Davis-Knuth dragon curve.
How to Dance with a Tree: Visualizing Fractals With Dance by Aarish Bhatia &
“ HERE’S A FUN project that my friend Upasana and I put together some weekends ago. It’s a visual exploration of fractals through dance, a piece of generative art that’s part performance and part mathematical exploration.
The two ingredients that went into creating this were the Microsoft Kinect sensor, which lets your computer track how your body moves, and Processing, a programming language that lets you create interactive visuals with code. Put the two together, and you can use your body to control virtual shapes and objects.
The idea for this project came about while I was walking home from work late October, idly watching the recently bare tree branches swaying in the wind. And for some reason that made me wonder, what would it be like to be a tree for an evening? Imagine lifting your arms, and a tree waves its branches.
And then I remembered reading about fractals in Daniel Shiffman’s book Nature of Code. Fractals are those wonderfully intricate structures that look the same as you keep zooming in to them. Benoit B. Mandelbrot was one of the earliest explorers of the fractal world. He coined the word fractal to mean a kind of geometric shape whose parts resemble “a reduced-size copy of the whole.” (Some fractal humor: What does the B in Benoit B. Mandelbrot stand for? Benoit B. Mandelbrot.)
At the heart of being a fractal is self-similarity, the idea that each piece appears similar to the whole. Think of how a coastline on a map appears similarly wrinkly across different levels of zoom. The same could be said of the jagged terrain of a mountain.”
First, how do you make a Steiner chain?
It is easy using inversion geometry. Just decide on the number of circles tangent to the inner circle (n). Then the ratio of the radii of the inner and outer circle will be .
The radii of the circles in the ring will be
and their centres are located at distance
from the origin. This produces a staid concentric arrangement. Now invert with relation to an arbitrary circle: all the circles are mapped to other circles, their tangencies preserved. Voila! A suitably eccentric Steiner chain to play with.
Since the original concentric chain obviously can be rotated continuously without losing touch with the inner and outer circle, this also generates a continuous family of circles after the inversion. This is why Steiner’s porism is true: if you can make the initial chain, you get an infinite number of other chains with the same number of circles.
Iterated function systems with circle maps
The fractal works by putting copies of the whole set of circles in the chain into each circle, recursively. I remap the circles so that the outer circle becomes the unit circle, and then it is easy to see that for a given small circle with (complex) centre z and radius r the map
maps the interior of the unit circle to it. Use the ease of rotating the original concentric ring to produce an animation, and we can reconstruct the fractal.
See more at: [https://vimeo.com/220080104] by
This is one of the first images I made with Processing.