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
@@-----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
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”.
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.
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"
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.
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
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
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.
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)
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.
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.
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?
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.
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.
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).
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.
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.
@@chopinyt Dude what? The practical application of this: A farmer has a field which is shaped irregularly (like a polygon). The state takes his fields and compensates him with a square field of equivalent area. This is how to convert the area from a polygon to a square.
Can't you directly convert the triangles into rectangles? Using a pair of triangles to change into a rectangle. Then the process will be slightly different for an odd number of triangles. We convert the final 3 triangles together in that case.
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.
Can you animate this using only straight line cuts? It is clear that you can create a square that has the same area of a polygon (just side lengths that are the square root of the area), but not as straight forward by only doing it with straight line cuts
hi.. great video! The only step I don't understand is step 1... turning the polygon into triangles....OK.. got it. BUT turning the triangles into rectangles without knowing the Height .... how do you do that? Nothing is labelled on the polygon so I don't know what you're using as starting info..... all the sides labelled?? What do yo need? thanks
This video seems to be using the rules for straightedge and compass construction alongside some basic geometric equations. Normally in this form of geometric manipulation you don't really deal with numbers much in the actual construction process, but if the shape given is arbitrary then it seems reasonable to assume that values such as the height and relevant side lengths/angles are given. The exact info you are given will probably be different for individual cases, but if you see this in a school environment you should always have enough info to solve, barring some mistake, and if in a practical environment you should be able to simply take additional measurements.
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.
For the sliding parts (ua-cam.com/video/9yoCbk_z_08/v-deo.html) 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?
I really like the fact that you chose a much more substantial topic for this one, but my only gripe is the fact that you didn't lay down the "rules" clearly (i.e. what actions are you allowed to do when "converting" one shape into another). For example, as mentioned in one of the other comments the equidecomposability theorem allows for only cuts while in this case it seems like there is more freedom (although I also have a feeling that the non-cut transformations you made can still be expressed as a series of cuts). I feel like for situations like these, prohibiting yourself from using a bunch of text can be detrimental at times. Nevertheless, your animations were on point as usual.
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
Is it possible to do this step by dividing the rectangle into finitely many parts that will make the square?
It is. Lemma 2 (and figure 3) of the following paper shows how to do this: rak.ac/files/papers/wallace-bolyai-gerwien.pdf
Thank you for this toy.
It is fun to slide along a hyperbola of constant area.
can you share us the code, great job by the way.
You did a great job. Keep making stuff like this. People need your creations!
This was dope as hecc
Thanks:)
Hi
Hi
If we take n-gon where n tends to infinity, so this problem can be named as..
Squaring the Circle!
But, are u sure u wanna construct infinite triangles and carry out the process ;p
It also reminds me trisection as infinite bisection 1-1/2+1/4-..
@@-----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
What does your name mean?
@@whatisthis2809 good q
beautiful animation, well done.
Thanks a lot.
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”.
as seen on numberphile
proof or it didn't happen
Does anybody know a simple proof of the solution of the third Hilbert's problem?
2:53 bottom corner of pink parallelogram: "am I a joke to you?"
Bullseye
Looks like insemination😏
Jeez, one pixel
Random Dude
Why, just why?
perfectionist brain : I HATE THISSS
ADHD Brain : Ahhh Perfection
I love your videos they inspire me to do more Mathematics in my free time!!!
Thank you;)
You are a luminary for disillusioned aspiring mathematicians like me.
That’s so cool.
Especially how you can turn the squares into a larger one using the simple Pythagorean theorem.
Love your work.
Thank you very much:)
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.
It's exactly the equidecomposability problem, no? Maybe viewers would like the numberphile video on the dehn invariant
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"
@@C4pungMaster yep. In some sense that is better, though this presentation is pretty good too
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.
The 1st thing that came to my mind when he started the proof 😂😂
I loved that video about the dehn invariant, it was nifty
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
Nice one! Enjoy this video with relaxing music
:)
@@ThinkTwiceLtu is there any link to the music video?
2:30 Best Pythagorean theorem visualization I've ever seen!
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
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.
I’ve never seen the skewing squares method shown here before, I’m glad I did because it looks beautiful.
Beautiful Animation, Proving with just visuals plus leading us with the intuition. Just Brilliant!!
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)
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:)
The catch is you should have a lot of patience and time ;)
It is amazing how how simple geometry tools you are using to achieve something like this. Thank you very much.
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.
Thanks for the kind words
What's agh?
@@telecorpse1957
Perhaps Applied Geometry something?
@@xCorvus7x That would be ags though :)
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.
I guess cinema 4d
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?
You need a calculator and a measuring tool for that. Ofc you can...
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.
That was actually a pretty nice visualisation of the Pythagorean theorem in your video. 👍
Elegance in simplicity...brilliant, as always.
Another beautiful video, as always.
I have to study for my geography exam instead I am watching this.
Excellent proof/video/demo as always!
👏 👏 ☺
Thank you!
@@ThinkTwiceLtu
No no no thank YOU!
Your videos are a source of delightful inspiration, keep going.
😁
Thanks for this! Amazing how you transform math into something approachable
1. Calculate the area of the polygon
2. Take the square root of the result and make a square of that size.
The step and the animation from rectangle to square was awesome!
Absolutely beautiful display of Euclidean Art & Smarts! Make me wish I could do the same in another area is math...
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
I adore the smooth animation and the awesome explanation!
Thank you very much:)
Would I be correct in thinking that this can be done with a ruler and a compass?
This was absolutely beautiful and ridiculously satisfying to watch
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.
2:25 oh my gosh thats the coolest and most easy to understand way of representing pythagorean theorem
Continues to be *the* most underrated math channel on UA-cam.
Amazing 😉 with calming music 🎶
:)
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).
BEAUTIFUL!
I love your videos. Keep up the great work!
00:44 made me hit like and subscribe immediately
I must say that the beauty of these visual proofs are tending to infinity.
2:29 why am i in love with this bit?
THe animations are excellent and very easy to follow. Great video! :D
Thanks!
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.
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.
Estos vídeos me dan la vida. Gracias por hacerlos.
This was lovely, thank you.
u got a new subscriber
So cool! Imagination at level 5000+ to create/find/animate that!
in one word ..... BEAUTIFUL !!!
This channel is pure gold
The moment at 0:50 is super awesome 👌👏
Why is this actually tough. So sick. Like the music too
Such a good video! I was just wondering if there are any practical uses for this?
It's not real math if there is a practical use for it
@@chopinyt
Dude what?
The practical application of this:
A farmer has a field which is shaped irregularly (like a polygon).
The state takes his fields and compensates him with a square field of equivalent area.
This is how to convert the area from a polygon to a square.
@@Invalid571 That's a joke from people studying maths ;)
@@chopinyt
Oh, I took it very literally. 😅
/rwoosh me
@@Invalid571 Sounds like something G.H. Hardy would say. G.H. Hardy hates applied mathematics.
0:49 here I was flattered
This is one of the best videos i have ever seen... Subscribed!
I love the aesthetics of your videos
Can't you directly convert the triangles into rectangles? Using a pair of triangles to change into a rectangle. Then the process will be slightly different for an odd number of triangles. We convert the final 3 triangles together in that case.
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)
so basically you can fold a granola bar into the shape of texas
I’ve never seen such a brilliantly simple proof of the Pythagorean theorem!
I love how you just casually proved the Pythagoras along the way.
This is delightful. :)
wow! what a great educational yet relaxing video. might just leave it on for the music alone :)
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.
I hate mathematics. Why is this channel still so damn satisfying?
Can you animate this using only straight line cuts? It is clear that you can create a square that has the same area of a polygon (just side lengths that are the square root of the area), but not as straight forward by only doing it with straight line cuts
This was beautiful. I didn’t know I wanted to know this.
A bit out of topic, I love the music you use in some of your videos. 😌
beautiful work sir :) thank you very much so inspiring
Nice!
Lucid Explanation. Keep up the good work.
*How many letters are there in the word **_unnecessary?_*
Can you attach the link with music from this video?
Thanks!
i done this with a square, it worked
If you don't mind, may I know which software do you use for making such animation videos???
Brilliant and beautiful as always
hi..
great video! The only step I don't understand is step 1... turning the polygon into triangles....OK.. got it.
BUT turning the triangles into rectangles without knowing the Height .... how do you do that? Nothing is labelled on the polygon so I don't know what you're using as starting info..... all the sides labelled?? What do yo need?
thanks
This video seems to be using the rules for straightedge and compass construction alongside some basic geometric equations. Normally in this form of geometric manipulation you don't really deal with numbers much in the actual construction process, but if the shape given is arbitrary then it seems reasonable to assume that values such as the height and relevant side lengths/angles are given. The exact info you are given will probably be different for individual cases, but if you see this in a school environment you should always have enough info to solve, barring some mistake, and if in a practical environment you should be able to simply take additional measurements.
@@nemo2803 Thanks a lot! Great explanation.. I was was definitely thinking about it the wrong way.
Do you develop your ideas from a book (i mean every single step) or you just elaborate what you understood after reading it?
How the rectangles become a square is a bit fuzzy, but the rest of the video is spot on.
Can't you just add the areas of the three rectangles and square-root the result to get the sides of the ending square?
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.
I came for the math but stayed for the music
Amazing, could it work with fractal polygons?
I don't think that's possible.
2:37 I can't believe I've never seen this before
Music? The Link doesn't work
Wonderful visualisation...😍
beautiful and very satisfying
I can't really understand what's happening at 2:40 when you stretch the shapes and then combine them.
what if you sum the perimeter of all the first 4 rectangles and then you divide by 4?
so elegant!!
Just beautiful
I think my brain had an or*asm while watching this... so smoothly and perfectly done. Thanks :D
Good ole' quadrature. Now do a circle! ;P
For the sliding parts (ua-cam.com/video/9yoCbk_z_08/v-deo.html) 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?
Me: *Watches the video*
Google maps: *Matt Parker wants to know you location*
Tenho minhas duvidas... Por exemplo: que?
I really like the fact that you chose a much more substantial topic for this one, but my only gripe is the fact that you didn't lay down the "rules" clearly (i.e. what actions are you allowed to do when "converting" one shape into another). For example, as mentioned in one of the other comments the equidecomposability theorem allows for only cuts while in this case it seems like there is more freedom (although I also have a feeling that the non-cut transformations you made can still be expressed as a series of cuts).
I feel like for situations like these, prohibiting yourself from using a bunch of text can be detrimental at times.
Nevertheless, your animations were on point as usual.
This was beautiful