Tupper's self-referential formula

The formula was defined by Jeff Tupper and appears as an example in his 2001 SIGGRAPH paper on reliable two-dimensional computer graphing algorithms.

[1] This paper discusses methods related to the GrafEq formula-graphing program developed by Tupper.

denotes the floor function, and mod is the modulo operation.

equal the following 543-digit integer: Graphing the set of points

which satisfy the formula, results in the following plot:[note 1]

The formula is a general-purpose method of decoding a bitmap stored in the constant

, the formula tiles a vertical swath of the plane with a pattern that contains all possible 17-pixel-tall bitmaps.

One horizontal slice of that infinite bitmap depicts the drawing formula since other slices depict all other possible formulae that might fit in a 17-pixel-tall bitmap.

Tupper has created extended versions of his original formula that rule out all but one slice.

is a simple monochrome bitmap image of the formula treated as a binary number and multiplied by 17.

is divided by 17, the least significant bit encodes the upper-right corner

; the 17 least significant bits encode the rightmost column of pixels; the next 17 least significant bits encode the 2nd-rightmost column, and so on.

It fundamentally describes a way to plot points on a two-dimensional surface.

is the number whose binary digits form the plot.

In the fourth subplot, the k-value of "AFGP" and "Aesthetic Function Graph" is added to get the resultant graph, where both texts can be seen with some distortion due to the effects of binary addition.