Tupper's self-referential formula

Tupper's self-referential formula is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Introduction

Jeff Tupper, in his 2001 paper "Reliable Two-Dimensional Graphing Methods for Mathematical Formulae with Two Free Variables", shows a set of methods to graph equations and inequalities with two variables in the cartesian plane.

One of the examples of the paper, refers to the inequality:

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

That inequality, once plotted in the range 0 ≤ x ≤ 106 and k ≤ y ≤ k + 17 for k = 960, 939, 379, 918, 958, 884, 971, 672, 962, 127, 852, 754, 715, 004, 339, 660, 129, 306, 651, 505, 519, 271, 702, 802, 395, 266, 424, 689, 642, 842, 174, 350, 718, 121, 267, 153, 782, 770, 623, 355, 993, 237, 280, 874, 144, 307, 891, 325, 963, 941, 337, 723, 487, 857, 735, 749, 823, 926, 629, 715, 517, 173, 716, 995, 165, 232, 890, 538, 221, 612, 403, 238, 855, 866, 184, 013, 235, 585, 136, 048, 828, 693, 337, 902, 491, 454, 229, 288, 667, 081, 096, 184, 496, 091, 705, 183, 454, 067, 827, 731, 551, 705, 405, 381, 627, 380, 967, 602, 565, 625, 016, 981, 482, 083, 418, 783, 163, 849, 115, 590, 225, 610, 003, 652, 351, 370, 343, 874, 461, 848, 378, 737, 238, 198, 224, 849, 863, 465, 033, 159, 410, 054, 974, 700, 593, 138, 339, 226, 497, 249, 461, 751, 545, 728, 366, 702, 369, 745, 461, 014, 655, 997, 933, 798, 537, 483, 143, 786, 841, 806, 593, 422, 227, 898, 388, 722, 980, 000, 748, 404, 719 produces a drawing that visually mimics the inequality itself, hence it is called self-referential.

Although the inequality is intended to be drawn on the continuum of the cartesian plane, the drawing can be performed iterating over the integer values of both the horizontal and vertical ranges.

Task

Make a drawing of the Tupper's formula, either using text, a matrix or creating an image.

This task requires arbitrary precision integer operations. If your language does not intrinsecally support that, you can use a library.

Epilogue

The value of k is an encoding of the bitmap of the image, therefore any 17-width bitmap can be produced, using its associated encoded value as k.

References

• Wikipedia: Tupper's self-referential formula
• Tupper, Jeff. "Reliable Two-Dimensional Graphing Methods for Mathematical Formulae with Two Free Variables"

Fōrmulæ

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.

Programs in Fōrmulæ are created/edited online in its website.

In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.

Solution

First, let us make the definition of k:

Tupper uses the notation mod(a, b) for the modulo operation, while Fōrmulæ uses the notation a Mod b. If we want to use the Tupper's formula exactly as it is, we can use the trick:

Now we can use a formula that looks exactly as Tupper's:

Visualization as a matrix. The following creates a 17×107 matrix. Values for which inequality is true are shown as an opaque black color, otherwise they are shown as an transparent color.

Visualization as an image. In the following script, a function to draw the Tupper's formula is used to generate the image with different sizes, because the standard 1x1 pixel per unit results to be too small in order to be appreciated:

Julia

import Images: colorview, Gray
import Plots: plot

setprecision(BigFloat, 8000)

""" get logical inverse of Tupper function values for white on black graphic """
function tupper_mat(k)
    tmatrix = trues(17, 106)
    for (i, x) in enumerate(0.0:1:105), (j, y) in enumerate(k:k+16)
        tmatrix[j, 107 - i] = 1/2 >= floor(mod(floor(y / 17) * 2^(-17 * floor(x) - mod(floor(y), 17)), 2))
    end
    return tmatrix 
end

const k = big"""960_939_379_918_958_884_971_672_962_127_852_754_715_004_339_660_129_306_651_505
                519_271_702_802_395_266_424_689_642_842_174_350_718_121_267_153_782_770_623_355
                993_237_280_874_144_307_891_325_963_941_337_723_487_857_735_749_823_926_629_715
                517_173_716_995_165_232_890_538_221_612_403_238_855_866_184_013_235_585_136_048
                828_693_337_902_491_454_229_288_667_081_096_184_496_091_705_183_454_067_827_731
                551_705_405_381_627_380_967_602_565_625_016_981_482_083_418_783_163_849_115_590
                225_610_003_652_351_370_343_874_461_848_378_737_238_198_224_849_863_465_033_159
                410_054_974_700_593_138_339_226_497_249_461_751_545_728_366_702_369_745_461_014
                655_997_933_798_537_483_143_786_841_806_593_422_227_898_388_722_980_000_748_404_719"""

function test_tupper()
    bmap = tupper_mat(k)
    display(plot(colorview(Gray, bmap), ylims = [1,17])) # invert black / white with .!
    savefig("tupper.png")
end

test_tupper()

Output: