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

History

The formula was defined by Jeff Tupper and appears as an example in his 2001 SIGGRAPH paper on reliable two-dimensional computer graphing algorithms. This paper discusses methods related to the GrafEq formula-graphing program developed by Tupper.

Formula

The formula is an inequality defined as:

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

Plots

equal the following 543-digit integer:

1. 960939379918958884971672962127852754715004339660129306651505519271702802395266424689642842174350718121267153782770623355993237280874144307891325963941337723487857735749823926629715517173716995165232890538221612403238855866184013235585136048828693337902491454229288667081096184496091705183454067827731551705405381627380967602565625016981482083418783163849115590225610003652351370343874461848378737238198224849863465033159410054974700593138339226497249461751545728366702369745461014655997933798537483143786841806593422227898388722980000748404719

Graphing which satisfy the formula, results in the following plot:

, 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 is divided by 17, the least significant bit; 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.

.

See also

References

Footnotes

Notes

Sources

• Weisstein, Eric W. "Tupper's Self-Referential Formula." From MathWorld—A Wolfram Web Resource.

• Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Natick, MA: A. K. Peters, p.

• "Self-Answering Problems." Math. Horizons 13, No. 4, 19, April 2006

• Wagon, S. Problem 14 in stanwagon.com

External links

• Extensions of Tupper's original self-referential formula

• Tupper's self-referential formula in Rosetta Code, implementation in several programming languages

• TupperPlot, an implementation in JavaScript

• Tupper self referential formula, an implementation in Python

• The Library of Babel function, a detailed explanation of the workings of Tupper's self-referential formula

• Tupper's Formula Tools, an implementation in JavaScript

• Trávník's formula that draws itself close to the origin