Tupper's Self-Referential Formula
Encode a 106×17 bitmap into the giant Tupper integer k, or decode k back into the image.
Tupper's formula plots a 106×17 pixel bitmap whose contents are encoded in a single giant integer k. Geocachers use it to hide coordinates inside a number that can only be read by plotting it. Encode an image, drawing, or piece of text below — or paste a k to reveal the image it represents.
Encode (Image → k)
Decode (k → Image)
About Tupper's Formula
Discovered by Jeff Tupper in 2001, this self-referential inequality plots itself when graphed over a specific tiny region of the plane. The trick is that the formula is a generic bitmap reader: the giant integer k is the picture, encoded in binary.
The bitmap is fixed at 106 pixels wide by 17 pixels tall. For pixel
(x, y), the formula extracts bit (x × 17 + y)
of k ÷ 17 and colors the pixel black if the bit is 1.
Tupper numbers y from the bottom, so this tool flips the canvas
coordinates accordingly.
In geocaching, puzzle setters encode coordinates as a bitmap, convert to k, and present the giant number as the puzzle. Solvers paste it into a Tupper plotter to reveal the coordinates.