Tupper's formula

Tupper's formula is an inequality established by the Canadian Jeff Tupper , which generates a pattern of 17 × 106 pixels from a given number (a so-called parameter ) . Depending on the parameter, all possible pixel patterns of size 17 × 106 can be generated. In particular, the formula itself can be displayed as a pixel pattern using the appropriate parameters. The sometimes used term "Tuppers self-referential formula" is wrong from a technical point of view.

Publication and statement

Jeff Tupper of the University of Toronto published the formula in 2001 at SIGGRAPH , a conference for computer graphics .

The formula is in the original . ${\ displaystyle {\ frac {1} {2}} <\ left \ lfloor {\ bmod {\ left ({\ left \ lfloor {\ frac {y} {17}} \ right \ rfloor 2 ^ {- 17 \ lfloor x \ rfloor - {\ bmod {(}} {\ lfloor y \ rfloor}, {17})}}, 2 \ right)}} \ right \ rfloor}$

According to Tupper, this inequality should be evaluated for all value pairs within a pixel field of size 17 × 106. The position of this pixel field depends on a parameter that defines the starting point of the values in the coordinate system. The coordinates of the pixel field are limited by and . Due to the design, it should be a natural multiple of 17. ${\ displaystyle (x, y)}$ ${\ displaystyle k}$ ${\ displaystyle y}$ ${\ displaystyle 0 \ leq x <106}$ ${\ displaystyle k \ leq y <k + 17}$ ${\ displaystyle k}$

The rounding function (Gaussian bracket) reduces the real variables and to their integer part. Since the whole right-hand side of the inequality is in Gaussian brackets , either the value or that of any whole number is reached. Thus the inequality is always fulfilled when the right-hand side assumes a positive value. If you color all pixels of a 17 × 106 pixel field that meet the inequality for a fixed parameter , you get a pixel pattern. If one interprets the formula as a decoder , one can say that the parameter contains the complete image information of the pixel field. Since there are pixels in the pixel field, there are theoretically possible pixel patterns. These are actually all in the coordinate system, which can be found using a suitable one . ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle 0}$ ${\ displaystyle k}$ ${\ displaystyle (x, y)}$ ${\ displaystyle k}$ ${\ displaystyle 17 \ cdot 106 = 1802}$ ${\ displaystyle 2 ^ {1802} \ approx 2 {,} 858 \ cdot 10 ^ {542}}$ ${\ displaystyle k}$

The localization of the individual pixel fields in the coordinate system is not accidental. The lowest field is on the same level as . In this section, the resulting pixel field is empty. On the other hand , if you set , there is a single pixel in the lower left or upper right corner, depending on the alignment of the axes. Each additional field can be found by writing down the pixels of the field from bottom left to top or from top right to bottom in binary notation ( for condition not met and for condition met ), converting the number into the decimal system and multiplying by 17 . ${\ displaystyle 0 \ leq y <17}$ ${\ displaystyle k = 0}$ ${\ displaystyle k = 17}$ ${\ displaystyle 0}$ ${\ displaystyle 1}$

presentation

Accordingly, one possible way to represent a value for using the formula is as follows: ${\ displaystyle k}$

Choose a natural multiple of 17 as . ${\ displaystyle k}$
Create a pixel field with a width of 106 pixels and a height of 17 pixels.
Choose the axes so that the pixel at the bottom left receives the coordinate and the pixel at the top right receives the coordinate . ${\ displaystyle (0, k)}$ ${\ displaystyle (105, k + 16)}$
Check for each pixel in the field whether the inequality applies: Take its coordinate (the axes result in and ) and check the inequality. If it is fulfilled, color the pixel, otherwise not. ${\ displaystyle (x, y)}$ ${\ displaystyle 0 \ leq x \ leq 105}$ ${\ displaystyle k \ leq y \ leq k + 16}$

Examples and supposed self-referentiality

There is a value for that results in a pixel pattern, the shape of which can be read well as the mathematical symbols of the formula itself (see graphic). Since Tupper himself stated this value and presented it with the formula, it is best known and has led to the often used designation "Tuppers self-referential formula". However, the formula would not be called self-referential even if this value and the formula were viewed as a fixed unit. The condition for self-referentiality is that something refers to itself. The formula (also in connection with the corresponding value, (see below)) does not do this per se. The pixel image created in this case is legible, but from a mathematical point of view (i.e. the one that applies to the formula) has no content beyond the pixel pattern and does not flow into the creation process of the pattern or represent the creation process. Both would be necessary to achieve the To be able to describe the formula as self-referential. ${\ displaystyle k}$ ${\ displaystyle k}$ ${\ displaystyle k}$

Graph of the Tupperware formula at the usual value for with negatively aligned axes.

{\ displaystyle k}

To get to the original formula in the coordinate system, the value for von Tupper was with ${\ displaystyle k}$

"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 “

specified. With the usual axis alignment (y-axis upwards and x-axis to the right), however, this generates the point-mirrored image of itself. The non-mirrored variant can be found at

"4 858 450 636 189 713 423 582 095 962 494 202 044 581 400 587 983 244 549 483 093 085 061 934 704 708 809 928 450 644 769 865 524 364 849 997 247 024 915 119 110 411 605 739 177 407 856 919 754 326 571 855 442 057 210 445 735 883 681 829 823 754 139 634 338 225 199 452 191 651 284 348 332 905 131 193 199 953 502 413 758 765 239 264 874 613 394 906 870 130 562 295 813 219 481 113 685 339 535 565 290 850 023 875 092 856 892 694 555 974 281 546 386 510 730 049 106 723 058 933 586 052 544 096 664 351 265 349 363 643 957 125 565 695 936 815 184 334 857 605 266 940 161 251 266 951 421 550 539 554 519 153 785 457 525 756 590 740 540 157 929 001 765 967 965 480 064 427 829 131 488 548 259 914 721 248 506 352 686 630 476 300 “

for . ${\ displaystyle k}$

For example, a stylized logo from Wikipedia can be found at

"255 953 791 972 924 990 661 370 909 067 266 714 667 432 752 223 029 927 796 475 868 890 692 855 670 245 876 073 408"

for . ${\ displaystyle k}$

Tupper's formula

contents

Publication and statement

presentation

Examples and supposed self-referentiality

Web links

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