Visualising Pythagoras: ultimate proofs and crazy contortions
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Finally, a Mathologer video about Pythagoras. Featuring some of the most beautiful and simplest proofs of THE theorem of theorems plus an intro to lots of the most visually stunning Pythagoranish facts and theorems from off the beaten track: the Pythagoras Pythagoras (two words :), 60 and 120 degree Pythagoras, de Gua's theorem, etc.
Things to check out:
The cut-the-knot-list of 121 beautiful proofs of the theorem (Marty and my new proof is number 118): www.cut-the-knot.org/pythagor...
The book featuring 371 proof is The Pythagorean Proposition by Elisha Scott Loomis. In particular, check out the "Pythagorean curiosity" on page 252, I only mention some of the curious facts listed here) files.eric.ed.gov/fulltext/ED...
The nice book by Eli Maor, The Pythagorean Theorem: press.princeton.edu/titles/93...
Marty and my new book The dingo ate my math book bookstore.ams.org/mbk-106/
Marty and my webstite www.qedcat.com/
As usual thank you very much to Marty and Danil for all their help with this video.
Enjoy!
![Pythagoras Would Be Proud: High School Students' New Proof of the Pythagorean Theorem [TRIGONOMETRY]](http://i.ytimg.com/vi/p6j2nZKwf20/mqdefault.jpg)
E=m(a²+b²).
:)
My Life has been a lie...
(fixed)
E²=m²c⁴+p²c² !
e^(i*pi) = -m(a^2 + b^2)/E
Did anyone notice that the Mathologer logo is a proof for the Pythagorean theorem?
I did right before I noticed you point it out.
I did then I scrolled, and there was the first comment pointing it out.
@@topherthe11th23 You're welcome.
Thanks
@@jamesolatunji5 you bet
pizza riddle: cut all three in half, arrange them to form a triangle with the cut sides. If the triangle is acute, the small pizzas is the best deal. If it is obtuse, the large pizza is the best deal
...or just count the number of pepperonis on each
You raise an excellent point. We haven't defined the "inherent value" of any given pizza to the customer; it will differ from person to person, and total surface area isn't necessarily what it will be. It could just be the total amount of toppings, if you don't care much about the cheese, the sauce, and the crust.
Fred
26+36 vs 81
Unfortunately you still have to pay for all three pizzas after cutting them up and playing with them :P
But yeah, I came to the same technique.
No. Big one is obviously the best, as it's perimeter is the smallest, so less wasted space for unnecessary crust.b
haha yes, and thats why pizza have a circle shape, to maximize the surface/perimeter ratio.
Pizzaiolo are really smart people.
The scaling proof is absolutely beautiful
by A, B and C? That was my favorite one
Ikr its the best one imo
Wow. The simplicity blew the hell out of me.
I agree. The general shape proof, that he favored was merely consistent but not conclusive
Agreed. The observation that if the sum of the areas of two similar figures attached to the sides of a right triangle equals the area of a third similar figure attached to the hypotenuse, then Pythagoras follows is clever. And the one-line construction that yields these similar figures is truly elegant. I am now convinced that for sheer utility and richness of results, Pythagoras's Theorem beats Euler's Identity as the most beautiful result in Mathematics.
I can't wait to share this proof, as well as de Gua's theorem and the quadrature sum of the areas of a right pyramid, with all my friends! Also looking forward to more fun and games when I get my hands on "A dingo ate my math book." Thanks Mathologer for making these entertaining and informative videos.
I never got to algebra in school. Never made it to college either. All that well over 40+ years ago. But I enjoy these videos like you wouldn't believe. I feel like I'm learning via osmosis. Wish we had this back in the day. No telling where I'd be today. I do work these problems and am understanding algebra a bit. So please continue making these and please keep in mind, some of us are old dogs but we are learning new tricks. Thanks guys.
And please continue to be interested :) It's always a great inspiration for us youngsters to see that curiosity is a quality you can have at any age !
Big Nasty If you have time just try learning from Khan academy; it has videos and workouts
This is geometry though
I am 54, I hear ya! There are a lot of university lectures here on TH-cam as well, Yale and Stanford and many others have their own channels with courses.
Here's the link to Yale where you can see on the side links to other universities, if you're interested.
th-cam.com/users/YaleCourses
Click their "Playlists" to find something of interest.
I am from Ireland and algebra is taught to students from 11 years up. As youngster, I had studied mechanical drawing at school and that has hugely helped me with mathematics. For example, it taught me to understand the theorems of circles, squares, rectangles, etc.
6:15 scaling of triangles. Absolutely beautiful
Oh my god it made me laugh out loud
This man always has the best shirts
Jonathan Hernandez your profile pic is great
I´m German and wondered, why i do understand perfectly the Englisch of Burkard. Well, now I know, Burkard is German ...
Really nice proofs an even better animation !!!
Some other-Pythagorean triples:
120 degree triples: (a,b,c) = (3,5,7), (5,3,7), (6,10,14) (10,6,14) (7,8,13) (8,7,13)
60 degree triples: (a,b,c) = (1,1,1) (2,2,2) (n,n,n) etc (3,8,7) (8,3,7)(5,8,7)(8,5,7)(6,16,14)(16,6,14)(7,15,13)(15,7,13)(8,15,13)(15,8,13)(10,16,14)(16,10,14)
Tehom
It would be better to remove all multiples and duplicates, and simply list the primitives.
e.g.
(3,5,7) (7,8,13)
(1,1,1) (3,7,8) (5,7,8) (7,13,15) (8,13,15)
based on what you provided
In general,
(a² - b²)² + (2ab - b²)² - (a² - b²)(2ab - b²) = (a² - ab + b²)²
(a² - b²)² + (2ab + b²)² + (a² - b²)(2ab + b²) = (a² + ab + b²)²
Integer-triangles with an angle of 60 degrees or 120 degrees are called "Eisenstein triples":
60 degrees: (3, 8, 7); (5, 8, 7); (5, 21, 19); (7, 40, 37); ...
120 degrees: (3, 5, 7); (7, 8, 13); (5, 16, 19); ...
Based on the name, I suppose there's a connection with solutions to those Diophantine equations and Eisenstein integers (i.e. complex numbers of the form a + b\omega + c\omega^2, where \omega is a primitive third root of unity and a, b, c are integers) similar to 3b1b's video on Pythagorean theorem and Gaussian integers.
simpler for 60deg: 1, 1, 1 (or any other Number, they just have to be equal) :D :P
UniTrader he says „nontrivial“ in the video
1 1 0 is a trivial solution. 1 1 1 is OK.
I absolutely love your T-shirts, can we get them anywhere?
icestork.com/product/pythagoras-vs-einstein-c2-shirt/ might get it myself.
Usually just a quick google of whatever is on the t-shirt will get you there :) By popular demand I've also put some that I designed myself here shop.spreadshirt.com.au/1035536/
I will not be surprised if they are soon out of stock.
shouldn't Pythagoras be wearing trousers?
Sleves sqaured plus hat squared equals trousers hoses squared?
Sleeves volumes squared plus hat volume squared plus trousers hoses squared equals muscle shirt volume squared? IDK I'm a flatlander with poor imagination.
The last six minutes were mind-blowing!!
The proof at 6:30 - 6:50 blew my mind. So neat and beautiful.
The Mathologer logo is the first proof of Pythagoras's theorem. ;)
That's it. In fact, I used 3-4-5 triangles for the triangles in my logo :)
Its a new logo
No, the channel logo.
Nvm false memory
My own favorite proof is based of that shape :: 4 identical copies of a right triangle of sides a, b and c organized in a square shape.Exterior square has side s = a+b and its area is s^2 = a^2 + b^2 + 2ab . But the area is also the sum of the four triangles plus the inner square of side c.So Area = 4* (ab/2) + c^2 = 2ab + c^2 = s^2 = a^2 + b^2 + 2ab. And, therefore a^2 + b^2 = c^2. Done.
I've never found Pythagoras so entertaining! A fascinating, fun and educational video! Many thanks, Mathologer!
I love your sense of humor so much, man... You are a delight.
Thank you for doing these - my world is enriched with you in it, in so many ways. ...Okay, I may have a bit of a crush.
I was asked to prove Pythagoras during a university interview back in 1983. I used the first proof with the big square in it, which seemed to go down well. However, I think the scaling proof is definitely my new favourite!
I absolutely appreciate the amazing visualizations in each of the videos here. Great work!
I hope you never stop making videos!
6:46 OHHHH WOW!!! That has got to be absolutely the most wonderful proof ever!
Great video, simple and very fun! Reminds me a bit of Byrne's edition of Euclid due to the sheer artistry of the whole ''coloured shapes manipulation'', while this may not be the most rigorous or useful way to do geometry, darn is it esthetically beautiful. Have to check out that book of 371 proofs of pythagoras, sounds like recreational heaven!
Yes, Byrne is great, here is an online version www.math.ubc.ca/~cass/euclid/byrne.html
I've also provided a link to the book of proofs in the description. Have a look :)
cut the pizzas in half
arrange them into a triangle
if the triangle is acute take the two smaller pizzas
if the triangle is obtuse take the large pizza
if its a right triangle then take either
Michael P it's*
I think that works. But the smaller two diameters don't have to add up too the diameter of the larger one. If that's the case they don't make a triangle.
Which obviously means the bigger one has more area than the smaller combined.
Nice one.
Awesome, a fellow intellectual who watches Rick and Morty with a IQ of 300
one scenario you forgot to mention is if the triangle couldn't be made. In that case again get the large pizza.
I'm only eating at pizza places that give the square root of the price from now on.
I really enjoy these amazing gems of math wisdom! Keep it up! It is so enjoyable to learn and relearn these interesting facts!!!
Thank you!
Mathologer is really amazing and I love it. Nice T-shirt!
My answer would be to cut the pizza's in half and put them together into a triangle. Putting the small and medium at a 90 degree angle to eachother and then fitting the large slice inbetween the remaining 2 vertices. If the diameter of the bigger pizza slice is smaller than the distance between the vertices, then it is a bad deal, if it is larger then it is a good deal. If it fits exactly then both are a great deal.
Jelle I'm not sure if this would work since you ONLY have a pizza knife...
Jelle (not then again, it's a math problem)
Jelle How do you cut the pizzas exactly in half?
We have to assume we can do at LEAST that, with a mathematical pizza knife.
mushookie man I suppose so. I asked because finding the center of a given circle is a common compass-and-straightedge exercise.
FINALLY, I got the proof of that 1/A²+1/B²=1/D² stuff.
Consider the area of the given triangle. Since it is a right triangle, the area (I call it F) can be calculated with F = 1/2*A*B. Since D is the height on C, the Area can be evaluated with F = 1/2*C*D aswell. Combining these two equations we get: C*D=A*B. Squaring both sides give us C²*D²=A²*B². But C²=A²+B². So we get the final equation: D²(A²+B²)=A²*B². Rearranging will give us what we initially wanted to show. q.e.d.
Good job, I just found out it is called the "inverse Pythagorean theorem"
Thanks man
Put together the same proof today. Using the equivalent areas is the key.
angeluomo exactly 👍
Nice! I solved it in a more pedestrian way :-(. I called the the two intervals into which D dissects C C1 and C2. Then we have three equations:
Two P-theorems:
(1) C1²+D²=A²
(2) C2²+D²=B²
And, as the two internal triangles have the same proportions:
(3) C1/D = D/C2,
which can be reerranged to obtain:
(3') C1*C2=D²
Express C1 and C2 from (1) and (2), plug them into (3') and after rearrangement you get the expression. I prefer yours though.
Congrats with the new book!!!
The best math channel ever.
OK, here's my answer to the pizza problem:
Cut all three pizzas in half and form a triangle with one half from each pizza--the cut edge (diameter) being used as each side. If the triangle is right, both deals are equal. If it's an acute triangle, the two smaller pizzas are a better deal, and if it's an obtuse triangle, the larger pizza is the best deal.
Great video; planning to watch it several times over.
Ken Haley How do you cut the pizza exactly in half?
Also, how do you measure angles if we've only given a pizza knife?
+John Chessant Cut it in half by "eyeballing" it. A small error in the angle of the cut would result in a much smaller error (percentage-wise) in the length of the cut.
+letao12 You can look at it and see if it's close to a right triangle; if so, you can consider the deals equal within a few pennies.
To both: If I can't make these assumptions, I don't think a solution exists.
Omg Pythagoras and Einstein fighting over c squared, and that's the first thing I see...
I want that T shirt
Thank you so much... I have shared this with many kids in lower school grades they have found this to be very helpful!!!
i love when simple demonstrations put a smile on my face :D
For 19:13, make the three lengths from the right angle a, b, and c. How we find the sum of the areas and use Heron’s formula for D^2. After some simplification the two expressions become the same.
I proved the 3D counterpart (square areas) when I was in high school. When I went to uni, I showed it to a classmate who then proved a 4D version. He showed it to one of his computer science tutors, Carroll Morgan, who then proved it for all dimensions, but I don't know if he ever published it. A few months later, I was stunned when I was browsing at the library and by chance I opened up a journal and saw another proof. That was around the late 1970s.
Is it a 3D version of the Pythagorean theorem?
@@T.R.-TRUTH.REASON more than that, people proved an nD version.
the 3D version is the sum of the squares of the areas of the three right-angular sides of a tetrahedron equals the square of the area of the largest side.
@@MusicalRaichuI think you're talking about a cuboid or a cube, right?
@@T.R.-TRUTH.REASON Yes, diagonally slice a tetrahedron from the corner of a cube. sum of squares of areas of the three rectangles in the corner equals square of area of diagonal side.
For 4D, it's squares of volumes of tetrahedra. And so on for higher dimensions.
@@MusicalRaichu Yes，because it is a square.I have got it.
Thank you, professor!! You really widen my eyes open. God bless you!!!
Why can't I stop smiling while watching a math video? Beautiful
Proving Pythagoras theorem by proving fermat's last theorem
This.
Geometers are always in love with Pythagoras...
Excellent! thank you so much for your presentation. I loved how you made it fun and the visuals were so easy to understand.
this is the best video I've seen in AGES. fantastic. pleeeeease make a part 2 vid
It's also pretty simple to prove in general Hilbert spaces:
||a+b||² = = + + + = ||a||² + 2 Re + ||b||². So if a and b are orthogonal, i.e. = 0, we have ||a+b||² = ||a||² + ||b||².
Yes, but Hilbert spaces all use the L2 metric, which is practically the same thing as taking the Pythagorean theorem as an axiom. Of course it's easy to prove, it's the very thing powering the entire space!
What are Hibert Spaces?
+tamara ciocan A generalized version of Euclidean vector spaces that allows for infinite dimensions and complex vectors.
a52Productions exactly!
i think I'd use the pizza cutter as a tachometer wheel and measure the circumference of the two smaller pizzas and if the square of both is greater than that of the square of the larger, I'd go for the two smaller :)
The best teacher and the best material on youtube ... thanks
Happy birthday, Dr. Polster!
ayyye sunday evening is saved :D
The text at 10:04 says: "Any shape with non zero area!!!" :)
The king of visualizations!
this video is long overdue. amazing 👏👏👏👏
Giving credit to the real inventors of the theorem. You earned yourself a subscriber.
For the pizza problem:
Go to the pizzaman with the knife, cut him half, see whether it creates an 90 degree triangle and then eat the pizzas
That simplest proof actually blew my mind the moment I saw it.
It was like Pythagoras just became as immediately obvious as common sense in an instant.
Wow.
This is video made me open my mouth in awe more than once.
It made me pause and run to my notes,
It made me think,
It was so beautiful. Thank you
What about A³+B³+C³=D³? Does this have a positive integer solution?
Yes, for example, 1³+2³+2³=3³ . Have a look at this en.wikipedia.org/wiki/Pythagorean_quadruple
Mathologer Umm, I mean its cubed, not squared. Sorry about that...
Sorry, my eyesight much be failing me. Of course, you were after cubes and not squares. Here you go 1³+6³+8³=9³ . These is an example of a cubic quadruple.
Mathologer Oh great. Thanks! I wonder if you can prove it, like you can prove Pythagoras, but using cubes and some sort of 3D shapes. Does it have any applications in geometry?
Nice video before watching it
Another glorious romp in geometry led by the ever-enthusiastical Mathologer! ABC-Delicious!
Oh wow, the proof with the scaled triangles is so beautiful.
9:53, What about fractals :p
Yep the super-theory is actually wrong "for all shapes". It needs to be "for all 2 dimensional shapes"
Fractals have fractional dimensions and don't scale the way one might normally expect.
8bitpineapple you forgot the question mark
Fractals don't always have to have fractional dimension. A filled-in koch snowflake has a fractal dimension of two, even though its boundary hasn't. The mandelbrot set even has a two-dimensional boundary and is still considered a fractal.
Jorge. I didn't forget the question mark, I intentionally didn't include it. I already figured that in the case of fractals it's not going to work when the Hausdorff dimension doesn't equal it's topological dimension, and that we'd need to replace "area" for something like "the sum of the area of all 2-faces" if we want to include shapes with a topological dimension not equal to 2.
So while it's not proper grammar I decided not adding a question mark would eliminate some confusion that I was actually asking "What about fractals?", my point was just to be cheeky and pedantic.
I would also like to know Donald Trump's tweeted proof of Pythagoras's theorem, immediately.
I would like to see how all Trump detractors also post a self discovered proof of Pythagora's theorem, seeing that all of them are much smarter than him.
John ... well we aren't the president, are we?
It begins "First, construct a wall W..."
"I know proofs, I have the best proofs, believe me..."
They are measuring him on another former president, not on themselves. Of course, they are judging him on a matter that hasn't got much to do with what a President needs to do, and I doubt he would be any more pleasent if he was more intelligent, but it seems like this was a joke.
Fanstastic video! It's amazing to see how beautiful and versatile the Pythagorean theorem is and how many generalizations it allows. One could even make the case that the Pythagorean theorem was in a sense the foundation for mathematics! I just think it's unfortunate that Pythagoras got too much credit for it, since he probably wasn't the first to prove it, but I guess that's how history goes.
Outstanding video Sir, great job.
Your t-shirt 😓 suuu goooddd make me 😳laugh so badly 🤤
I wonder if Trump has ever heard of Pythagoras. He probably thinks that Pythagoras is a refugee who wants to steal all his money :P
You are like the best math teacher I never had
i just realized i conjectured de gua's theorem when i was 15.
great video as always, congrats on the book
r/iamverysmart
Pi ta go Ra s
Well, more like pi - tha - gO - ras
The intonation is in the "ό" letter (πυθαγόρας)
Here is how we pronounce it in modern Greek
translate.google.com/#auto/en/%CF%80%CF%85%CE%B8%CE%B1%CE%B3%CF%8C%CF%81%CE%B1%CF%82
I wonder how the actual pronounciation of his name, was in ancient Greek...
You mean Pi Ta Go Ra Su I Cchi, right?
That Trump joke was VERY disrespectful, Burkard.
Don't you know that Trump invented ALL the proofs of this theorem thousands, millions, billions of years ago???
So ignorant.
JustOneAsbesto Yeah, and he made the babylonians pay for them. He used them to build the numerals great wall of china and crooked pythagoras stole them. Then the christians stole alternate facts and named them lies and orange was outlawed in europe by fake news. He fled to america.aa
Less is more.
I hate the fuckiiiin tao. Modern symbol sucks!
Omg what a roller coaster of a video, I had no clue on how many levels was Pythagoras cool!
Brilliant.I hope some of my ex-pupils were watching and some of the proofs and ideas come back to haunt them.I also
was made aware of so much more.Thank you
The guy in the background sounds a bit like a sitcom laugh track - I think he adds nothing to an already great video
okarakoo That guy is the cameraman... He's just doing unscripted friendly interaction.
erm ... no.
The guy in the background in Marty Ross, one of Burkhard's collaborators.
I enjoy the spontaneous reactions from the guy behind the camera: to me he kind of represents all of us. He is also kind of Burkard’s ‘Greek Chorus’. Their interactions are real, relaxed and charming.
ABSOLUTELY BEAUTIFUL
Thanks....thanks...thanks...
When I first saw this video title I thought it was something trivial that I was supposed to know ....but after watching it...I'm still in awe !....the elegant universe of mathematics....😎
Mindblowing thank you for this!
this channel is great.
Animation is a really powerful tool for explaining these things. I have seen several of these proofs before, understood them, and promptly forgotten them. I think I will remember two of these.
That's great, mission accomplished then as far as you are concerned :)
Amazing the number of demos and facts you show here ! Thanks !!!
NB : There's a funny similar theorem with parallelograms too...
My mind exploded when you showed the "make three copies of the ABC triangle and scale each by A, B, and C" proof; that's my new favorite proof. Though, oddly enough, I already knew about the "square to parallelogram to rectangle" proof, but I had no good way of describing what I knew.
I also knew about how you can extend the Pythagorean Theorem to ANY triangle and it involved adding or subtracting something multiplied by a trig function to compensate; I just happen to stumble upon it while watching TV, and that's how I first learned about trigonometry as early as 6th grade.
Congrats on the book, I’m excited to read it! Any plans to release it in kindle format?
This video gave me a new appreciation of geometry.
I guarantee it, this video will become one of the Mathologer classics
(also 1000th comment!)
That final proof of pythagoras is amazing.
This is Fantastic!
brilliant and amazing proofs.. really enjoyed..
Hey mathologer, I just wanted to say that i am a big fan of your videos and have been a fan for over a year and a half.Recently my uncle gave me a book for my birthday and coincidentally it was written by you.It was Sciencia and your book (QED Beauty in mathematical proofs) was written very well and was informative.I just wanted to say thank you for spreading math on youtube and also writing great books.
Glad you like what I do and thank you very much saying so :)
Thank you!!
That subtle morph at 10:34 literally hurt my brain.
Thank you for this, more math proofs please!
When I was revising for my maths exam this year I found that out before I watched this video.
Thanks a lot, I hard a really bad time memorizing al-Kashi theorems but thanks to you it made it more clear
One of the best videos..
When angle = 60*. 5^2 + 8^2 - (5*8*) = 49 = 7^2. When angle = 120*. 7^2 + 8^2 + (7*8) = 169 = 13^2. Thanks you Mathologer, because of you I find some really interesting new stuff.
It blows my mind that nobody has pointed out that your shirt is actually even funnier when you realize that Einstein's equation E =mc2 is actually a very specific application of pythagorean's theorem.
Because the full form of it is E^2 = (mc^2)^2 + (pc)^2
The version we normally see is for a stationary object with Rest Mass. So p is equal to 0, canceling the second term.
I always enjoy your videos, but this one is very nice. Just bought your book too...
Excellent man! I like the squares of areas stuff :)
Excellent as always! Lots of homework! I'm getting the book next week! Going to try cubes. A^3+B^3+C^3+D^3 = ( ? ) after watching this a couple more times. Thank you Doc!
Of course I enjoyed this video! What a question.
Außergewöhnlich!
Even I as a nonmathematician have now a grasp (a slight one perhaps -- and probably very imperfect at that) of Pythagoras & the wider implications.
Vielen Dank, Herr Professor Mathologer!
That's great :)
Hey, bist du auch Deutscher?
Every time he says "most of you know this" I am learning something completely new to me.
Very nice video guys keep it up, greetings from Uruguay
அருமையான விளக்கம்.நன்றி... fantastic explanation.Thanks...
Awesome
I really enjoyed this video!
Awesome
I really enjoyed this video!
That shirt is more brilliant than it lets on, because the extended form of E=mc^2 basically turns mass and momentum into the lesser sides of a right triangle, with E being the hypotenuse!