Squaring the Circle by Rolling (animated visual proof)
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- Опубликовано: 4 окт 2024
- This is a short, animated visual proof that we can square the circle IF we allow the circle to roll. Unfortunately, this is not a solution to the squaring the circle problem from antiquity because that requires it to be done with only a straightedge and compass. #mathshorts #mathvideo #math ##geometry #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #circle #mathematics #thales #squaringthecircle
This animation is based on a visual proof by Thomas Elsner from the May 1977 issue of Mathematics Magazine (doi.org/10.230... - page 162).
To learn more about animating with manim, check out:
manim.community
Fun fact: the area of a cycloid (also made by rolling a circle) is exactly three times the area of the circle
The correct statement is that the area under a cycloid is three times the area of the circle with the same radius, not exactly three times the area of the circle.
According to chatgtp after I copy and pasted this comment...
Math is confusing.
Thank you for putting this together! It’s perfect. So much better as an animation than a plate.
Thanks! I agree that one is hard because there are a lot of things going on. Also, the proof without words really should probably have some labels at least.
that is almost magical! thanks for animating it so beautifully!!
Wow, very intuitive and demonstrative! now I know how to convert a given circle into a equivalent square.
day 52 of waiting for this channel to blow up. good luck!
Thanks as always for watching :)
Thank you for making this! This is one of the early examples in the book "Proofs Without Words". I just couldn't figure out how the figure in the book was showing that the square had the same area as the circle and it was driving me crazy. The part I didn't see was that we could construct a triangle with the diameter of the larger circle which, in turn, is split into two similar triangles. In my opinion the best depiction of this proof without words would include the two lines making the similar triangles apparent which were not present in the book's illustration (but were included in yours).
Agree! That’s the visual I cite in the video (and description). One of the reasons for these videos is to help put “assumed results” in context for various PWWs.
@@MathVisualProofs Thank you very much! On a related note, I was reading an article on the "art of problem solving" website about "How to Write A Math Solution", and it expressed the following notion which I very much agree with and think applies in this situation: "The experienced reader should never have to wonder where you are headed, or why any claim you make is true." I think this notion is applicable far beyond competition problems and extends to PWW's (obviously), proofs given by professional mathematicians, text books, and perhaps even to fields outside mathematics. In fact, I might go so far as to say this should be considered a foundational principle of argument itself. Anyway, keep up the great work!
@@gerhardtfunk4463 This is great. I know of aops, but I hadn't seen that article. I agree with you for sure. Even sometimes I wish I could say more here, but I am still just learning the animation software.
So it is theoretically possible to square the circle, if rolling out a curve was allowed. Very interesting
I believe u are referring to the problem of squaring a circle. Yes, there exists a square that has same area of a circle. But in geometry, it is impossible to square a circle by using only compass and ruler. There is a proof on the same as well.
This is a very elegant way to import the transcendental pi into a geometric construction.
Seems like the sort of things the ancients could have come up with but were not satisfied with as this rolling motion is too... irrational for their liking.
Did you just square the circle? 😱 wasn't this supposed not able to do?
It is impossible to square a circle with only starightedge and compass. Here we have the rolling. Of the circle for half a turn to get pi.. which we cannot do with only straightedge and compass
You can do it with a straight edge and compass, but you'd have to set the plane / paper parallel to the compass rather than perpendicular, to transfer the half-circumference to the paper.
That's cheating though.
I would not have mentioned that the triangles are similar. Rather, that in the semicircle, mn=h², therefore pi.r×r=h², and x=h
Multiply the diameter by ( square root of pi divided by 2). Make that the length of the square. And the areas. will be same. Thus squaring the circle mathematically.
r + (1/9)r approximates the square root of ((L²)/2)
Where the circumference of a circle with radius "r" approximates the perimeter of a square with side length "L". This is squaring the circle. Always approximates because of pi.
Now circle a square HAHA
The problem is to measure out π; to geometrically construct a line segment of length proportinal to π, which could be a symmetric closed figure whose side is of length proportional to π. But a circle is nothing but the limiting case of an n sided symmetric closed figure as n tends to infinity. As to whether this latter fact implies, the only n sided symmetric closed figure whose side is of length proportional to π is a circle and that such a construction, in other words, squaring of a circle is impossible, is hard to tell. It may or it may not be true.
Also, this raises the question, if a quantity has an endless decimal (fractional) part, then is it to be considered or is it not to be considered as a precise and as an exact value ?
If it is to be considered as exact value then what is it's value exactly ?
Rightly, are such numbers called as Irrational.
Thanks for that but it is well known that
there is no square can have an area equal
the circle ...because the. Pi is transcedental. Number..
There is a square with area of circle but you can’t construct it with a straightedge and compass. Here I’ve shown you can construct it if you roll the circle, not part of the constructions allowed in antiquity.
Rather than roll the circle 1/2 way, roll it 1/4 the way. Turn the circle 90°, roll 1/4 the way, turn the circle by 90°...
What types of triangles are these that are formed in this way? They are not 'Kepler triangles' but they appear to be facsimiles of one another. Are they '30-60-90' triangles?
It’s way simpler than that but yea that’s one way to show it. The other is… ya know… Pythagorean theorem and a pencil and ruler. Better yet just a calculator or piece of paper.
I'm curious, can this be done with a compass and straightedge?
Nope.
No, because there is no way to accurately measure the circumference of the circle. We only know the radius is "r", which can be anything. IRL of course, we could make a circle out of some material and mark a point on the edge and another point directly across, and then roll it on a straight line. But a circle is not a straight edge in the classic definition.
But to do it with just a straight edge and a compass?
Can't be done unfortunately. You need some special tool (marked ruler, Archimedean spiral, etc.)
@@MathVisualProofs
go to this link, I guess he did it with ruler and compass and, working over areas got the exact value of π = 3.1446:
3.141592653589793238462643383279502884197169399375105820974944592.eu/wp-content/uploads/2017/09/Geometrisches-Pi.pdf
@@MathVisualProofsI think it’s possible to construct a square which differs from the circle by an arbitrarily small area, using only ruler and compass. which is something at least.
Good in theory if pi wasn’t a transcendental number. This was tried in the late 1800’s but it was found that pi was given the value of 3.2 which can be a derived number. The area of a square is finite and the area of a circle has no endpoint because of pi not having an endpoint.
Yes, it's impossible to create that square with only a ruler and a compass. This video cheats a little by making that line the same length as half the circumference which is impossible to do with only a ruler and a compass.
Woo!
Relavent numberphile video (starting at 2:33 of ruclips.net/video/9VVPBS_flOI/видео.html). This numberphile video shows how you can construct the square root of any length, which is essentially what is done here to construct the square root of pi.
But the square root of pi is not constructible using classic techniques while square roots of integers are :)
@@MathVisualProofs Yes, I guess because you can't construct pi given a length of 1 without constructing a circle and rolling it like in the video.
@@Mr_Happy_Face you can if you use the Archimedean spiral :) it’s another video: ruclips.net/video/_e4Yn5uGznI/видео.html
wow amazing
Pretty cool right?!
@@MathVisualProofs ofcourse more than what i can excepected
Why do we know that the two triangles created by cutting along the "x" line are similar?
Their top angles make up a 90 degree angle, so they are complementary. Both are right triangles, so their other angles are complementary too. That means the satisfy AAA similarity.
Which part you did that Euclid couldn't do?
Rolling the circle is not part of the classical construction techniques. You can only use straightedge and compass for classical problem.
@@MathVisualProofs Thanks
Why is this impossible to construct with classical tools? Can you not construct pi times a given length?
Deep theorems involved. But you cannot construct any transcendental number with straightedge and compass.
@@MathVisualProofs
the real, exact value of π is not transcendental. To see how it comes you can go to this link now:
ruclips.net/video/ccxVW2MIbxA/видео.html
Remember that a "straight edge" is not a ruler. There are no distance markings! I'd have to experiment, but I think it could be done if we're allowed to tilt the plane of the paper so that it's parallel to the compass, then draw a half-circumference onto it, using the straight edge as a pivot point for the compass.
@@padraicbrown6718 I know it's not a ruler, but you can still copy arbitrary lengths
I don't get it. π r is clearly larger than x. So how can π r squared be the same as x squared?
If r = 1, x = √π. Therefore, π * 1² = x² = π
It doesn't work that way if you take the approximated value of π to construct your figure. 3.14159 is the perimeter of polygon and a polygon has 2 main diameters, which means that 3.14159 was not elicited according to the theorem of π=P/D.
This whole construction is basically illicit and this because a geometric construction is "exact" whereas 3.14159 is a gross approximation.
When did they use an approximate value for pi?
@@ttmfndng201 Which value of π would you use to solve A=x^2=πr^2?
And, besides, what's the meaning of rolling the circle to print a straight line = πr? How can we know that line is = πr?
In the same way we could draw just a square, with sides √π/4 and solve it much easier.
@@O-Kyklop That equation is already solved, it says x^2 (the area of the square) = pi*r^2 (the area of the circle). The equal sign says the two areas are equal, so they found a circle with the same area as a square, which is what they were trying to do.
If you wanted to construct this, you can't just use a straight edge and compass, since they "rolled" a circle on a line, but maybe you could use a specially made gear or something like that. In that case, you don't need to use approximate values for pi, since you get it by turning an arc of a circle into a straight line.
We know the length of this line is pi*r because the circumference of a circle is 2pi*r, so half of that is pi*r.
Drawing a square with side sqrt(pi)/4 would not work, as that would give you a square with area pi/16, and not the pi*r^2 we were looking for. Instead you need to draw a square with side length sqrt(pi)*r. This would work, but is not that much easier or much more difficult than the solution in this video, since you still need to find a way to draw a line of length sqrt(pi) using only a straight edge, compass, and "circle rolling". finding pi is pretty easy, but getting the square root is not totally straightforward.
@@ttmfndng201 I meant sqrt(π/4).
But anyway. Why do you think it is easier to draw a line rolling the circle than to take a square with side sqrt(π/4). Mechanical means are not valid to determine exact lengths. The rolling of the circle is just for show, we can just draw the line and write down the length. To roll the circle is even a problem. Because you're using the exact value of π but, to solve numerically the values of the figure you are going to use the approximated value of π.Which is not the same.
Hermozo, nos proporcionas el codigo???
Gracias! El código está desordenado y no estoy seguro de que sea legible. Pero aquí está: github.com/Tom-Edgar/MVPS/blob/main/RollingCircle.py
@@MathVisualProofs me va a servir mucho, gracias
It's hip to be square...
👍
osm
Can you please share source code :)
At some point I am trying to make the code public. It is not well-documented and not as efficient as it could be.
@@MathVisualProofs oh, ok.
@@MathVisualProofs isn't this 3b1b's program? Or does he just use it?
@@nomekop777 he wrote a package for python that anyone can use to animate. But you still have to write the code.
As long as there's a pi you can't make it... pi is transcendental
Rolling a circle is literally the definition of pi
You still require Pi to perform this. And as Pi is an irrational number what you have demonstrated is that the 'circle' cannot be 'squared'!
It’s not just that pi is irrational (root 2 is irrational but can be constructed with straightedge and compass). Pi is transcendental (but that is perhaps way more than needed ) :)
Hi what do you will do if i prove to that pi is not the well known 3.1415
@@dribrahimaldhaify4469 I'll nominate you for a Nobel Prize in mathematics.
Hhhhhhuuuuh
Thanks
Not as exciting as I thought haha
give me code?
There is a link in a response to one of the comments.
For real they look the same!! !
x = pi r
😀
in this video x = sqrt(pi) r
loklol
I don’t see how this is useful, because it’s basically just assuming you are able to rectify a circumference, which is equally as challenging as squaring the circle.
If you are true then the PI VALUE IS NOT 3.14
PI IS THE VALUE THAT I GOT (???) AFTER WORKING FOR ONE YEAR?? ?
Wrong in the first step. You can't construct that rolling circle.
Fake
where's the flaw
Oh! That was a good one. I'm surprised I haven't seen that one before.
But wait! Creating the line "𝝅r" seems easy in an animated graphic but how would you do that with a straight-edge and compass on flat paper? It would seem that straightening the circle is as difficult as squaring the circle. We're back to square one.
Can’t do it with straightedge and compass for sure. Have to roll the circle
Did you just square the circle? 😱 wasn't this supposed not able to do?
Haha! Yes, I squared the circle, but as I note in the description, this doesn't violate the impossibility theorem because rolling the circle wasn't allowed by the Greeks (they only wanted to use straightedge and compass). There are lots of ways you can square the circle if you use other tools; this is just one. I recommend Dave Richeson's recent book "Tales of Impossibility" for a few others.