Squaring the Circle with the Archimedean Spiral (animated visual proof)

This is a short, animated visual proof that we can square the circle IF we use the Archimedean spiral. 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 #squaringtherectangle
This animation is based on a visual proof from Roger Nelsen's and Claudi Alsina's wonderul book "Charming Proofs: A Journey into Elegant Mathematics" (Theorem 9.5 from page 145). You can find the book and other information about it at the links below
bookstore.ams....
www.maa.org/pr...
www.amazon.com...
Here is another video that squares the circle with a nonclassical technique: • Squaring the Circle by...
To learn more about animating with manim, check out:
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[@j.s.mathew3140]

A really beautiful visual proof! Coupled with the background music, watching the video passes for great recreation. Wish I had animations like this during my school days.

[@MathVisualProofs]

Thanks!

[@blandle12]

Low-key one of the best channels on KZbin.

[@MathVisualProofs]

Hey thanks! 😀

[@xCorvus7x]

This is magic. Somewhere on that half circle you've found the geometric mean of XY and XZ.

[@MathVisualProofs]

Yes! Check this one for a more concrete construction of geometric mean : kzbin.info/www/bejne/hqWbeouVhpqIZ6M

[@xCorvus7x]

@@MathVisualProofs There's also a way to construct the square root of a number using a half circle. It appears that half circles are pretty useful for multiplicative tasks and now I marvel what else one might do with them. Edit: On second thought, that's just a special case of the geometric mean, namely the geometric mean of a number and one.

[@MathVisualProofs]

@@xCorvus7x Yes! Excellent. The half circle definitely can do a lot for you :)

[@JaceSomers]

The virgin compass and straightedge alove vs with archimedian spirals among other powerful things. Jokes aside, beautiful proof. And yes, I am a bit biased against using just compass and straightedge for construction. Why? Idk. It's limited. The archimedian spiral used definitely shows what can be done if you're allowed to draw more than just lines and circles.

[@MathVisualProofs]

👍

[@minimalrho]

Is there a physical tool that can draw out the Archimedean spiral of a given parameter? As mentioned previously, straightedge and compass aren't enough to do this (proved by Galois theory), but if Archimedes was interested in this curve, then there ought to be a way of mechanically constructing it.

[@MathVisualProofs]

Good question. I think something like the guy does here might be essentially the Archimedean spiral though I haven’t verified it : kzbin.info/www/bejne/pp3SoIF5hs94q6M

[@minimalrho]

@@MathVisualProofs It seems like the Archimedean spiral is the pedal curve of the circle involute. I don't know much about pedal curves (this is the first time I've heard of it!), but maybe it's constructible at least in this case?

[@MathVisualProofs]

@@minimalrho yes. If you use a single point as the center (the video I shared uses something larger than a point) then it seems you could square the circle with such a device. This isn’t surprising as there are lots of devices that let you square the circle (check out Dave Richeson’s Tales of Impossibility for other examples).

[@11am2]

With some crazy compass techniques you probably could

[@yurisich]

kzbin.info/www/bejne/gXLFgoysatNpnKc

[@omargaber3122]

I can't understand why just 10k subscribe in this channel !!!! Why not 5 million?!!!!!

[@MathVisualProofs]

I appreciate this :) I just animate short snippets/proofs because they are things I want to get out there. I think the types of channels that really get big create better narratives around the results and tell a bit longer stories. People need to know about the background of squaring the circle to see the interestingness of this particular short video. I am not sure I have the time and editing skills to really tell those longer narratives, but I will keep improving. Thanks!

[@vadimpavlov6037]

This solution is similar to that using quadratrix of Hippias in a sense that it's also a vicious circle (not a pun). For if you're able to construct a spiral r(theta) = a*theta, then, after first revolution, the spiral has radius r = 2pi*a, the circumference of the given circle. But if you're able to construct a circumference of any given circle, then the very problem of finding its quadrature is trivial. Neverheless, since Archimedes in his work "On spirals" gives a (very cumbersome) method of approximating a spiral with an arbitrary accuracy, this solution may be viewed as an approximate one, and only in this sense it is valid. Though the method he describes is quite tedious

[@MathVisualProofs]

👍

[@DitDede]

Archimedes gives us pi and the geometric mean square-roots it :-)

[@PASHKULI]

the part with the spiral coming in was a bit obscure (the segment aπ/2 has not been prooved)

[@potaatobaked7013]

well, the spiral is defined as r=aθ where r is the distance from the center, and at the vertical point, θ=π/2, therefore, r=aπ/2. And the length of the segment is the distance from the center of the spiral to that point

[@PASHKULI]

@@potaatobaked7013 Thanks, but should have been mentioned in the video, I think. When θ is 90°, straight angle (π/2 'rad' = 90°), then the vertical segment is indeed r = a × π/2 = aπ/2.

[@CARLESIUS]

How can the Archimedean spiral be constructed using only a ruler and compass?

[@MathVisualProofs]

It can’t. That’s why I say using the spiral to square the circle. You can’t square the circle using straightedge and compass. If you could create the spiral with these tools you could square the circle with them. :)

[@CARLESIUS]

@@MathVisualProofs Thanks for the prompt response.

[@TherealElijhawatkins]

12:02

[@MathVisualProofs]

?

[@oliverrhodes3679]

I have squared a circle and have circled a square within my business model (the Corporate Health Dynamics circle).The square and the circle both possess an equal surface area to a circle and vise versa using nothing more than my eyes. I used no compass or straight edge.

[@abogadojon]

~0:56: I suggest extending the line results in "extending" using your terms the DIAMETER not the "radius" as you call it. Generally, the sound quality varies too much -- distracting. Further, no musical accompaniment is necessary. And certainly not at the upper decibel level at which you have it. Plz repost after you have corrected your language and plugged yr narration (sans distracting music) through an equalizer and, importantly, a compressor. Until then I am withholding a thumbs up. `