guidodipietro / python_art

Some math drawings using Python

Some math drawings using Python

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About

Some math drawings done in Python, using the OpenCV library, or Matplotlib for the Amazing Graph series.
Inside each folder you'll find all the images I have produced, as well as the source code to generate them yourself.
Have fun tweaking the constant values!

Only made for fun.

_{The Trapped Knight was coded by a less-smart version of myself, therefore the code is pretty bad. I may or may not re-do it.}

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Sections

Drawings

Plots

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Plots

All these images were generated by plotting a scatter plot of an integer sequence with a pretty graph. These are all the sequences featured on the main videos of the Amazing Graphs series by Neil Sloane on Numberphile. Make sure to check those videos, because they are amazing as well.

Fly Straight, Dammit!

Defined as:
a(0) = a(1) = 1; for n > 1, a(n) = a(n-1) + n + 1 if a(n-1) and n are coprime, otherwise a(n) = a(n-1)/gcd(a(n-1),n) (OEIS).

Seems to be a bit chaotic, but after something happens... the graph becomes 4 straight lines (2 are almost overlapped!)

Wisteria

I love this one because of how simple it is to generate, and how pretty it turns out.
Defined as N - prod(N), where N is the index, and prod() is the product of its digits. So simple.

Rémy Sigrist's Sequence

This is my favourite image of all these plots, by far. The picture is astonishingly similar to a mountain landscape; it's just amazing.
Defined as the lexicographically earliest sequence such that, for any distinct i and j, a(i)=a(j) implies (i AND j)=0 (where AND stands for the bitwise AND operator) (OEIS).

Balanced Ternary (Star Wars)

I really like how this triforce-looking thing comes out of nowhere.
Defined as the resulting integer if you change all 2s on the ternary representation of the index number with a -1 (for example, 14 is 112, so you write it as [1,1,-1] and compute it as 1*3² + 1*3 + (-1) = 11, therefore the 15th element in the sequence is an 11 (first index is 0)).

Prime Parallelograms

Another one with a super simple definition, this time even super easy to code and compute, and with a really neat graph.
The definition is the shortest one of all these graphs: "n-th prime minus its binary reversal" (OEIS).

The sad thing about this is that the graph looks basically identical if you use all the odd numbers instead of the primes, but nevermind.

Stern's Sequence (Cathedrals)

This one is a bit fuzzy to code, but at least the graph is cool.
Defined as what you get after doing this, if you read all the resulting rows, row by row:

1 1  
1 (1+1) 1  
1 (1+2) 2 (2+1) 1  
1 (1+3) 3 (3+2) 2 (2+3) 3 (3+1) 1
...  

Forest Fire

This one is hard to compute, and the graph you get is a bit too populated and hard to appreciate. Check the one on OEIS for a better picture.

I really like the definition, because it is extremely easy to state, but not necessarily immediate to program efficiently. (In fact, I used a program listed in the OEIS to create this plot because mine was too slow).
Defined as the secuence of positive integers where each is chosen to be as small as possible subject to the condition that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic progression (OEIS).

Hofstadters' Sequence

This one is maybe my least favourite one, because its definition is super messy, hard to follow, and the final image isn't too rewarding.
Defined as this thing: a(n) = a(n-a(n-1)) + a(n-a(n-2)), with a(1)=1 and a(2)=1 initial conditions (OEIS).

For an interpretation of what that means, check the video on Numberphile.

Drawings

Other drawings, using the OpenCV library to "paint on a canvas".

Dragon Curve

Read about this fractal here or watch this Numberphile video.
Evidently, my favourite self-similar fractal, which I use as my profile picture for several websites, including GitHub.

Trapped Knight

Learn more about this drawing on this video featuring Neil Sloane.

Sierpinski Triangle constructed by chaos

Sierpinski Triangle constructed using something similar to this method (changing the initial point from something within the triangle to just anywhere).

Emerges from chaos. It's very interesting.

Fibonacci Spiral

Maybe the simplest of them all, a traditional Fibonacci Spiral.

Mandelbrot Set

Get a minuscule taste of what this mind-baffling structure is all about here.
I didn't do this program all by myself, but I don't remember where I got the code from.