Tupper's self-referential formula

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 Tupper's 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.^[2]

Although the formula is called "self-referential", Tupper did not name it as such.^[3]

Formula

The formula is an inequality defined as:

${\frac {1}{2}}<\left\lfloor \mathrm {mod} \left(\left\lfloor {\frac {y}{17}}\right\rfloor 2^{-17\lfloor x\rfloor -\mathrm {mod} \left(\lfloor y\rfloor ,17\right)},2\right)\right\rfloor$ or, as plaintext,

1/2 < floor(mod(floor(y/17)*2^(-17*floor(x)-mod(floor(y),17)),2))

where ⌊ ⌋ denotes the floor function, and mod is the modulo operation.

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

960939379918958884971672962127852754715004339660129306651505519271702802395266424689642842174350718121267153782770623355993237280874144307891325963941337723487857735749823926629715517173716995165232890538221612403238855866184013235585136048828693337902491454229288667081096184496091705183454067827731551705405381627380967602565625016981482083418783163849115590225610003652351370343874461848378737238198224849863465033159410054974700593138339226497249461751545728366702369745461014655997933798537483143786841806593422227898388722980000748404719.

If one graphs the set of points $(x,y)$ in $0\leq x<106$ and $k\leq y<k+17$ satisfying the inequality given above, the resulting graph looks like this (the axes in this plot have been reversed, otherwise the picture would be upside-down and mirrored):

The formula is a general-purpose method of decoding a bitmap stored in the constant k, and it could actually be used to draw any other image. When applied to the unbounded positive range 0 ≤ 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 itself, but this is not remarkable, 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.^[4]^[5]^[6]

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 binary number that forms the plot in base 10. The following plot demonstrates the addition of different values of k. In the fourth subplot the k value of "AFGP" and "Aesthetic Function Graph" are added to get the resultant graph, where both the 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.^[7]

References

Notes

^ * Tupper, Jeff. "Reliable Two-Dimensional Graphing Methods for Mathematical Formulae with Two Free Variables" Archived 2019-07-13 at the Wayback Machine
^ "Pedagoguery Software: GrafEq". www.peda.com. Archived from the original on 2021-02-24. Retrieved 2007-09-09.
^ Narayanan, Arvind. "Tupper's Self-Referential Formula Debunked". Archived from the original on 24 April 2015. Retrieved 20 February 2015.
^ "Archived copy". Archived from the original on 2015-02-20. Retrieved 2012-01-23.{{cite web}}: CS1 maint: archived copy as title (link)
^ "Archived copy". Archived from the original on 2015-06-20. Retrieved 2012-01-23.{{cite web}}: CS1 maint: archived copy as title (link)
^ "Archived copy". Archived from the original on 2015-06-20. Retrieved 2012-01-23.{{cite web}}: CS1 maint: archived copy as title (link)
^ Tupper's-Function, Aesthetic Function Graphposting, 2019-06-13, retrieved 2019-07-07

Sources

Weisstein, Eric W. "Tupper's Self-Referential Formula." From MathWorld—A Wolfram Web Resource. Archived 2021-02-05 at the Wayback Machine
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. Archived 2016-12-21 at the Wayback Machine
"Self-Answering Problems." Math. Horizons 13, No. 4, 19, April 2006
Wagon, S. Problem 14 in stanwagon.com Archived 2007-02-02 at the Wayback Machine

External links

Official website
Extensions of Tupper's original self-referential formula
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
A video explaining the formula

[1] * Tupper, Jeff. "Reliable Two-Dimensional Graphing Methods for Mathematical Formulae with Two Free Variables" Archived 2019-07-13 at the Wayback Machine

[2] "Pedagoguery Software: GrafEq". www.peda.com. Archived from the original on 2021-02-24. Retrieved 2007-09-09.

[3] Narayanan, Arvind. "Tupper's Self-Referential Formula Debunked". Archived from the original on 24 April 2015. Retrieved 20 February 2015.

[4] "Archived copy". Archived from the original on 2015-02-20. Retrieved 2012-01-23.{{cite web}}: CS1 maint: archived copy as title (link)

[5] "Archived copy". Archived from the original on 2015-06-20. Retrieved 2012-01-23.{{cite web}}: CS1 maint: archived copy as title (link)

[6] "Archived copy". Archived from the original on 2015-06-20. Retrieved 2012-01-23.{{cite web}}: CS1 maint: archived copy as title (link)

[7] Tupper's-Function, Aesthetic Function Graphposting, 2019-06-13, retrieved 2019-07-07

[1]

[2]

[3]

[4]

[5]

[6]

[7]