here are a few links that you may want to check out:
www.etudes.ru/en/etudes/drilli...
Drilling a square hole with rounded corners using a Reuleaux triangle (click on the video !!)
www.etudes.ru/en/etudes/reulea... Same (wonderful) Russian site. An animated intro to shapes of constant width.
www.etudes.ru/en/etudes/wheel-... An animation of the cart with non-circular wheels that I talk about in the video.
www.qedcat.com/articles/waterw... Preprint of my write-up of all the stuff I talk about in this video. This was published in the Mathematical Intelligencer.
demonstrations.wolfram.com/Dr... An interactive demo illustrating how a perfect square hole (NO rounded corners) can be drilled using a special shape of constant width.
Probably the most accessible intro to shapes of constant width is the chapter on these shapes in the book "The enjoyment of mathematics" by Rademacher and Toeplitz.
This article which I also mention in the video is behind a paywallMasferrer Leon, C. and Von Wuthenau Mayer, S. Reinventing the Wheel: Non-Circular Wheels, The Mathematical Intelligencer 27 (2005), 7-13.
Just found a Japanese toyshop the other day that sells wooden Nothing grinders global.rakuten.com/en/store/g... and a wooden Reuleaux triangle that can be rotated inside a square global.rakuten.com/en/store/g...
I didn't mention them in the video but there are also 3d shapes of constant width which are also very much worth checking out. All the touching stuff I talk about in this video generalises to these 3d shapes.
www.teepublic.com/t-shirt/626...
Today's t-shirt
The tune you can hear in the video is from the free audio library that UA-cam provides to creators. ua-cam.com/users/audiolibrary... . It's called Morning_Mandolin and it's by Chris Haugen.
As usual thank you very much to Danil for his Russian translation and to Marty for all his help with the script for this video.
Enjoy :)
This is so satisfying!! I usually ask my calc students to see if they think square or triangular wheels make any sense to them and many of them would think no! Then I would show them UA-cam videos of bicycles or tricycles with actual square or triangular wheels and explain the math with my badly drawn pictures on the board. Now I can share this video with wonderful explanation and beautiful animations with my students. Thank you so much for the video!
Hello there, have been recommending your videos for ages :) Good to finally make contact. I seem to remember that you are based in Singapore, is that right? I am there quite often, so maybe meet up for a chat next time I visit.
@@Mathologer
Hello Burkard, I actually live in the US now and I am originally from Taiwan. However, it would be so awesome if I get to meet you one day!
On the other hand, I notice you wear a lot of math shirts. I personally like to wear a lot of math shirts too and I create them as well. In fact, I would like to gift you some shirts that I designed and I would be really really happy and honored if you wear them in your videos. If you would like, then I just need your t-shirt size and mailing address and I will take care of the rest. You can send the info to me at blackpenredpen@gmail.com Thank you!
That sounds great, I am always on the lookout for new math t-shirts :)
@@Mathologer Yay!!
YaY!~ 😆
11:57 “Why blue? Because I ran out of green.”
Everyone, donate to Mathologer so he can keep his ink supplies up! Buy Mathologer Kart 8 Deluxe today!
*Mathologer Kart ∞ Deluxe
I miss my old 83 Mazda RX-7. That was the most fun to drive car I've ever owned. Rotary Power!
Haha...
Thought for sure we would see Wankel power at some point
Mazda rx7❤️❤️❤️
@@heartbreakandmotorjive1867 I'm kind of disapointed it is not even mentioned, as it is practical use of this. And additionally because Wankel was German.
Yes, the Wankel rotary engine also came to my mind as I was watching this. I was half expecting him to mention it.
"Why are manhole covers circular?"
"Because manholes are circular."
I wish I had the balls to give that answer.
That's actually one possible answer. Manufacturing a tunnel shaped like a different shape of constant width for the manhole cover to fit onto would be much more difficult than a simple cylinder.
Hmmm... Circles can not be tiled in such a way as to fill the plane (the closest you can get is about 78% with the hexagonal packing method) - can you improve on that 78% with a different shape of constant width, or are all shapes of constant width equally or less 'packable' than the circle?
Great question. I did stumble across a paper about this: arxiv.org/pdf/1504.06733.pdf Have not looked in detail but it seems like Reuleaux triangles can be packed denser than circles (of the same width :) arxiv.org/pdf/1504.06733.pdf
Given the fact that they have a smaller area than a circle of the same radius, they either have a different smaller packing method available to them or simply take up a smaller portion of their container. I know this is a disappointing result since the answer is "either more or less" but at least it eliminates the possibility of being the same.
Smaller area would suggest that it is possible to have less mass at the same amount of shapes. On the other hand same radius, instinctively says that you cannot patch it denser. I have to find my notes from 2014 and perhaps do some checking.
@@jannegrey593 this is false, because we're concerned about packing density, not perimeter. Consider that squares pack perfectly, despite having more perimeter/area than circles.
@@adamweishaupt3733
It doesn't eliminate the possibility of "the same" at all, the two effects could occur simultaneously and cancel out.
3:32 cries in heptagon
I wish the seven was gone.
i used to play gd too
Lol, I see Burkard photoshopped his face over Mario’s face in the thumbnail
Respect +1
@@Mr3344555 we're here for Maths.. its expected.. but the thumbnail is not.. but more importantly..
Developing social skills will get you much further in life, gl with that
(or to riff.. you could have a billion people in front of you and you'd still be an asshat)
Évariste Galois are you a ghost? ;)
In the video, too.
3:36 You can't do it with an even-sided shape because from any vertex at which you choose to start, there will always be a vertex that is opposite (farthest away) from it. So a circle drawn through any other points would fall within the shape, and a circle that is attached to that opposite vertex would be bigger than the shape itself, so you'd just have a circle.
Thank god you said 2010. After i was expelled from my studies at the beginning of 2014, for half a year I thrown myself in research of Shapes of constant width etc. I also came with both ideas of Ferris Wheel and of applications of this kind of shapes (or rather their 3D versions) in Wheel design. And I did it on my own. Always dreamed of doing something new. So when you said recently I thought I blew my chance. Thankfully others beat me to it. Of course I didn't know their papers and I though that that was well known since it was so simple..... And I am just hobbyist Physicist/Engineer who likes math. Thank you for that video.
:)
+1 for every time Mathologer says 'widts'
J Halson
+1
These are some satisfying animations. I liked it!
Each time I think there has been a pinnacle video, a new eye-opener arrives. Thank you for the ever-entertaining and mind-expanding work.
Thank you Burkard for, again, an amazingly beautiful video.
Thank you for another inspiration Geometry Lesson. You have a refreshing logic with your use of such wonderful visual material. Congratulations on a fine effort yet again!
Quite interesting! Thank you for making this.
Homer’s torso looks like a reuleaux triangle
Ed Jones His entire lower torso could be described as a SOLID of constant width. :)
In your challenge at 3:30, you are assuming to deal with regular polygons. The answer in that case is that opposite sides of an even-sided polygon are parallel, thus the intersecting points, and the corresponding radius go to infinity. However, if you extended the construction to irregular polygons, you could have parallel sides even in an odd-sided polygon; think of a pentagon for instance, drawn like a house, and you have the same problem as for regular, even-sided polygons. But it's possible to extended the Reuleaux construction to irregular-sided polygons, with an arbitrary number of mutually intersecting lines. One has to select the starting vertex and the minimum radius carefully, since a radius that is too small would cause the construction to end somewhere inside the bunch of lines. I have written with a friend a corollary to calculate the minimum radius. Excellent video by the way, thank you!
3:37
What goes wrong, if we try to apply Reuleaux's method of construction to an even-sided polygon, is that such a polygon also has an even number of corners.
Thus, each corner has exactly one opposite corner.
But Reuleaux's method requires to connect a pair of neighbouring opposite corners with a circle centered in the corner we are standing on, which is now impossible, as any circle centering in one corner and going through the single opposite corner goes through no other corner.
Therefore, this circle does not give us an aptly rounded arc to go between two corners of our polygon so that it would have the same width in all directions.
Pure beauty!
This is amazing!
WOW, WOW, WOW. I have known about constant shapes width for years. I first heard about it when measuring submarines and the Wankel engine. What has been done here is nothing new, but an amazing way of putting them together.
It is easy to think all new things need to be complicated, but this can be understood with basic high school geometry. I lost focus on your presentation and started to think of gearing, water wheels, and motors.
Absolutely jam-packed with WOWs and AHAs! Thank you @Mathologer
:)
One t-shirt more epic than the other!!! 😂
Just in case you are interested here is where I got it from: www.teepublic.com/t-shirt/626201-schrodingers-surprise
This mechanism is used in film projectors, to drive the prongs that protrude into, and grab, the holes in the edge of the film, and jerk it to the next picture (quickly, while the light is blocked). One Reuleaux cam drives the film clamp/unclamp and claw retract/engage, and the other drives the claw movement (to pull the film and return to starting point). Many different cams could be used, but would require spring loading on the follower. By having the follower being 2 parallel surfaces, the cam is always "tight" to the follower. If the rotation axis of the cam is at one of the corners of the triangle, then the motion will be move, dwell, move, ...just what is needed for the film mechanism.
While pure math does not need real-world uses to justify it, it *is* cool to see this abstract concept employed in a physical device!
@@prependedprepended6606 ...but real-world applications pay the bills for the real-math people! Even if only 1% of the math developed is practically useful, that justifies doing the math!
Sewing machines also use such cam mechanisms to increase the dwell & speed the move.
Very, very beautyful! Thanks a lot for all those wonderful math videos, Burkhard.
Top Video! Having done my maths degree over half a lifetime ago and having not used those skills for a similar time, I loved this video to remind me of the absolute beauty of mathematics
I love these videos!
this was a perfect presentation about maths concepts. great work!
10:20 because you assumed that the fifth postulate holds true and don't want to deal with the headache that is noneuclidean geometry at the moment.
LMAO You're right
One of the most interesting and unimaginable episodes.
Truly amazing ! Thank you so much !
The tall unicycle proposal immediately reminded me of a similar one by my former juggling partner, Jack Kalvan. Back in college or thereabouts he suggested a 3-stacked-wheel unicycle (like the one you pictured), but with each wheel being an identically eccentric ellipse. If they're arranged with the major and minor axes alternating (so if the middle wheel is "wide", the other two are "tall"), they can rotate on normal bicycle axes and they'll always maintain contact. Of course, actually building one would be extremely challenging (spokes??).
16:27 caption says :"were still mad students"!
Should I fix this or leave it ? :) Almost too funny to get rid of it.
Thank you for this delightful video!
Back in the 1960s, I remember reading a science fiction story about a society where the circle is too scared to be reproduced in anyway, meaning that the wheel is not kosher, i.e. forbidden. The hero engineer of the story saves the day and gets the girl by using machines based on figures of constant width instead of circles!
"The Three-Cornered Wheel" by Poul Anderson, appearing in the collection "The Trouble Twisters" (1966).
The sum of the inner angles of the Stars is 180 degrees and since each arc of the constant width shape is subtended (is that the right wording) by one of these angles and the arc measures exactly twice of the subtending angle, the sum of the arc measures must be 360 degrees. That means that the perimeter is the same as the circle QED.
Awesome presentation !
This is so awesome!!!!!!!!!!!! ♥
Awesome dude!
Very nice video. Thank you so much.
Excellent
I want that Bermuda coin it's beautiful!!
I've got a few, gold and silver :)
Agreed -- I'm British, so I knew about our 20p (not shown in the video) and 50p coins, and knew they are constant width. But I'd never seen the Bermudan Reuleaux triangle coin before.
Utterly fascinating.
My favorite guitar pick!!
Great video, really enjoyable
Sweet! I loved this. I've seen something similar on numberphile. Thank you. And geometric form interests me. Especially non-euclidean dimensionality.
Such a beautiful video
I saw this demonstrated in the early 1960s by Bob Brown, a traveling school science entertainer, similar to Bill Nye. I was in junior high school at the time. Totally fascinating!
I wish I could be your student forever and just be bathed in golden math presentation. How could anyone in the world not crave mathematical knowledge this rich after having it so delightfully served to them.
Even number polygons have no two oposit points with no other point between them so no shape with constant width.
A non-circular manhole cover might be harder to get out. When pulling/pushing to get it out there'd be more places for it to get stuck so making sure force is being applied at the right angle/direction might be more important than with a circle. Circle is easier to manufacture/slightly less precision needed than with other constant-width shapes
It can also fall down into the manhole.
2 videos in a month! Today is a good day ;)
Love you sir
I have not watched the vid in it's entirety yet, as dad needed sleep, & headphones die in my presence, so I have none.
But I watched enough to thank you for teaching me the constant width pentagram. It is a hybrid between my straight edged pentagram, & the standard round one. It is more balanced than the other two, & therefore of superior beauty. I like that it takes a bit more effort to craft. Extra effort leads to extra capacity for majik, in my humble opinion.
Oh, by the way, when I make even pointed mandalas, I like to put the first/last point in the center, hub & spoke style. This allows the overall structure to behave odd. I hope someone finds this useful.
A statement : Even only for the T-shirts, this channel is worth subscribing.
and a question : Is (in this sense) the start of a circle a polygon with only 1 corner, a polygon with ∞ corners, or are both possible ?
this might be the most interesting video ive seen on here. Interesting and intuitive/simple enough to follow.
This youtube channel is a point of constant witz.
A wonderful accomplishment. Congratulations. Thx for sharing.
Sir your music for maths is fantastic just showing your "goùt raffiné"...congrats and thanks a lot
BTW, maybe worth mentioning that while it is possible to eliminate the bumpiness of the ride (up and down), at a constant rotational speed of the wonky wheels you'd still get quite some shakiness in the back-and-forth direction, no?
AWESOME!!!!!!
Thank you for the video! All of you friends are super awesome!
I thought Reuleaux was French because of his name, but actually you pronounce it in German as Rouleaux, which means rollers in french. And naming them rollers triangles is so satisfying!
A great video! Also, a great T-shirt you're wearing!
3:30 Hey, that’s our 50p!
And that triangle coin later on in the video looks very interesting.
Holy crap, he did it again. You teased my curiosity/intuition to keep my attention thru-out the whole video. During the video you identified a chain of essential/basic consequential facts both you & I know to be absolutely valid & transient. Thus because of transience across that chain, I have to agree that your premises & conclusion is absolutely TRUE no matter what I initially thought of your initial conjecture. Note I didn't say intuitively true (because intuition is no-more than an euphoric guess). I said absolutely TRUE. I didn't know math guys existed that could be so competent/complete on an interest/practical topic. Keep up the good work.
:)
Hello Burgkart, great video as usual. I want only suggest that the story can drive even more odd and beautiful going 3D. I suggest the video "solids of constant width" to see these amazing behaviours on real made solids. I just wonder which kind of gears could one make from them.
I've watched only 30 seconds of this video and I hope I'll make it to the end, but I can't make any promises. However, what I REALLY want to tell you: LOVE your t-shirt man!
You make me smile.
100% great shirt!
Schrödinger's surprise is flawless! :-DD
(9:30) The perimeter of a circle of width W is W*pi, so we have to show that all constant-width-shapes have such perimeter. First, consider the Reuleaux equilateral triangle. The circle arcs drawn to build such triangle are part of a disk of radius W and width 2*W; then, if A is the angle in degrees between two sides of the generating equilateral triangle, the length of the circle arcs are (A/360)*2*W*pi, since they are the A/360th fraction of a 2*W*pi disk. A is 60 degrees and there are 3 of those arcs, then the perimeter of the Reuleaux equilateral triangle is (3*60/360)*2*W*pi = (180/360)*2*W*pi = 1/2*2*W*pi = W*pi.
Now we need a little "sideways" consideration. The sum of every "pointy" angle of a star built by connecting by lines of constant width an ODD number of points is 180 degrees. Since the number of points is odd, we can start a "walk" from point 1 and end in point 1 traversing all 2*N+1 points by traveling on the lines that form the star (this is also how those kind of stars can be built). Starting from point 1 we can go straigth to point 2, then we have to "rotate" by 180 - A_1 degrees, where A is the angle corresponding to corner 2 (the rotation angle is the supplement of angle A); this applies to every point, and since we have to re-align ourselves to the initial starting position, even when we travel from point 2*N to point 2*N+1 we have to "rotate" afterwards by 180 - A_(2*N+1) degrees. In total, we have to do 2*N+1 rotations of 180 - A_i degrees for every point i, for a total of 180*(2*N+1) - \sum_i A_i degrees. Considering as 0 degrees the angle "we have" being in the starting position facing the same direction as we started, then obviously the total 180*(2*N+1) - \sum_i A_i must be a multiple of 360 degrees. Then: 180*(2*N+1) - \sum_i A_i = 360*k ---> 180*2*N + 180 - \sum_i A_i = 360*k ---> \sum_i A_i = 180 + 360*N - 360*k ---> \sum_i A_i = 180 + 360*(N - k). Since this equation must hold for the triangle as well, then 2*N+1 = 3 and A_1 + A_2 + A_3 = 180, we can deduce that N = k; hence, \sum_i A_i = 180. This result also implies that for every 2*N+1 points we make exactly k = N full 360 degrees rotations, traveling along the star.
Given that, we can now easily extend the aforementioned Reuleaux equilateral triangle consideration to all star-based shapes built by lines of constant width W connecting an ODD number of points. Every angle A of the star generates a circle arc of length (A/360)*2*W*pi for the reasons given above, and summing all arcs is ((\sum A \in STAR)/360)*2*W*pi = (180/360)*2*W*pi = 1/2*2*W*pi = W*pi.
QED
awesome as always........cover the 3D versions in a follow up !?
Probably :)
Wow! So much awesome info about shapes of constant width, and he didn‘t even get to the 3D versions like the Meissner Tetrahedrons!
Nice t-shirt!
THx--- the square holdiong the axle of the wheel is the part that is HUGE! once u see that, it makes sense
I usually watch UA-cam while cooking but not mathologer videos. These require more concentration. Although this one was a bit easy yet verry elegant. Thanks
:)
I would play the hell out of mathologer kart
I love your shirt!
Interesting! The constant-width shapes are constructed of circles of the same width. In all cases, even the weird star ones, moving from edge 1 to edge 2 will require a change of theta, which will be connected by the perimeter of the circle. So, once you connect all the thetas that sum to 360, you will have the arc length of a circle from 0 to 360 degrees.
For example, in the simplest case of the regular triangle, you will have 3 arc-lengths equal to 1/3 of a circle that sum to the perimeter of a circle
I noticed you explained a different method for the solution in the video. Does it lead to the same answer? Is this valid logic for a proof?
A kinetic way to think about the equal perimeter proof- if the circle was a loop of paper, then putting creases in the right places would let you turn it into one of the reuleaux shapes. You'd have to prove that, then, but for some it might end up being easier.
My guess for even polygons is that the two points the circle connects must be opposite the corner the circle starts from. Impossible on an even cornered shape.
Doing it with an even number of vertices will form a circle - which means you can use the method with any number of vertices to get a shape of constat width, it's just that half of them will be circles.
Solids for the next episode!
I was always fascinated with shapes of constant width. Thank you for that.
Do you think there’s some sort of a generator function for these curves?
Well, yes, for various interpretations of the term "generating function". For example, I am pretty sure that any shape of constant width can be approximated arbitrarily close using the star based ones that I talk about in the video :)
From 3:00 we can follow that: Though we dont know how many sides a circle have (for being too many) we know that it is an odd number
Circles have 2 sides. The inside and the outside
11:56 how can someone have such a serious tone in his voice and yet still surprise us with such a ridiculous joke? haha.
This discovery is so fascinating yet so trivial that it's kinda amazing that it hasn't been discovered before
Yes, and it shows that it's still possible to discover really fundamental things about mathematics that people have been doing to death over hundreds and sometimes even thousands of years :)
Did you read the book Alex in Wonderland? Because this is not the first topic from that book.
With the shape within a shape, as you increase the ratio between their sizes to infinity, the larger shape tends towards a circle I think. So if you had a very small rouleax triangle axle on your car, it would be approximately comfortable.
The next problem for the car tires is that the contact patch slides forwards and backwards relative to the car, which would handle quite poorly. Think we can do better?
Schrodinger's surprise (on his T-shirt):
there's no surprise per se in the video, but everything he taught me in the video is a pleasant surprise.
Mission accomplished then :)
This just makes me miss my old 13B setup
This is interesting. but I think the key to wrapping your head around is that it is a shape of constant width, not constant diameter. What blew me away was the idea that there were shapes other than a circle that had a constant diameter. you could could draw a straight light through the center and it would be the same. but the way the widths of these shapes are measured seems to require that it be measured from one "corner" to any point on the opposite side. it doesn't seem like you could measure the width from a point on a side to a point on the opposite side and get constant widths. like the star in the five sided figure, it couldn't rotate around a center point and create that shape (constant diameter). the star has to pivot around one corner to the next corner. @8:16
The problem with even sided polygons is that corners will be opposite from corners, and sides will be opposite from sides. Reuleaux require every corner to be opposite a side in order for the curve of its circle to be represented.
This might be a stupid question, but: All shapes of constant width shown in the video are comprised of circular arcs, right? Is it possible to have shapes of constant width with a continuous change of curvature rather than segments with constant curvature?
Yes, that's also possible :)
So, I guess that the odd-shaped "impossible" gear chains exploit this fact.
@@inyobill To give you a more concrete example, check out the wiki page on "shapes of constant width" for an example! en.m.wikipedia.org/wiki/Curve_of_constant_width
@@Jordan-zk2wd Interesting stuff, but not directly related to my speculation. On 24 hr reflection, and review of some of the weird (non-circular) gears, I've decided that, once again, I was wrong. Appreciate the lead.
@@inyobill Could you please explain exactly what these "impossible" gear chains are, or at least post a url? I did a web search, but nothing that seems to correspond turned up.
9:18 I think I solved it. Rotate the star-shape around itself in the following way. First, pick a side and rotate until the next side has the same orientation. The angle of rotation is the the angle of one of the corners. Next, rotate until the next side has the same orientation, which will be the angle of one of the opposite corners. When the shape has done one complete rotation, you went over each corner twice. That means that the sum of the corners is always 180°.
I also want to mention that the game "The Witness" has a minecart with Reuleaux wheels.
I agree, It is difficult to believe these shapes really exist.
So this is why when drilling sheet metal, the hole will be somewhat triangular. More obvious with longer drills. The tip of the drill can be seen as a line and the hole will be a shape of constant width. Preferably a circle, possibly something else.
15:00 You can also make it by taking the Reuleaux triangle and rolling another shape of constant width around the outside, filling in the area it sweeps out.
Yep, that's correct :)