Squaring the Circle by Rolling (animated visual proof)

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

To learn more about animating with manim, check out:

[columbusmyhw]

Fun fact: the area of a cycloid (also made by rolling a circle) is exactly three times the area of the circle

[benmathewson]

Thank you for putting this together! It’s perfect. So much better as an animation than a plate.

[abhay]

that is almost magical! thanks for animating it so beautifully!!

[gerhardtfunk]

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).

[giantteshorelone]

Wow, very intuitive and demonstrative! now I know how to convert a given circle into a equivalent square.

[alvarol.martinez]

day 52 of waiting for this channel to blow up. good luck!

[cobusvanderlinde]

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.

[shivanga]

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.

[Catman_]

So it is theoretically possible to square the circle, if rolling out a curve was allowed. Very interesting

[christopherellis]

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

[l.ahlgren]

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?

[bobbycarroll]

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.

[sundareshvenugopal]

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.

[joeeeee]

Did you just square the circle? 😱 wasn't this supposed not able to do?

[danielroy]

I'm curious, can this be done with a compass and straightedge?

[carlimiller]

But to do it with just a straight edge and a compass?

[dribrahimaldhaify]

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..

[jacobkebe]

Why do we know that the two triangles created by cutting along the "x" line are similar?

[mohammedal-haddad]

Which part you did that Euclid couldn't do?

[jamesongarnett]

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.