This is Tupper’s Self-Referential Formula:

\displaystyle \frac12 < \left\lfloor \mathrm{mod} \left( \left\lfloor{\frac{y}{17}}\right\rfloor 2^{-17\lfloor x \rfloor - \mathrm{mod}(\lfloor y \rfloor, 17)}, 2 \right) \right\rfloor

what makes it interesting is that the set of points [x, y], for which the inequality holds, in the rectangle 0 \leq x \leq 106, n \leq y \leq n+16, you will get this:

Turns out that the graph of the formula resembles the formula itself, which at first sounds pretty incredible. Oh, wait: I didn’t specify the value of n. It should be equal to:

960939379918958884971672962127852754715004339660129306
651505519271702802395266424689642842174350718121267153
782770623355993237280874144307891325963941337723487857
735749823926629715517173716995165232890538221612403238
855866184013235585136048828693337902491454229288667081
096184496091705183454067827731551705405381627380967602
565625016981482083418783163849115590225610003652351370
343874461848378737238198224849863465033159410054974700
593138339226497249461751545728366702369745461014655997
933798537483143786841806593422227898388722980000748404719.

Read more about it in Wikipedia.

I spent a couple of evenings trying to figure out how does it work, and intended to write a long post about it, but I see that it has already been done here much better than I could, so I’ll only give you the main idea: the graph of the formula contains all possible combinations of pixels of size 106 \times 17 and what you need to do is to locate the right fragment of the graph by choosing the value of n. Fortunately, the formula is constructed in such a way that it can be done rather easily. For instance, if you set n to be equal to:

770137614616740349659457383109219250729889213023196245
754997397474539522884572693040838328048207848222847384
255800207585680609541957604841348860558024608884092791
814186507612666983299083984568308963916568266347954249
445046515417964282372279838338100838479512127675315663
107944860681436794491747199511098181076928

you will get a fragment of the graph saying “hello world!”. (Oh, and it will be upside down, as pointed out in the blog post I already mentioned)

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3 Responses to

  1. christopherolah says:

    Very clever.

  2. christopherolah says:

    Someone had some fun and made a really self referential formula… http://jtra.cz/stuff/essays/math-self-reference/index.html

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