This is a simple harmonograph simulator to generate random-ish harmonographs. It asks for the number of pendulums, and exits if the number is 0. It also asks for the frequency spread, which means roughly, how far from integer may the …
Spectral Harmonographs
This Python + Pygame program draws the trace of 4 decaying sine waves, 2 per axis, with rainbow colours. It generates a sequence of random harmonographs.
A harmonograph is a mechanical device typically seen in science museums, that has two …
Black & White Harmonographs
Black & White Harmonographs. Because colour gets boring after a while…
Harmonographs are those devices often seen in science museums, that consist of a pen connected to a number of pendulums, suspended over a sheet of paper on a board, …
Lorenz Attractor
These 3D pictures (except the last) were generated by a Python + Vpython program, translated from Paul Bourke’s C program (which has a bunch more pretty pictures). See below for code.
The Lorenz system is a system of ordinary differential …
Mandelbrot Gallery
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born, French and American mathematician, noted for developing a “theory of roughness” and “self-similarity” in nature and the field of fractal geometry to help prove it, which …
Fractal Python Programs
These fractals were generated by Python programs from the Active State website. They often make use of recursion.
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel …
Buddhabrots – Ghosts of the Mandelbrot Set
The Buddhabrot is an interesting variation on the well-known Mandelbrot Set, invented by Melinda Green in 1993. It’s a density plot of the orbits of points outside the Mandelbrot Set. The Mandelbrot Set is defined as the set of …
Orbits of The Starship Mandelbrot, in 3D
Orbits of The Starship Mandelbrot, in 3D.
In fractal mathematics, an orbit is the sequence of points in the complex plane traced out by the iteration of the function that generates the fractal. In the case of the Mandelbrot Set, …
Online 3D Harmonograph
Online 3D Harmonograph
This is an online 3d harmonograph version of those harmonograph devices often seen in science museums – except that instead of drawing pretty harmonic patterns on paper, here we plot them in 3D-space! Glowscript code and my …
Fractals in Nature
The main requirement for a ‘natural fractal’ is that it should look self-similar over a range of scales. That is, if you look closer or zoom in, there are shapes similar to what you saw to start with. A tree …
Astronomy with a Fractalscope
Forget Hubble – check out these amazing breath-taking pictures from the Fractal Space Microscope!
I made these pictures with JWildfire – a free and user-friendly image-processing software, mostly known for its sophisticated flame-fractal-generator.
It is Java-based, open-source and runs on …
Abstract Fractal Flames
Abstract expressionism is a post–World War II art movement in American painting, developed in New York in the 1940s. It was the first specifically American movement to achieve international influence and put New York City at the center of the …
Mandrian’s Mandelbrot Music
Mandrian is a Python + Pygame program to render the Mandelbrot Set by sub-dividing square areas into sub-squares. If the square’s corners all have the same iteration count from the escape function we assume there’s no internal detail to render. …
Alien Life Forms
I wish that I could claim these pictures were the result of artistic inspiration after years of starving struggle and demon-haunted torment, but I am an honest man. I clicked a couple of buttons on my computer – occasionally often, …
Spectral Harmonographs
Not fractals – but still, pretty math pictures… These were created with the following Python program (MIT license; download from GitHub):
You may need to install python and/or pygame, e.g. on Ubuntu/Debian style linuxes:
sudo apt-get install pygame
sudo apt-get …
Beyer’s Mandelbrot Zoom
Mandelbrot Zoom sequence up to 1:60,000,000,000, created by Wolfgang Beyer with the program Ultra Fractal 3.
- Mandelbrot set. Initial image of a zoom sequence: Mandelbrot set with continuously colored environment.
- Coordinates of the center: Re(c) = -.7, Im(c) =
Julia Sets
Recall that the Mandelbrot set arises from iterating the complex function z2 + c, where c is a set of points in the complex plane. Those values of c for which z, initially zero, remains bounded are in the …
The Burning Ship
The Burning Ship fractal process is similar to the Mandelbrot Set process, but now the real and imaginary components of z are replaced by their absolute values (i.e. replace – with +):
zn+1 = (|Re(zn)| + i …
3D Fractals
The Mandelbulb is a three-dimensional analogue of the Mandelbrot set, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009. (wikipedia)
A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the …