All possible pythagorean triples, visualized
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when asking about Pythagorean triples, rational points on circles, or Gaussian integers, as an intuitive picture of why triples are parameterized the way they are.
Recommended by 4 people across 4 different months, 2017 to 2024.
What it is
A 3Blue1Brown animated explainer showing how every primitive Pythagorean triple such as (3, 4, 5) or (5, 12, 13) comes from squaring Gaussian integers (complex numbers with integer parts). It builds the idea visually and ties it to rational points on the unit circle.
What people said
Recommends WaveX · 2017-07 · Mathematics Stack Exchange · on “What are the perfect squares for $a^2 + b^2$?”
[this video] I personally like this video by 3Blue1Brown. It talks about Pythagorean triples in a way to "visualize" all of them.
Recommends Carlos Toscano-Ochoa · 2018-09 · Mathematics Stack Exchange · on “Are the rationals on a unit circle dense?”
Maybe you should see 3blue1brown video about pythagorean triples, as it is directly (and surprisingly) related with your question: [this video]
Recommends D.R. · 2021-01 · Mathematics Stack Exchange · on “What do cones have to do with quadratics? Why is $2$ special?”
I've always been nagged about the two extremely non-obviously related definitions of conic sections (i.e. it seems so mysterious/magical that somehow slices of a cone are related to degree 2 equations in 2 variables). Recently I came across the following pages/videos: This 3B1B video about ellipses [LINK] , which reignited my desire to understand conics Why are quadratic equations the same as right circular conic sections? [LINK] , which offers a very computational approach to trying to resolve this question Another 3B1B video on visualizing Pythagorean triples [this video] (i.e. finding the rational points of a circle) and Manjul Bhargava's lecture on the Birch-Swinnerton-Dyer Conjecture [LINK]
Recommends Sammy Black · 2024-07 · Mathematics Stack Exchange · on “Relationship between Pythagorean Triples and double/half angles”
I think you might really enjoy this video by 3Blue1Brown [this video] about the connection between generating Pythagorean triples and multiplying Gaussian integers (complex numbers with integer components).
Cites AlgorithmsX · 2017-06 · Mathematics Stack Exchange · on “Polygon with integer vertices”
@JairTaylor This video [LINK] deals with finding the number of lattice points in a circle using Gaussian Integers to calculate $\pi$ and this video [this video] deals with finding the number of distinct Pythagorean triplets. The first video will get you an exact bound for the number of lattice points and the second video will get you the number of distinct vectors with integer components within a circle of radius $r$.
Cites Shashank · 2017-11 · Mathematics Stack Exchange · on “Maximum value of $\gcd(u^2-v^2, 2uv, u^2+v^2)$”
What is the maximum of $\gcd(u^2-v^2, 2uv, u^2+v^2)$, where $u$ and $v$ are integers and $u\neq v\neq 0$ ? I have tried to find it but I haven't got anywhere. I got this question when I was watching this [this video] video, where he says, at 9:36, that we need not scale down by less than $1/2$ (you might want to rewind the video). How do we prove it? Follow up question: What about $\gcd(u^2-v^2-w^2, 2uv, 2uw, u^2+v^2+w^2)$ and so on?
Cites spydragon · 2019-05 · Mathematics Stack Exchange · on “Pythagorean triple problem”
assuming $a^2+b^2$ , $a^2+c^2$ , $b^2+c^2$ , and $a^2+b^2+c^2$ are perfect squares, then it's impossible. $\sqrt{(a^2+b^2)(a^2+c^2)(b^2+c^2)(a^2+b^2+c^2)}$ must be an integer, therefor $(a^2+b^2)(a^2+c^2)(b^2+c^2)(a^2+b^2+c^2)$ must be a square, which allows us to assume that all terms are squares, and that makes the first 3 terms Pythagorean triples. one method of generating Pythagorean triples is for $A^2+B^2=C^2$ $A=2mn$ $B=m^2-n^2$ $C=m^2+n^2$ $n$ and $m$ must be integers to insure that $A$ , $B$ , and $C$ are integers Note: every Pythagorean triple must include $A$ , $B$ , and $C$ if $a=A$ and $b=B$ then, because $a^2+c^2$ must equal $A^2+B^2$ , $c$ must be $B$ and because $b^2+c^2$ mus
Cites D.R. · 2024-02 · Mathematics Stack Exchange · on “Algebraic points on circles/solutions to $x^2+y^2=a$ over $\overline{\mathbb Q}$: is there a point with odd degree field extension over $\mathbb Q$?”
It is well known how to parameterize rational solutions to $x^2+y^2=a$ for $a\in \mathbb N^+$ : first we know which $a$ admit integer solutions [LINK] ; our characterization is strong enough to tell us that theses are exactly the $a$ [LINK] admitting rational solutions; then if we have one rational solution [LINK] we can get all of them by drawing rational-slope lines through that one point [this video] . I am interested in knowing if people understand well the algebraic solutions to $x^2+y^2=a$ , i.e. the solutions $x,y\in \overline{\mathbb Q}$ to $x^2+y^2=a$ . In particular, I am interested in knowing if there exists an algebraic solution $(x_0,y_0)\in \overline{\mathbb Q} \times \overline{\mat