# The Equation That Produces a Graph of Itself

## Tupper’s Self-Referential Formula describes itself, along with pretty much everything else.

WIkipedia

Everything in physics is described by an equation. Equations can describe the shape of lines, curves, surfaces, and just about any object you can think of. In fact, you’d be hard-pressed to think of anything not described by some equation or another.

So this leads us to an interesting question: Is there a mathematical equation that describes itself? The answer, it turns out, is yes. It’s called Tupper’s Self-Referential Formula, and it looks like this:

Tupper's self-referential formula is a formula that visually represents itself when graphed at a specific location in the (x, y) plane.

— Fermat's Library (@fermatslibrary)

The top part of that image is the equation itself. The bottom part is a graph of the equation, which, as you can see, is exactly the same.

To be fair, this equation cheats a little bit. You may notice that the y-axis of that graph begins at ‘k’. What’s k? In this case, it’s a 543-digit number, so this graph is actually pretty high up the y-axis.

So what’s going on? Well, Tupper’s Self-Referential Formula doesn’t only describe itself. It actually describes everything. More specifically, the graph created by the formula features every possible 106 by 17 pixel grid, arranged one after another along the y-axis. All you have to do in order to find a specific grid is travel far enough up the axis.

This Numberphile video has more background and context on Tupper’s Self-Referential Formula: