Tupper's self-referential formula

Template:Short description 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:

$$\frac{1}{2}<\lfloor \mathrm{m}\mathrm{o}\mathrm{d}(\lfloor \frac{y}{17}\rfloor 2^{-17\lfloor x\rfloor -\mathrm{m}\mathrm{o}\mathrm{d}(\lfloor y\rfloor ,17)},2)\rfloor$$ where $\lfloor \dots \rfloor$ denotes the floor function, and mod is the modulo operation.

Plots

Let $k$ equal the following 543-digit integer:

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Derivation of k

Graphing the set of points $(x,y)$ in $0\le x<106$ and $k\le y<k+17$ which satisfy the formula, results in the following plot:Template:NoteTag

The formula is a general-purpose method of decoding a bitmap stored in the constant $k$, and it could be used to draw any other image. When applied to the unbounded positive range $0\le y$, 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.

The constant $k$ is a simple monochrome bitmap image of the formula treated as a binary number and multiplied by 17. If $k$ is divided by 17, the least significant bit encodes the upper-right corner $(k,0)$; 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. The value of $k$ is the number whose binary digits form the plot. The following plot demonstrates the addition of different $k$ values. 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. The information regarding the shape of the plot is stored within $k$.

Addition of different values of k

See also

• Template:Annotated link
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References

Sources

• Weisstein, Eric W. "Tupper's Self-Referential Formula." From MathWorld—A Wolfram Web Resource. Template:Webarchive
• 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. 289, 2006. Template:Webarchive
• "Self-Answering Problems." Math. Horizons 13, No. 4, 19, April 2006
• Wagon, S. Problem 14 in stanwagon.com Template:Webarchive

External links

• Template:Official
• 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