## Thursday, August 17, 2017

### The Formula That Plots (Almost) Everything

Hold onto your logic hats! In this article we're going to explore one of the most amazing formulas in maths: Tupper's self-referential formula.

The protagonist of our story is the following inequality:

1/2<\floor{mod(\floor{\frac{y}{17}}2^(-17\floor{x}-mod(\floor{y},17)),2))

The plot works by either coloring a square or not coloring it: a square with coordinates (x, y) is colored if the inequality is true for x and y. If not the square is left blank.

If you plot the plot for many values of and , the outcome is the following:

I'll let that sink in a moment. No, your eyes are not deceiving you, the formula plots a bitmap picture of itself! Hence the name Tupper's self-referential formula (though Tupper never called this function that himself in his 2001 paper).

There is one missing detail, however. I haven’t told you the value of the number N on the y-axis.