All possible pythagorean triples, visualized

To understand all pythagorean triples like (3, 4, 5), (5, 12, 13), etc. look to complex numbers.
An equally valuable form of support is to simply share some of the videos.

Regarding the brief reference to Fermat's Last Theorem, what should be emphasized is that it refers to *positive* integers. You can of course have things like 0^3 + 2^3 = 2^3, or (-3)^3 + 3^3 = 0^3.

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld

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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).

Various social media stuffs:

[bluebrown]

As to the "you're" typo at 1:20, I keep telling that second blue pi creature (Randolph is his name) to learn his grammar, but for whatever reason, he just never listens and focuses only on his math lessons.

[vibng]

imagine being a 1st grader doing their shapes homework and searches up “triangles” and gets this

[onlynamelefthere]

At some point you think you have seen everything, which is to say about a "simple" topic like pythagorean triples. And then comes this video and blows your mind with the elegance and simplicity of it all. And you will be reminded, there is no such thing as "simple topics" and "everything to know".

[felely]

This is hella interesting when you have an English essay due

[primephoenix.]

Special Thanks to
1. Pythagoras
2.Reńe Descartes
3.Bernhard Riemann
4.Grant Sanderson
For this Marvellous Video😄

[generalralph]

I needed this today. I’m building a house made entirely of Pythagorean Triples.

[johnrickert]

Absolutely beautiful! I have a Ph.D. in Mathematics and have never seen a discussion of Pythagorean Triples in terms of complex numbers before. Thanks for this great video!

[jacheto]

I LOVE THE FACT THAT YOU ARE POSTING VIDEOS EVERY TIME PLEASE NEVER STOP

[theseal]

You should make an ”Essence of topology” series. Topology is very visual but can be hard to describe with just numbers. I think ur animations would make a great fit for teaching topology

You could cover topics like: Projective space, Equivalance relations or quotient space, affine geometry, hyperbolic geometry.

And then u can end of the series by briefly giving an understanding to the poincaré conjecture.

[maane]

"The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.”
- Henri Poincaré -

[nathanielsharabi]

>has final exam in 2 days
>*sees 3blue1brown uploaded new vid*
>"the bloody exam can wait"

[Joe]

Does anyone ever feel saddened by the beauty of these videos? It's not just, "I wish math was taught to me this way", it's that I now think there's got to be this beauty in so much more, and my eyes are just not open to seeing it.

[shiladri]

This is quite simply the best Maths learning resource on the interent...a service to humanity!

[soheilsanati]

In Euclid’s Elements there is a description of all the possible Pythagorean Triples. Here’s a modern paraphrase of Euclid.

Take any two Odd Numbers m and n, with m < n, and relatively prime (that is, no common factors). Let A = m x n; B = (n^2 – m^2)/2, and; C = (n^2 + m^2)/2. Then A:B:C is a Pythagorean Triple.

For instance, if you take m = 1, and n = 3, then you get the smallest Pythagorean triple 3:4:5.

[OskarElek]

The beauty of maths is that you can take something seemingly trivial and boring, and make it extremely intersting by digging deep enough.
The beauty of 3b1b is that he does it for us :)

[MegaMoh]

For anyone who wants to graph the intersecting parabola, the general equation for each parabola is x=[+/-](y^2 / 4(n)^2 - n^2) where "[+/-]" is plus or minus and "n" represents the nth parabola away from the origin. In latex, it's written as:



for those who want it written neatly. The straight line equations are as simple as taking each coordinate that from the intersection (a, b) and making the equation y=b/a * x or y= \frac{b}{a}x in latex

NOTICE: A parabola written in the form of ax^2+bx+c has a=1/(4f) where f is the focus. I noticed that the focus for those parabolas using the equation is n^2 so that the focus of all of these parabolas is it's number squared. then noticed that the focus changes when the "c" term changes in the equation, then the focus get translated by "c" and what turned out is that the "c" term in the above equation is also n^2! so n^2(the focus) - n^2(translation by "c" term) gives 0. so that all of those parabolas have their focus at the origin and each one is away from the origin by n^2 distance! Let's work together to figure out why this equation works with these givens

[joefagan]

Grant, you are simply amazing. I've a life long passion for maths and took an M.Sc in maths just for fun. Thank you so much for these videos. Imagine if Einstein or Feignman or even Euler or Pythagoras could have seen your videos, they would have been blown away. You're taking the beauty and structure that they could see and shown it to the masses. You are the ultimate pedagogue. Thank you.

[nitinmadan]

What an amazing visualization. A few years back, I tried coming up with a proof to find an elegant proof for finding Pythagoras triplets. Didn’t succeed.
But this video just gave me a whole new perspective.

Cheers!

[blockyhour]

The fact that I finally understand what he's talking about makes it SO much more interesting

[macmos]

You have an incredible intuition and perspective on mathematics. Please never stop sharing your knowledge with us!