Funny Math
Having fun with Pythagoras
How Pythagorean triples are arranged in space
Some days ago I was playing around with Plotly and JavaScript.
I remembered the high school and the funniest Math competitions. I always had some weird ideas in my mind about problems and exercises. Everything I needed was a pencil and a piece of paper.
This is what I believed. When I discovered Computer Science, I leveled up! 👾👾👾
A question that’s always been on my mind was:
How are Pythagorean triples arranged in space?
In this article you can get the answer and it’s really awesome. Read further!
Let’s take a step back
What are Pythagorean triples?
They are special triads of numbers. If you take each number of a Pythagorean triple and create a triangle with such sides, you get a right triangle. 😎 Alright?
Why are Pythagorean triples so beautiful?
First of all, because there are fews. The smallest one is (3,4,5).
Each number of a triple is coprime with the other two. It means that the greatest common divisor is always one.
I believe the triples are consonant and in music they would sound like a major triad (e.g. C-E-G). 🎶 Three perfect individualities that come together in a mathematical masterpiece.
My high school idea
If we have three numbers in a Pythagorean triple, and three dimensions in space… How are these triples arranged, if we represent them in a Cartesian three axis space?
Of course paper and pencil were no use at all. So the problem remained a beautiful mystery to me for so many years. Like an Egyptian tomb covered with sand.
My Codepen
Out of the blue, some days ago, I decided to plot several Pythagorean triples in a Codepen (I always use it as a sandbox for my evil experiments).
My JavaScript algorithm is really simple. There are just two nested for cycles. Indeed, you only need two numbers to get the third of the triple and check if it’s a Pythagorean one.
After generating three arrays of x,y and z coordinates, I draw them in Plotly.
I also put some rainbow colors to spice up the situation! 🥳
An interesting optimization: when I find a Pythagorean triple, I also add the other 5 permutation of that triple. Indeed, if (a,b,c) is a valid Pythagorean triple, then also (a,c,b), (b,a,c), (b,c,a), (c,a,b), (c,b,a) are Pythagorean triples.
In this way, I don’t need to iterate from 1 to n in both my cycles, but I can stop before in the second cycle.
I even avoid checking the couples in the (a,a) form. They surely aren’t Pythagorean triples.
for (let i = 1; i < max; i++) {
for (let j = 1; j < i-1; j++) {
// k^2 = i^2 + j^2
// check if (i,j,k) is a Pythagorean triple
}
}Let’s test our cycle. For example, at the fourth step of our loop, we check if the following couples form Pythagorean triples:
- (1,2) → Not a triple (1+4=5) ❌
- (1,3) → Not a triple (1+9=10) ❌
- (2,3) → Not a triple (4+9=13) ❌
- (1,4) → Not a triple (1+16=17) ❌
- (2,4) → Not a triple (4+16=20) ❌
- (3,4) → Triple (9+16=25) ✔
Ok, some triples will be repeated many times, but nevermind, the points will be overlapped. Here is the final result in my Codepen.
Hope you enjoyed the article! 💙
