Constructing a Square of Equal Area to a given Polygon

Explore the world of Euclidean geometry by solving geometry puzzles at:
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About the video:
1. Pick any polygon
2. Split it up into triangles (While it is trivial to triangulate any convex polygon it is has been proven that any polygon, convex or concave, can be decomposed into triangles and there are many triangulation techniques).
3. Construct a rectangle of equal area for each triangle.
4. Construct a square of equal area for each rectangle.
5. Construct a larger square equal to the sum of each smaller squares via Pythagorean Theorem.
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Any further questions:
► [e-mail]
Programs used:
► Cinema 4D
► Processing
Music by
► Lee: • (FREE) Lo-fi Type Beat...

[@ThinkTwiceLtu]

For anyone who is interested, I have also made a little interactive sketch with p5.js library where you can play around by squaring a rectangle. I am still learning so let me know if you encounter some bugs, also it will not work that well on your mobile phone so open it up on your PC. Here is the link: editor.p5js.org/psyduck/present/HtpaSry_c

[@mik000]

Is it possible to do this step by dividing the rectangle into finitely many parts that will make the square?

[@mik000]

It is. Lemma 2 (and figure 3) of the following paper shows how to do this: rak.ac/files/papers/wallace-bolyai-gerwien.pdf

[@xCorvus7x]

Thank you for this toy. It is fun to slide along a hyperbola of constant area.

[@adolfocarrillo248]

can you share us the code, great job by the way.

[@shreyasd4663]

You did a great job. Keep making stuff like this. People need your creations!

[@TimTom]

This was dope as hecc

[@ThinkTwiceLtu]

Thanks:)

[@-thanawat-8296]

Hi

[@ira6391]

Hi

[@-----bk7le]

If we take n-gon where n tends to infinity, so this problem can be named as..

[@ChiefVS]

Squaring the Circle! But, are u sure u wanna construct infinite triangles and carry out the process ;p

[@-----bk7le]

It also reminds me trisection as infinite bisection 1-1/2+1/4-..

[@ChiefVS]

@@-----bk7le Don't judge me, but I tried to trisect 179° angle with compass :p I was so close, damn it... Btw, took me like 30 steps to get to 59.6582325 degrees lol I still got the workings haha

[@whatisthis2809]

What does your name mean?

[@randomdude9135]

@@whatisthis2809 good q

[@vaprin2019]

beautiful animation, well done.

[@ThinkTwiceLtu]

Thanks a lot.

[@4AneR]

2:53 bottom corner of pink parallelogram: "am I a joke to you?"

[@aiksi5605]

Bullseye

[@randomdude9135]

Looks like insemination😏

[@ianprado1488]

Jeez, one pixel

[@bolson42]

Random Dude Why, just why?

[@randombanana640]

perfectionist brain : I HATE THISSS ADHD Brain : Ahhh Perfection

[@erfanshekarriz4707]

I love your videos they inspire me to do more Mathematics in my free time!!!

[@ThinkTwiceLtu]

Thank you;)

[@laurahoughton1289]

You are a luminary for disillusioned aspiring mathematicians like me.

[@Carl-Gauss]

There is also a 3D version of this problem: can you cut given polyhedron into finite number of pieces and make cube with equal volume of them? It even was in the list of Hilbert’s problems (№3). Interestingly, it turns out that unlike 2D version answer for this one is “no”.

[@1996Pinocchio]

as seen on numberphile

[@alvarol.martinez5230]

proof or it didn't happen

[@mst7155]

Does anybody know a simple proof of the solution of the third Hilbert's problem?

[@redsalmon9966]

That’s so cool. Especially how you can turn the squares into a larger one using the simple Pythagorean theorem. Love your work.

[@ThinkTwiceLtu]

Thank you very much:)

[@alexandersanchez9138]

You guys, this is technically different from the numberphile video: that one said you can cut up a polygon and re-assemble the pieces into a square of equal area; this one says that you can *construct* a square of equal area (in the sense of straight-edge/compass constructions). The upshot is that from each regular n-gon one can construct a square of equal area. This is kind of interesting, since in some certain sense, the circle is almost like a limit of regular n-gons. So, if we weren't being careful, we might begin to suspect that squaring the circle is possible; of course, that's nonsense because we couldn't do the first step (of triangulation) on a circle which was assumed at the outset. However, this does show that to square a shape, it is sufficient to triangulate it, which is cool in its own right.

[@HuslWusl]

That's such a beautiful visualization of pythagoras theorem a^2 + b^2 = c^2 I never realized the potential of it. I always thought like "yeah whatever I'll never use it for anything else other than finding c" but damn, you opened my eyes

[@madhuragrawal5685]

It's exactly the equidecomposability problem, no? Maybe viewers would like the numberphile video on the dehn invariant

[@C4pungMaster]

I was thinking about that too, but i think the method that was used here also uses other geometric transformations (skewing, stretching, i dont know what the mathematical term is...). While Dehn variant strictly uses only "cutting"

[@madhuragrawal5685]

@@C4pungMaster yep. In some sense that is better, though this presentation is pretty good too

[@martinepstein9826]

This video is about ruler and compass construction, not just cutting and rearranging. In particular, when he constructs the square with equal area to the rectangle at no point does he cut, move, or stretch the rectangle.

[@randomdude9135]

The 1st thing that came to my mind when he started the proof 😂😂

[@Maniclout]

I loved that video about the dehn invariant, it was nifty

[@VibingMath]

Nice one! Enjoy this video with relaxing music

[@ThinkTwiceLtu]

:)

[@grumpydinosaur1728]

@@ThinkTwiceLtu is there any link to the music video?

[@ivarangquist9184]

2:30 Best Pythagorean theorem visualization I've ever seen!

[@mrkhunt.]

Beautiful Animation, Proving with just visuals plus leading us with the intuition. Just Brilliant!!

[@DiegoMathemagician]

Before my hollidays I challenged myself to "square" a triangle, like Leonardo DaVinci did. Then I forgot about it and you just upload this video. Not sure if I should watch it :(( I want to figure it out by myself but I have a lot of work to do because I also want to find out so many things on my own. A little bit frustrating :/ Anyway, I love your content man! If you see this, Think Twice, i am legenddaryum from twitter (we chatted a while ago about geometry and stuff)

[@ThinkTwiceLtu]

Hey, nice hearing from you. Take your time, it's definitely more rewarding when you figure out things on your own. As always thanks for the support:)

[@randomdude9135]

The catch is you should have a lot of patience and time ;)

[@mathemaniac]

Although I have seen on Numberphile a similar problem, seeing the animation is a lot more satisfying. By the way, which software do you use? I might put some animations in my videos using it.

[@nazishahmad1337]

I guess cinema 4d

[@neerajnandan3519]

I have to study for my geography exam instead I am watching this.

[@0FG0]

There's also the mathematical way: I assume we know the height and bottom length of the triangles as h1,h2,h3 and a,b,c A1 = (h1*a)/2 A2 = (h2*b)/2 A3 = (h3*c)/2 And of course, the area of the polygon is Ap = A1+A2+A3 while the area of the square will be Ap = As = B*B, or B*(d+e+f) where d,e,f are the lengths of the individual rectangles. Each rectangle must have the same area as the respective triangle, and the end result must be a square: (1): A1 = (h1*a)/2 = B*d (2): A2 = (h2*b)/2 = B*e (3): A3 = (h3*c)/2 = B*f (4): B = d+e+f Rearranging the first three and substituting into the fourth: B = (h1*a)/(2*B)+(h2*b)/(2*B)+(h3*c)/(2*B) = (h1*a+h2*b+h3*c)/(2*B) Moving the B under the division to the left hand side means it's squared on the left, so the value of B is: B = sqrt((h1*a+h2*b+h3*c)/2) Where all variables are known. Then from (1),(2) and (3): d = (h1*a)/(2*B) e = (h2*b)/(2*B) f = (h3*c)/(2*B) And yeah, then you're done really. Proof: Ap = A1+A2+A3 = (h1*a+h2*b+h3*c)/2 As = B*(d+e+f) = B*((h1*a+h2*b+h3*c)/(2*B)) = (h1*a+h2*b+h3*c)/2 And as we can see: As = Ap

[@oliverhoare6779]

I’ve never seen the skewing squares method shown here before, I’m glad I did because it looks beautiful.

[@ich6885]

That was actually a pretty nice visualisation of the Pythagorean theorem in your video. 👍

[@mohammedal-haddad2652]

It is amazing how how simple geometry tools you are using to achieve something like this. Thank you very much.

[@tetsi0815]

One of the saddest things in modern middle school mathematics is, that the Pythagorean theorem is mostly taught in the context of triangles and side lengths and not in the context of squares and their area that add up to a square of the same size - or at least that's what most people remember. Putting it in this context makes so much more sense I think and shows why it was such a center piece of "old" mathematics.

[@Ymitzna]

Your videos are just perfect! Short and sweet, showing how beautiful agh can be. Just incredible, always with simple and to the point explanations and fantastic music. Just incredible.

[@ThinkTwiceLtu]

Thanks for the kind words

[@telecorpse1957]

What's agh?

[@xCorvus7x]

@@telecorpse1957 Perhaps Applied Geometry something?

[@telecorpse1957]

@@xCorvus7x That would be ags though :)

[@famicom_guy]

Would I be correct in thinking that this can be done with a ruler and a compass?

[@NotEnoughMs]

Why can't we just transform the polygon in triangles, get each triangle area, add the areas, take the square root and that would be the side of the square?

[@nirorit]

You need a calculator and a measuring tool for that. Ofc you can...

[@SomeoneCommenting]

Because the trick is to do it with the physical sizes of the things that you have. Straight line and compass only, which is what the ancient mathematicians wanted to show. You use Pythagoras as a way to know where to extend lines, it is _not_ to compute any actual numerical value with roots. They only wanted it to know the proportions so that they could find _h_ by sketching its size alone.

[@captainsnake8515]

Continues to be *the* most underrated math channel on KZbin.

[@rahul7270]

BEAUTIFUL! I love your videos. Keep up the great work!

[@spandansaha5663]

This was absolutely beautiful and ridiculously satisfying to watch

[@Alex-xk6sx]

My OCD liked these math animations very much, thank you.

[@farisakmal2722]

Another beautiful video, as always.

[@luisvictoria9542]

Thanks for this! Amazing how you transform math into something approachable

[@syedmraza99]

Absolutely beautiful display of Euclidean Art & Smarts! Make me wish I could do the same in another area is math...

[@cjrm15macpherson20]

2:25 oh my gosh thats the coolest and most easy to understand way of representing pythagorean theorem

[@thesilenttraveller7]

Elegance in simplicity...brilliant, as always.

[@bostash8442]

This is one of the best videos i have ever seen... Subscribed!

[@baixado4ever]

You deserve so many more subscribers for such quality content. I'll do my best to share your channel to as many math enthusiasts I can

[@swankitydankity297]

THe animations are excellent and very easy to follow. Great video! :D

[@ThinkTwiceLtu]

Thanks!

[@arsicjovan9171]

I love how you just casually proved the Pythagoras along the way.

[@dockwonder2278]

The step and the animation from rectangle to square was awesome!

[@BlazingshadeLetsPlay]

Why is this actually tough. So sick. Like the music too

[@karrensusan4825]

I adore the smooth animation and the awesome explanation!

[@ThinkTwiceLtu]

Thank you very much:)

[@chirayu_jain]

Amazing 😉 with calming music 🎶

[@ThinkTwiceLtu]

:)

[@Invalid571]

Excellent proof/video/demo as always! 👏 👏 ☺

[@ThinkTwiceLtu]

Thank you!

[@Invalid571]

@@ThinkTwiceLtu No no no thank YOU! Your videos are a source of delightful inspiration, keep going. 😁

[@ivarangquist9184]

1. Calculate the area of the polygon 2. Take the square root of the result and make a square of that size.

[@AdamGhatta]

This isn’t necessary isn’t necessary for irregular polygons, as 0.5(apothem)(Perimeter) works to find the area of any n-gon, and taking the square root of that formula will give you the side length of the congruent square respectively . In mathematics, they usually don’t give you a shape with no numerical values and ask that you turn it into an equivalent square. This video does something very abstract that was interesting to watch, but not really applicable. Vey cool video, nice job on the solution.

[@user-vn7ce5ig1z]

Each step is clear and makes sense, but taken as a whole, it's insane and doesn't look like it makes sense. Brilliant. 👍 (pun intended)

[@gaurangagarwal3243]

I must say that the beauty of these visual proofs are tending to infinity.

[@theotherihd]

This is the jewel from the second book of the Elements. Its other achievements are the two cases of the law of cosines, and the construction of the extreme and mean ratio (although it isn't given that name until Book VI).

[@alhdlakhfdqw]

beautiful work sir :) thank you very much so inspiring

[@iceiceisaac]

This channel is pure gold

[@Xammed]

This was lovely, thank you.

[@geeteevee7667]

2:29 why am i in love with this bit?

[@denelson83]

And if you do this with a regular polygon, with all those triangles meeting at the centroid of the regular polygon, you get a process that converges toward squaring the circle when the number of sides of the regular polygon increases towards infinity.

[@flavafee]

wow! what a great educational yet relaxing video. might just leave it on for the music alone :)

[@gabrielpacheco2000]

So cool! Imagination at level 5000+ to create/find/animate that!

[@Rockpablosky]

Estos vídeos me dan la vida. Gracias por hacerlos.

[@zraven2931]

I hate mathematics. Why is this channel still so damn satisfying?

[@chiragchandel2749]

0:49 here I was flattered

[@maciek252]

I love the aesthetics of your videos

[@itswakke]

Brilliant and beautiful as always

[@harikrishna2k]

in one word ..... BEAUTIFUL !!!

[@artcool5800]

Can you attach the link with music from this video? Thanks!

[@derekhasabrain]

I think my brain had an or*asm while watching this... so smoothly and perfectly done. Thanks :D

[@joelformica8344]

This was beautiful. I didn’t know I wanted to know this.

[@mcsquidinc.4648]

the cool thing is that all you need to duplicate this in real life is an unmarked piece of rope, some right angles, and a compass

[@mathamatics5384]

For the sliding parts (kzbin.info/www/bejne/b6rSdJWhlN-SZpo) are we using the fact that the area of of slanted square doesn't change since the area of the parallelogram is still base * height?

[@he110h3LLo]

A bit out of topic, I love the music you use in some of your videos. 😌

[@donielf1074]

The problem itself is pretty trivial if you use an algebraic or trigonometric solution, but I’m glad I watched this geometric one if only for the visual proof of the Pythagorean Theorem toward the end.

[@TarekAlShawwa]

00:44 made me hit like and subscribe immediately

[@myselfmono]

so elegant!!

[@musiclibrary894]

If you don't mind, may I know which software do you use for making such animation videos???

[@light.236]

Can you give the link of this music

[@CoryMck]

*How many letters are there in the word **_unnecessary?_*

[@cyancoyote7366]

So nice! I first read about the Dehn Invariant a few months ago. You can do this with any polygon in a finite amount of steps! However, the same is not true for 3 dimensions and volume.

[@grumpydinosaur1728]

Music? The Link doesn't work

[@samisiddiqi5411]

This animation sums up the entirety of Euclid's book 1 lmao

[@erikavagyan4353]

Anyone knows the music's name?

[@TheGiantHog]

Anybody have a link to the proof of 1:45?

[@KNOWLEDGE-lm4re]

Wonderful visualisation...😍

[@SWAGCOWVIDEO]

so basically you can fold a granola bar into the shape of texas

[@Max-sd8vm]

what if you sum the perimeter of all the first 4 rectangles and then you divide by 4?

[@SiiKiiN]

How the rectangles become a square is a bit fuzzy, but the rest of the video is spot on.

[@npip99]

This was beautiful

[@dailywebmoments]

u got a new subscriber

[@jorgeluismonteseljach7980]

I came for the math but stayed for the music

[@JuanIgnacioAlmenaraOrtiz]

Brillante e impecable!

[@lagduck2209]

Just beautiful

[@emanuellopez8578]

Do you develop your ideas from a book (i mean every single step) or you just elaborate what you understood after reading it?

[@ettorefagioli1012]

I love your videos! They make me love math always more and more. So relaxing music 🎶

[@ThinkTwiceLtu]

I'm happy to hear that:)

[@TimMeep]

beautiful and very satisfying

[@hauntedmasc]

This is delightful. :)

[@TheFoolishSamurai]

Can't you just add the areas of the three rectangles and square-root the result to get the sides of the ending square?

[@xCorvus7x]

Algebraically, yes. Geometrically, you need to turn them into squares (or at least into similar polygons) first, since there is no addition theorem such as the Pythagorean Proposition for arbitrary rectangles.

[@kshitijgupta1664]

Lucid Explanation. Keep up the good work.

[@AlexanderRafferty]

I’ve never seen such a brilliantly simple proof of the Pythagorean theorem!

[@PregmaSogma]

Me: *Watches the video* Google maps: *Matt Parker wants to know you location*

[@TheGiantHog]

This channel is beautiful! SO AMAZING

[@KStar-iv4kp]

I feel like this whole thing is just fucking with me

[@adolfocarrillo248]

The basis of all modern technology is mathematics, and the basis of all mathematics is geometry.

[@Invisible-cs24]

gorgeous!

[@ThePharphis]

I can't really understand what's happening at 2:40 when you stretch the shapes and then combine them.