Water retention on mathematical surfaces

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Water being poured over a Lego surface.
Illustration of water retention on a surface.

Water retention on mathematical surfaces refers to the water caught in ponds on a surface of cells of various heights on a regular array such as a square lattice, where water is rained down on every cell in the system. The boundaries of the system are open and allow water to flow out. Water will be trapped in ponds, and eventually all ponds will fill to their maximum height, with any additional water flowing over spillways and out the boundaries of the system. The problem is to find the amount of water trapped or retained for a given surface. This has been studied extensively for two mathematical surfaces: magic squares and random surfaces.

Magic squares[edit]

Retention on a 5x5 magic square.
A 5x5 magic square with the maximal retention.

Magic squares have been studied for over 2000 years. In 2007, the idea of studying the water retention on a magic square was proposed.[1] In 2010, Al Zimmermann's programing contest[2] produced the presently known maximum retention values for magic squares order 4 to 28.[3] Computing tools used to investigate and illustrate this problem are found here.[4][5][6]

There are 4,211,744 different retention patterns for the 7x7 square. A combination of a lake and ponds is best for attaining maximum retention. No known patterns for maximum retention have an island in a pond or lake.[1]

Maximum-retention magic squares for orders 7-9 are shown below:[3]

The figures below show the 10x10 magic square. Is it possible to look at the patterns above and predict what the pattern for maximum retention for the 10x10 square will be? No theory has been developed that can predict the correct combination of lake and ponds for all orders, however some principles do apply. The first color-coded figure shows a design principle of how the largest available numbers are placed around the lake and ponds. The second and third figures show promising patterns that were tried but did not achieve maximum retention.[1]

Several orders have more than one pattern for maximum retention. The figure below shows the two patterns for the 11x11 magic square with the apparent maximum retention of 3,492 units:[3]

The most-perfect magic squares require all (n-1)^2 or in this case all 121 2x2 planar subsets to have the same sum. ( a few examples flagged with yellow background, red font). Areas completely surrounded by larger numbers are shown with a blue background.[7][8]

Most-perfect magic square.jpg

Before 2010 if you wanted an example of a magic square larger than 5x5 you had to follow clever construction rules that provided very isolated examples. The 13x13 pandiagonal magic square below is such an example. Harry White's CompleteSquare Utility [4] allows anyone to use the magic square as a potter would use a lump of clay. The second image shows a 14x14 magic square that was molded to form ponds that write the 1514 - 2014 dates. The animation notes how the surface was sculptured to fill all ponds to capacity before the water flows off the square. This square honors the 500th anniversary of Durer's famous magic square in Melencolia I.

13 x 13 Pandiagonal Magic Sqauare.png
Magic square water retention.gif

The figure below is a 15x15 bordered magic square with zero water retention.[4] This figure also provides an example of a square and its complement that have the same pattern of retention. There are 137 order 4 and 3,254,798 order 5 magic squares that do not retain water.[1]

16 x 16 associative magic square retaining 17840 units. The lake in the first image looks a little uglier than common. Jarek Wroblewski notes that good patterns for maximum retention will have equal or near equal number of retaining cells on each peripheral edge ( in this case 7 cells on each edge) [2] The second image is doctored, shading in the top and bottom 37 values.

Magic square pattern.jpeg
Associative magic square 2013.jpeg

The figure below is a 17x17 Luo-Shu format magic square.[9] The Luo-Shu format construction method seems to produce a maximum number of ponds. The drainage path for the cell in green is long eventually spilling off the square at the yellow spillway cell.

The figure to the right shows what information can be derived from looking at the actual water content for each cell. Only the 144 values are highlighted to keep the square from looking too busy. Focusing on the green cell with a base value 7, the highest obstruction on the path out is its neighbor cell with the value of 151 (151-7=144 units retained). Water rained into this cell exits the square at the yellow 10 cell.

The computer age now allows for the exploration of the physical properties of magic squares of any order. The figure below shows the largest magic square studied in the contest. For L > 20 the number of variables/ equations increases to the point where it makes the pattern for maximum retention predictable.

L = 28: 219822 units of retention
1 5 259 659 257 713 712 282 256 283 657 255 656 284 726 725 254 285 654 253 286 55 673 674 471 645 7 3
9 640 664 25 717 26 27 716 715 714 28 668 29 744 30 31 730 743 50 681 680 679 51 52 678 206 646 11
265 665 355 722 496 618 71 484 95 121 721 400 774 418 130 176 293 541 749 479 106 175 389 148 230 682 43 644
663 14 724 356 313 513 75 189 198 449 213 775 87 478 539 139 326 60 451 750 461 566 141 442 638 477 677 276
266 720 164 572 354 226 491 171 512 117 776 247 244 503 435 85 629 406 144 634 751 592 462 125 134 514 44 672
711 15 565 116 100 357 579 112 637 777 108 469 433 546 80 559 525 468 526 227 146 752 368 557 328 212 46 671
710 16 153 174 222 119 353 627 778 64 297 456 544 474 178 473 410 563 515 331 403 387 753 402 569 304 45 670
709 17 70 325 168 509 445 779 166 366 401 83 92 482 129 338 408 492 585 529 369 298 424 754 582 519 676 275
267 719 156 103 455 531 780 391 358 537 76 142 367 309 522 245 320 437 632 386 545 497 224 123 755 161 675 277
264 718 444 600 508 781 196 553 65 352 488 344 624 104 216 551 98 616 370 294 233 101 416 490 109 756 47 652
662 18 723 417 782 310 564 606 420 483 359 518 548 246 475 58 628 385 571 69 149 223 335 235 86 113 733 274
263 669 218 783 127 429 581 77 399 136 88 351 602 538 636 635 371 220 74 570 99 633 543 498 502 173 48 727
661 19 784 407 179 184 195 609 393 495 203 567 360 576 394 384 388 137 625 154 523 229 489 485 219 314 738 279
268 748 597 307 505 615 441 315 583 562 194 542 446 350 372 588 316 443 120 162 89 102 560 317 110 329 737 272
729 20 521 177 232 340 128 411 152 122 334 241 605 383 361 412 578 202 619 73 611 549 589 587 432 568 736 278
262 746 68 580 242 187 558 183 398 601 594 182 373 296 460 349 332 556 205 419 614 323 547 586 207 114 735 273
269 745 458 131 111 78 337 610 532 612 622 382 59 365 554 448 362 613 82 574 172 493 466 126 145 630 734 280
261 747 158 465 598 221 459 214 524 167 374 608 533 409 319 330 595 348 181 428 305 453 584 199 61 765 33 651
660 21 773 536 561 94 345 165 204 381 621 528 447 211 500 135 452 342 363 301 396 527 185 225 764 306 666 281
270 694 517 772 392 431 312 240 375 190 617 151 91 324 333 520 231 215 511 347 540 238 97 763 413 707 49 650
260 693 105 405 771 550 295 380 302 336 311 620 234 133 427 197 516 150 90 607 364 425 762 486 67 530 703 271
53 692 300 163 631 770 376 191 157 552 414 415 555 422 626 590 339 507 79 188 147 761 430 308 436 132 702 54
683 22 397 423 535 379 769 155 421 494 322 454 390 217 510 623 107 200 591 186 760 341 346 593 237 115 24 696
684 23 228 118 377 575 303 768 327 534 487 573 438 472 457 599 464 439 143 759 604 138 160 72 395 124 32 697
480 691 209 378 440 504 140 501 767 81 201 159 404 210 467 577 57 169 758 193 426 470 93 596 639 180 701 499
648 258 695 299 192 208 481 321 318 766 463 96 63 506 84 236 239 757 343 708 450 243 170 434 603 706 62 641
10 649 38 690 39 37 689 688 687 40 732 36 742 741 740 739 731 41 667 35 705 42 34 13 704 66 643 12
2 6 647 287 686 685 288 252 251 658 289 728 250 249 290 248 291 655 292 653 56 698 699 700 476 642 8 4
Jarek Wroblewski March 24, 2010

Random surfaces[edit]

Water retention on a random surface.
Water retention on a random surface of 10 levels.
Five levels

Another system in which the retention question has been studied is a surface of random heights. Here one can map the random surface to site percolation, and each cell is mapped to a site on the underlying graph or lattice that represents the system. Using percolation theory, one can explain many properties of this system. It is an example of the invasion percolation model in which fluid is introduced in the system from any random site.[10][11][12]

In hydrology, one is concerned with runoff and formation of catchments.[13] The boundary between different drainage basin (watersheds in North America) forms a drainage divide with a fractal dimension of about 1.22.[14] [15] [16]

The retention problem can be mapped to standard percolation.[17][18][19] For a system of five equally probable levels, for example, the amount of water stored R5 is just the sum of the water stored in two-level systems R2(p) with varying fractions of levels p in the lowest state:

R5 = R2(1/5) + R2(2/5) + R2(3/5) + R2(4/5)

Typical two-level systems 1,2 with p = 0.2, 0.4, 0.6, 0.8 are shown on the right (blue: wet, green: dry, yellow: spillways bordering wet sites). The net retention of a five-level system is the sum of all these. The top level traps no water because it is far above the percolation threshold for a square lattice, 0.592746.

The retention of a two-level system R2(p) is the amount of water connected to ponds that do not touch the boundary of the system. When p is above the critical percolation threshold p c, there will be a percolating cluster or pond that visits the entire system. The probability that a point belongs to the percolating or "infinite" cluster is written as P in percolation theory, and it is related to R2(p) by R2(p)/L2p − P where L is the size of the square. Thus, the retention of a multilevel system can be related to a well-known quantity in percolation theory.

To measure the retention, one can use a flooding algorithm in which water is introduced from the boundaries and floods through the lowest spillway as the level is raised. The retention is just the difference in the water level that a site was flooded minus the height of the terrain below it.

Besides the systems of discrete levels described above, one can make the terrain variable a continuous variable say from 0 to 1. Likewise, one can make the surface height itself be a continuous function of the spatial variables. In all cases, the basic concept of the mapping to an appropriate percolation system remains.

A curious result is that a square system of n discrete levels can retain more water than a system of n+1 levels, for sufficiently large order L > L*. This behavior can be understood through percolation theory, which can also be used to estimate L* ≈ (p - pc)−ν where ν = 4/3, p = i*/n where i* is the largest value of i such that i/n < pc, and pc = 0.592746 is the site percolation threshold for a square lattice. Numerical simulations give the following values of L*, which are extrapolated to non-integer values. For example, R2 < R3 for L ≤ 51, but R2 > R3 for L ≥ 52:[17]

n n+1 L* Retention at L*
2 3 51.12 790
4 5 198.1 26000
7 8 440.3 246300
9 10 559.1 502000
12 13 1390.6 428850
14 15 1016.3 2607000

As n gets larger, crossing become less and less frequent, and the value of L* where crossing occurs is no longer a monotonic function of n.

The retention when the surface is not entirely random but correlated with a Hurst exponent H is discussed in.[19]

Algorithms[edit]

The following time line shows the application of different algorithms that have expanded the size of the square that can be evaluated for retention

2007 Define all neighbor-avoiding walks from each interior cell to the exterior and then sort all those paths for the least obstruction or cell value. The least obstruction value minus the interior cell value provides the water retention for that interior cell (negative values are set to a retention value of 0). The number of neighbor-avoiding walks to be evaluated grows exponentially with the square size and thus limits this methodology to L < 6.[1]

2009 Flooding algorithm - water is introduced from the boundaries and floods through the lowest spillway as the level is raised. The retention is just the difference in the water level that a site was flooded minus the height of the terrain below it. The flooding algorithm allows for the evaluation of water retention up to L < 10,000.[17] This algorithm is similar to Meyer's flooding algorithm that has been used in analysis of topographical surfaces.

2011 With the realization that an n-level system can be broken down into a collection of two-level systems with varying probabilities, standard percolation algorithms can be used to find the retention as simply the total number of sites at the lower level minus the draining regions (clusters of low-level sites touching the boundary). A novel application of the Hoshen-Kopelman algorithm in which both rows and columns are added one at a time allows L to be very large (up to 109), but computing time considerations limits L to the order of 107.[20]

Paths that drain water off the square, used in the neighbor-avoiding walk algorithm

The panel below from left to right shows: 1) the three unique interior positions for the 5x5 square; 2 & 4) correct paths off the square in grey for the interior corner cell in red; 3) incorrect path in grey as the water cannot travel on the diagonals; 5) this path is correct but there is a short-circuit possible between the grey cells. Neighbor-avoiding walks define the unique or non-redundant paths that drain water off the square.

See also[edit]

References[edit]

  1. ^ a b c d e Craig Knecht, http://www.knechtmagicsquare.paulscomputing.com
  2. ^ a b Al Zimmermann http://www.azspcs.net/Contest/MagicWater/FinalReport
  3. ^ a b c Harvey Heinz, http://www.magic-squares.net/square-update-2.htm#Knecht
  4. ^ a b c Harry White, http://budshaw.ca/Download.html
  5. ^ Walter Trump http://www.trump.de/magic-squares/
  6. ^ Johan Ofverstedt,http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-176018
  7. ^ Harry White, http://budshaw.ca/Most-perfect.html
  8. ^ 2x2 planar subsets,https://oeis.org/A270205
  9. ^ Harvey Heinz,http://www.magic-squares.net/square-update.htm
  10. ^ Chayes, J. T.; L. Chayes; C. M. Newman (1985). "The stochastic geometry of invasion percolation". Communications in Mathematical Physics. 101 (3): 383–407. Bibcode:1985CMaPh.101..383C. doi:10.1007/BF01216096. 
  11. ^ Damron, Michael; Artëm Sapozhnikov; Bálint Vágvölgyi (2009). "Relations between invasion percolation and critical percolation in two dimensions". Annals of Probability. 37 (6): 2297–2331. doi:10.1214/09-AOP462. 
  12. ^ van den Berg, Jacob; Antal Járai; Bálint Vágvölgyi (2007). "The size of a pond in 2D invasion percolation". Electron. Comm. In Probab. 12: 411–420. doi:10.1214/ECP.v12-1327. 
  13. ^ Tetzlaff, D.; McDonnell, J. J.; Uhlenbrook, S.; McGuire, K. J.; Bogaart, P. W.; Naef, F.; Baird, A. J.; Dunn, S. M.; Soulsby, C. (2011). "Conceptualizing catchment processes: simply too complex?". Hydrological Processes. 22 (11): 1099–1085. Bibcode:2008HyPr...22.1727T. doi:10.1002/hyp.7069. 
  14. ^ Fehr, E.; D. Kadau; N. A. M. Araújo; J. S. Andrade Jr; H. J. Herrmann (2011). "Scaling Relations for Watersheds". Physical Review E. 84 (3): 036116. Bibcode:2011PhRvE..84c6116F. arXiv:1106.6200Freely accessible. doi:10.1103/PhysRevE.84.036116. 
  15. ^ Schrenk, K. J.; N. A. M. Araújo; J. S. Andrade Jr; H. J. Herrmann (2012). "Fracturing Ranked Surfaces". Scientific Reports. 2: 348. Bibcode:2012NatSR...2E.348S. PMC 3317236Freely accessible. PMID 22470841. arXiv:1103.3256Freely accessible. doi:10.1038/srep00348. 
  16. ^ Fehr, E.; D. Kadau; J. S. Andrade Jr; H. J. Herrmann (2011). "Impact of Perturbations on Watersheds". Physical Review Letters. 106 (4): 048501. Bibcode:2011PhRvL.106d8501F. PMID 21405368. arXiv:1101.5890Freely accessible. doi:10.1103/PhysRevLett.106.048501. 
  17. ^ a b c Knecht, Craig; Walter Trump; Daniel ben-Avraham; Robert M. Ziff (2012). "Retention capacity of random surfaces". Physical Review Letters. 108 (4): 045703. Bibcode:2012PhRvL.108d5703K. arXiv:1110.6166Freely accessible. doi:10.1103/PhysRevLett.108.045703. 
  18. ^ Baek, Seung Ki; Beom Jun Kim (2012). "Critical Condition of the Water-Retention Model". Physical Review E. 85: 032103. Bibcode:2012PhRvE..85c2103B. arXiv:1111.0425Freely accessible. doi:10.1103/PhysRevE.85.032103. 
  19. ^ a b Schrenk, K. J.; N. A. M Araújo; R. M. Ziff; H. J. Herrmann (2014). "Retention capacity of correlated surfaces". arXiv:1403.2082Freely accessible. 
  20. ^ Hoshen, Joseph (1998). "On the application of the enhanced Hoshen-Kopelman algorithm for image analysis". Pattern Recognition Letters. 19 (7): 575–584. doi:10.1016/S0167-8655(98)00018-x. 

Further reading[edit]

  • Pickover, Clifford (2002). The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions. Princeton, NJ: Princeton University Press. ISBN 0-691-11597-4. 
  • Stauffer, Dietrich; Aharony, A. (1994). Introduction to Percolation Theory. London Bristol, PA: Taylor & Francis. ISBN 978-0-7484-0253-3. 

External links[edit]