Self Drawing Equation

The project started with two goals: have fun, and create a 2D equation that could draw its own written representation on a Cartesian plane.

The motivating thought was delightfully impractical: if an equation described the whole universe, then a written copy of it would exist inside its own solution. Could a much smaller equation at least draw itself?

Drawing an alphabet

I first defined equations for every symbol required by the final expression:

0123456789+*-/^=()xyteansigfwhrdlo.b,

Each symbol became a small implicit curve. A selector function could then enable the appropriate curve for every encoded character.

Encoding the equation

The expression had to contain both the drawing machinery and its own encoded text. I associated each character with a digit and concatenated those digits into a very large base-38 integer.

A decoder inside the equation extracts the character at each horizontal position. The corresponding curve is enabled and shifted into place, turning one large number back into a line of mathematical text.

Plotting it

A conventional plotter could not handle the resulting expression well. I wrote a Matlab plotter, which was extremely slow but eventually produced part of the equation.

I never completed the full render. Discovering Tupper’s self-referential formula also made my improvised approach feel somewhat less novel, although building it remained worthwhile.

The original source is preserved in this Google Drive folder.

Other equation experiments

I also experimented with implicit equations that create fractal-like structures:

sin(log(((4*x+sqrt(16*x^2-12*(x^2+y^2)))/4)*pi)/log(1.25))
* sin(log(((4*x-sqrt(16*x^2-12*(x^2+y^2)))/4)*pi)/log(1.25)) = 0

One final equation was made as a birthday present. It may not be mathematically elegant, but it was certainly original.