# Does this simple pattern show that all integers are part of a fractal set?

## The simple pattern

Count up through nine, then on ten you wrap to the next line and keep going until you've counted nine more integers - then wrap to the next line. Repeat ad infinitum.

1  2  3  4  5  6  7  8  9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27...


Now we can easily see that the digital roots of these numbers are all the same as their first-row primitives. Any integer's digital root can be located immediately if we just list everything nine at a time.

I'm curious as to whether there is a name for this pattern. When searching for information on digital roots I did not find any mention of it, which is surprising because it's really elegant and interesting, and I think very likely to have been observed before due to its simplicity.

It makes a lot of sense that zero acts as a modifier, shifting up like a musical octave - or should I say 'decave', since it resets on the 10. A reminder of how the 0 multiplies by ten, while the . (decimal) divides by ten.

What I would like to know is - can this self-similarity be considered "fractal"?

For your viewing convenience, here are numbers 1 through 999:

001 002 003 004 005 006 007 008 009
010 011 012 013 014 015 016 017 018
019 020 021 022 023 024 025 026 027
028 029 030 031 032 033 034 035 036
037 038 039 040 041 042 043 044 045
046 047 048 049 050 051 052 053 054
055 056 057 058 059 060 061 062 063
064 065 066 067 068 069 070 071 072
073 074 075 076 077 078 079 080 081
082 083 084 085 086 087 088 089 090
091 092 093 094 095 096 097 098 099
100 101 102 103 104 105 106 107 108
109 110 111 112 113 114 115 116 117
118 119 120 121 122 123 124 125 126
127 128 129 130 131 132 133 134 135
136 137 138 139 140 141 142 143 144
145 146 147 148 149 150 151 152 153
154 155 156 157 158 159 160 161 162
163 164 165 166 167 168 169 170 171
172 173 174 175 176 177 178 179 180
181 182 183 184 185 186 187 188 189
190 191 192 193 194 195 196 197 198
199 200 201 202 203 204 205 206 207
208 209 210 211 212 213 214 215 216
217 218 219 220 221 222 223 224 225
226 227 228 229 230 231 232 233 234
235 236 237 238 239 240 241 242 243
244 245 246 247 248 249 250 251 252
253 254 255 256 257 258 259 260 261
262 263 264 265 266 267 268 269 270
271 272 273 274 275 276 277 278 279
280 281 282 283 284 285 286 287 288
289 290 291 292 293 294 295 296 297
298 299 300 301 302 303 304 305 306
307 308 309 310 311 312 313 314 315
316 317 318 319 320 321 322 323 324
325 326 327 328 329 330 331 332 333
334 335 336 337 338 339 340 341 342
343 344 345 346 347 348 349 350 351
352 353 354 355 356 357 358 359 360
361 362 363 364 365 366 367 368 369
370 371 372 373 374 375 376 377 378
379 380 381 382 383 384 385 386 387
388 389 390 391 392 393 394 395 396
397 398 399 400 401 402 403 404 405
406 407 408 409 410 411 412 413 414
415 416 417 418 419 420 421 422 423
424 425 426 427 428 429 430 431 432
433 434 435 436 437 438 439 440 441
442 443 444 445 446 447 448 449 450
451 452 453 454 455 456 457 458 459
460 461 462 463 464 465 466 467 468
469 470 471 472 473 474 475 476 477
478 479 480 481 482 483 484 485 486
487 488 489 490 491 492 493 494 495
496 497 498 499 500 501 502 503 504
505 506 507 508 509 510 511 512 513
514 515 516 517 518 519 520 521 522
523 524 525 526 527 528 529 530 531
532 533 534 535 536 537 538 539 540
541 542 543 544 545 546 547 548 549
550 551 552 553 554 555 556 557 558
559 560 561 562 563 564 565 566 567
568 569 570 571 572 573 574 575 576
577 578 579 580 581 582 583 584 585
586 587 588 589 590 591 592 593 594
595 596 597 598 599 600 601 602 603
604 605 606 607 608 609 610 611 612
613 614 615 616 617 618 619 620 621
622 623 624 625 626 627 628 629 630
631 632 633 634 635 636 637 638 639
640 641 642 643 644 645 646 647 648
649 650 651 652 653 654 655 656 657
658 659 660 661 662 663 664 665 666
667 668 669 670 671 672 673 674 675
676 677 678 679 680 681 682 683 684
685 686 687 688 689 690 691 692 693
694 695 696 697 698 699 700 701 702
703 704 705 706 707 708 709 710 711
712 713 714 715 716 717 718 719 720
721 722 723 724 725 726 727 728 729
730 731 732 733 734 735 736 737 738
739 740 741 742 743 744 745 746 747
748 749 750 751 752 753 754 755 756
757 758 759 760 761 762 763 764 765
766 767 768 769 770 771 772 773 774
775 776 777 778 779 780 781 782 783
784 785 786 787 788 789 790 791 792
793 794 795 796 797 798 799 800 801
802 803 804 805 806 807 808 809 810
811 812 813 814 815 816 817 818 819
820 821 822 823 824 825 826 827 828
829 830 831 832 833 834 835 836 837
838 839 840 841 842 843 844 845 846
847 848 849 850 851 852 853 854 855
856 857 858 859 860 861 862 863 864
865 866 867 868 869 870 871 872 873
874 875 876 877 878 879 880 881 882
883 884 885 886 887 888 889 890 891
892 893 894 895 896 897 898 899 900
901 902 903 904 905 906 907 908 909
910 911 912 913 914 915 916 917 918
919 920 921 922 923 924 925 926 927
928 929 930 931 932 933 934 935 936
937 938 939 940 941 942 943 944 945
946 947 948 949 950 951 952 953 954
955 956 957 958 959 960 961 962 963
964 965 966 967 968 969 970 971 972
973 974 975 976 977 978 979 980 981
982 983 984 985 986 987 988 989 990
991 992 993 994 995 996 997 998 999


## More interesting observations - symmetry and digit-flipping

In the following image, numbers highlighted in blue represent mirror points where the digits are flipped in the numbers above and below.

Chartreuse highlights mark vertical pairs that share a common middle digit. You can think of these as doubled-up if counting vertically downward: 01234556789, for example.

What's really interesting is the points where the blue symmetry centers and chartreuse symmetry centers connect. Example: ←023 032→ 041 050 059 068 ←077→ 086 095... so five is their common point. And of course it is also their digital root.

The digits repeat in patterns that are self-similar in multiple directions - or multiple dimensions if you think of it that way. So could this set be described as fractal?

$$a_n10^n + a_{n-1}10^{n-1} + \dots + a_0$$
with all $a_i < 10$
Now consider the number modulo $9$:
$$a_n10^n + a_{n-1}10{n-1} + \dots + a_0 \equiv \\ a_n(9 + 1)^n + a_{n-1}(9 + 1)^{n-1} + \dots + a_0 \equiv \\ a_n + a_{n-1} + \dots + a_0$$
That is, the number itself is congruent to the sum of the digits, mod 9. So the remainder from dividing by $9$, which dictates the column in your picture, uniquely determines the digital root.