10000 14000 12000 8000 10000 6000 10000 5000

10000 14000 12000 8000 10000 6000 10000 5000 0 0 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 30 x 30, 2% 0 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 20000 10000 7000 8000 0 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 9000 6000 5000 4000 3000 1000 0 0 40 x 40, 4% 3500 3000 5000 4000 3000 2500 4000 3000 2000 6000 3000 0 0 102 104 106 108 110 112 114 116 118 120 122 124 126 128 130 132 134 136 138 50 x 50, 2% 8000 12000 7000 6000 5000 4000 60 x 60, 2% 7000 10000 6000 2500 2000 1500 1000 500 0 0 60 x 60, 8% 3000 4000 3000 2500 4000 2500 2000 3000 2000 1500 1000 0 0 1000 510 512 514 516 518 520 522 524 526 528 530 532 534 536 538 540 542 544 546 548 550 552 554 556 558 560 562 564 566 568 570 572 574 576 578 2000 60 x 60, 10% 3500 3000 250 252 254 256 258 260 262 264 266 268 270 272 274 276 278 280 282 284 286 288 290 3000 4500 0 753 756 759 762 765 768 771 774 777 780 783 786 789 792 795 798 801 804 807 810 813 816 819 822 825 828 831 834 837 4000 3500 60 x 60, 6% 3500 5000 6000 0 60 x 60, 4% 12000 8000 3500 1000 50 x 50, 10% 4000 1500 515 517 519 521 523 525 527 529 531 533 535 537 539 541 543 545 547 549 551 553 555 557 559 561 563 565 567 569 571 573 575 577 0 172 174 176 178 180 182 184 186 188 190 192 194 196 198 200 202 204 206 208 50 x 50, 8% 2000 340 342 344 346 348 350 352 354 356 358 360 362 364 366 368 370 372 374 376 378 380 382 384 386 388 390 392 394 396 158 160 162 1000 0 2500 3000 2000 0 3000 4000 5000 2000 50 x 50, 6% 6000 4000 1000 50 x 50, 4% 14000 1500 1000 0 2000 1500 312 314 316 318 320 322 324 326 328 330 332 334 336 338 340 342 344 346 348 350 352 354 356 358 360 362 364 366 368 370 372 374 376 378 1000 2500 2000 6000 40 x 40, 10% 5000 7000 270273276279282285288291294297300303306309312315318321324327330333336339342345348 40 x 40, 8% 3500 2000 0 40 x 40, 6% 4000 7000 8000 210213216219222225228231234237240243246249252255258261264267270273276279282285288 4000 12000 10000 0 4500 8000 4000 1000 4500 14000 6000 0 2000 5000 9000 8000 162 164 166 168 170 172 174 176 178 180 182 184 186 188 190 192 194 196 198 200 202 204 206 208 210 212 214 216 218 220 222 224 2000 102 104 106 108 110 112 114 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144 146 148 150 152 154 156 158 2000 16000 10000 3000 2000 40 x 40, 2% 30 x 30, 10% 30 x 30, 8% 6000 4000 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 0 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119 121 123 125 7000 6000 212 214 216 218 220 222 224 226 228 230 232 234 236 238 240 242 244 246 248 250 252 254 256 258 260 262 264 0 1000 8000 5000 2000 30 x 30, 6% 12000 10000 3000 30 x 30, 4% 25000 15000 4000 2000 5000 4000 6000 8000 7000 418 420 422 424 426 428 430 432 434 436 438 440 442 444 446 448 450 452 454 456 458 460 462 464 466 468 470 472 474 476 478 480 482 484 486 15000 8000 673 676 679 682 685 688 691 694 697 700 703 706 709 712 715 718 721 724 727 730 733 736 739 742 745 748 751 754 757 760 20000 9000 2000 1500 1000 500 0 0 984 988 992 996 1000 1004 1008 1012 1016 1020 1024 1028 1032 1036 1040 1044 1048 1052 1056 1060 1064 1068 1072 1076 1080 1084 1088 1092 1096 1100 25000 9000 102 104 106 108 110 112 114 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144 146 148 150 152 154 156 158 16000 10000 513 516 519 522 525 528 531 534 537 540 543 546 549 552 555 558 561 564 567 570 573 576 579 582 585 588 591 594 597 600 603 12000 20 x 20, 10% 828 831 834 837 840 843 846 849 852 855 858 861 864 867 870 873 876 879 882 885 888 891 894 897 900 903 906 909 912 915 918 18000 30000 20 x 20, 8% 20 x 20, 6% 1200 1204 1208 1212 1216 1220 1224 1228 1232 1236 1240 1244 1248 1252 1256 1260 1264 1268 1272 1276 1280 1284 1288 1292 1296 1300 1304 1308 1312 1316 1320 20 x 20, 4% 20 x 20, 2% Figure S 1. Exploring the boundary limits of the normal approximation. Boundary limits of the normal approximation were evalauted across a factorial of five levels of landscape dimensionality (20 x 20, 30 x 30, 40 x 40, 50 x 50, 60 x 60) and five levels of proportion coverage of a minority class (2%, 4%, 6%, 8%, 10%) in a two-class landscape. The normal approximation is a very good fit for landscapes of at least 30 x 30 pixel and at lest 6% coverage by a minority class. Departures from normality of the distribution of total edge length are seen in landscapes of very small extent and very small proportion of minority class, in which the distribution is truncated at the upper end.

1000 900 800 700 600 500 400 300 200 100 0 0 1200 1000 800 600 400 200 1000 900 800 700 600 500 400 300 200 100 0 15135 15104 15073 15042 15011 200 200 0 0 3000 2500 2000 1500 500 0 0 1200 m, Rep 4 0 1800 m, Rep 4 0 2400 m, Rep 4 1000 1200 1000 800 600 400 3000 m, Rep 4 0 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400 404 408 412 416 4000 1752 1759 1766 1773 1780 1787 1794 1801 1808 1815 1822 1829 1836 1843 1850 1857 1864 1871 1878 1885 1892 1899 1906 1913 3500 6018 6029 6040 6051 6062 6073 6084 6095 6106 6117 6128 6139 6150 6161 6172 6183 6194 6205 6216 6227 6238 3500 4500 10088 10109 10130 10151 10172 10193 10214 10235 10256 10277 10298 10319 10340 10361 10382 10403 10424 10829 10808 10787 10766 10745 10724 10703 10682 1000 515 520 525 530 535 540 545 550 555 560 565 570 575 580 585 590 595 600 605 610 615 600 m, Rep 4 14661 14682 14703 14724 14745 14766 14787 14808 14829 14850 14871 14892 14913 14934 14955 14976 14997 3000 m, Rep 3 14980 3000 m, Rep 2 14949 1200 400 10661 1200 600 14918 400 1400 800 10619 1800 m, Rep 3 10640 600 1200 1000 2220 2246 2218 2253 2237 2257 2304 2260 2279 2219 2275 2267 2279 2311 2255 2263 2277 2259 2244 2256 2248 2261 2257 2256 1200 m, Rep 3 14887 2400 m, Rep 3 10598 800 575 581 587 593 599 605 611 617 623 629 635 641 647 653 659 665 671 677 683 689 695 600 m, Rep 3 14856 2400 m, Rep 2 14825 200 10556 400 10577 600 2000 1800 1600 1400 1200 1000 800 600 400 200 0 14794 800 1600 1400 1200 1000 800 600 400 200 0 14763 1800, Rep 2 2900 2909 2918 2927 2936 2945 2954 2963 2972 2981 2990 2999 3008 3017 3026 3035 3044 3053 3062 3071 3080 3089 3098 3107 0 10514 0 10535 0 2000 1800 1600 1400 1200 1000 800 600 400 200 0 14732 500 11195 500 11269 500 11271 1000 11230 1000 11216 1500 11204 1500 11219 2000 11233 2000 11229 1200 m, Rep 2 11154 2000 11278 2500 11180 2500 2228 2236 2244 2252 2260 2268 2276 2284 2292 2300 2308 2316 2324 2332 2340 2348 2356 2364 2372 2380 2388 1000 6138 6149 6160 6171 6182 6193 6204 6215 6226 6237 6248 6259 6270 6281 6292 6303 6314 6325 6336 6347 2000 11207 5452 5431 600 m, Rep 2 14701 3000 m, Rep 1 5410 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 11182 2400 m, Rep 1 10636 0 10656 200 10686 400 5389 600 10678 800 5368 1000 10656 1200 5347 1400 10637 1600 5326 1600 10700 1000 1800 5305 1800 m, Rep 1 10658 1000 5284 1200 m, Rep 1 10624 1500 5263 0 10706 500 5242 3000 10640 2000 5221 4000 10711 2500 349 366 361 363 369 361 340 362 352 372 350 369 358 357 348 361 357 356 380 359 368 346 344 1000 2318 2325 2332 2339 2346 2353 2360 2367 2374 2381 2388 2395 2402 2409 2416 2423 2430 2437 2444 2451 2458 2465 2472 2479 1500 5200 3000 10610 3500 10660 506 512 518 524 530 536 542 548 554 560 566 572 578 584 590 596 602 608 614 620 6000 17117 17138 17159 17180 17201 17222 17243 17264 17285 17306 17327 17348 17369 17390 17411 17432 17453 17474 11828 11807 11786 11765 11744 11723 11702 11681 11660 11639 3000 2500 2629 2636 2643 2650 2657 2664 2671 2678 2685 2692 2699 2706 2713 2720 2727 2734 2741 2748 2755 2762 2769 2776 2783 4000 17622 17643 17664 17685 17706 17727 17748 17769 17790 17811 17832 17853 17874 17895 17916 17937 17958 1600 1400 1200 1000 800 600 400 200 0 4293 4304 4315 4326 4337 4348 4359 4370 4381 4392 4403 4414 4425 4436 4447 4458 4469 4480 4491 4502 0 11618 0 14709 14730 14751 14772 14793 14814 14835 14856 14877 14898 14919 14940 14961 14982 15003 15024 15045 15066 15087 600 m, Rep 1 600, Rep 5 2500 1000 1200 m, Rep 5 2500 2000 1500 1000 500 1800 m, Rep 5 1600 1400 1200 1000 800 600 400 2400 m, Rep 5 3000 m, Rep 5 1200 1000 800 600 400 200 Figure S 2: Normal approximation for test landscapes. Match between normal probability density function (dotted line) and observed frequency (solid line) of total edge lengths In the distribution of 100, 000 permutations of the 25 sampled landscapes. X-axis is total edge length. Y-axis is frequency of that total Edge length out of 100, 000 permutations. The first row is the 5 replicate landscapes of 600 m size (20 x 20 cells at 30 m cell size). Second row - Landscapes 40 x 40. Third row – landscapes 60 x 60. Fourth row – landscapes 80 x 80. Fifth row – landscapes 100 x 100.

0 -5000 -10000 Rep 1 -15000 -20000 -6000 -8000 -10000 -12000 Rep 1 -14000 -16000 -18000 5000 0 -2000 cs 3000, Rep 1 ln. W** -5000 6470 6260 6050 2 -3000 2000 -4000 -6000 Rep 2 -8000 -10000 -12000 -1000 -2000 -3000 -4000 -5000 -6000 cs 2400, Rep 2 ln. W** 0 -2000 -4000 cs 3000, Rep 2 ln. W** 5000 0 0 -5000 -20000 -25000 6510 6300 6090 5880 5670 5460 5250 5040 4830 Rep 3 -4000 2000 -6000 -8000 Rep 3 -14000 -16000 -18000 -100 -800 -1000 -1200 -1400 cs 1800, Rep 3 ln. W** cs 1800, Rep 4 ln. W** 500 1000 0 -500 0 -1000 -1500 -2000 Rep 4 cs 2400, Rep 3 ln. W** 2000 0 -2000 -4000 -3000 -10000 -6000 -5000 -12000 -8000 Rep 4 -10000 -12000 cs 3000, Rep 3 ln. W** -1000 -2000 -10000 -3000 Rep -15000 Rep 2 -15000 Rep 3 -4000 -5000 4 -1000 -2500 -4000 Rep 5 -3000 -5000 -3500 -6000 -4000 -7000 cs 2400, Rep 4 ln. W** 0 -1000 -2000 cs 3000, Rep 4 ln. W** 1000 2000 0 0 -2000 -4000 -8000 -10000 Rep -6000 -16000 -7000 -18000 5910 -600 5700 -600 Rep 3 5490 0 Rep 4 5280 -400 5070 -400 4860 -200 4650 0 4440 0 4230 200 4020 1200 m, Rep 4, ln. W** 3810 200 3600 400 3390 1200 m, Rep 3 ln. W** 3180 -50 200 215 230 245 260 275 290 305 320 335 350 365 380 395 410 425 440 455 470 485 500 515 530 545 560 575 -50 1200 1240 1280 1320 1360 1400 1440 1480 1520 1560 1600 1640 1680 1720 1760 1800 1840 1880 1920 1960 2000 2040 2080 2120 2160 2200 2240 2280 2320 2360 -400 0 2970 -150 340 355 370 385 400 415 430 445 460 475 490 505 520 535 550 565 580 595 610 625 640 655 670 685 700 715 730 745 760 775 790 0 1100 1170 1240 1310 1380 1450 1520 1590 1660 1730 1800 1870 1940 2010 2080 2150 2220 2290 2360 2430 2500 2570 2640 2710 2780 2850 2920 2990 -300 950 1040 1130 1220 1310 1400 1490 1580 1670 1760 1850 1940 2030 2120 2210 2300 2390 2480 2570 2660 2750 2840 2930 3020 3110 3200 3290 3380 3470 3560 cs 1800, Rep 2 ln. W** 4620 400 1250 1310 1370 1430 1490 1550 1610 1670 1730 1790 1850 1910 1970 2030 2090 2150 2210 2270 2330 2390 2450 2510 2570 2630 2690 2750 2810 2870 2930 1200 m, Rep 2 ln. W** 4410 -2000 4200 -1500 3990 -2000 3780 Rep 2 -200 3570 -500 50 10980 5630 -100 340 360 380 400 420 440 460 480 500 520 540 560 580 600 620 640 660 680 700 720 740 760 780 800 820 840 860 880 900 920 940 100 Rep 10560 cs 2400, Rep 1 ln. W** 5840 500 -200 Rep 4 10140 -2000 Rep -100 9720 -1000 0 9300 0 5420 50 Rep 2 150 8880 0 5210 100 8460 200 8040 -4500 200 7620 -4000 200 7200 -3500 1000 5000 100 600 m, Rep 4 ln. W** 6780 -3000 200 6360 -2500 250 5940 Rep 1 250 5520 -2000 300 5100 -1500 250 Rep 3 4820 5130 5440 5750 6060 6370 6680 6990 7300 7610 7920 8230 8540 8850 9160 9470 9780 10090 10400 10710 11020 11330 -1000 4790 150 5000 5310 5620 5930 6240 6550 6860 7170 7480 7790 8100 8410 8720 9030 9340 9650 9960 10270 10580 10890 11200 11510 11820 12130 12440 500 4370 1800 m, Rep 1 ln. W** 4580 -3000 4160 -2500 3950 -1000 3740 Rep 1 3320 1200 m, Rep 1 ln. W** 3530 -1500 3110 -1000 0 2900 500 2480 -300 2690 -200 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 375 390 405 420 435 450 465 480 495 510 525 540 555 570 -50 1250 1320 1390 1460 1530 1600 1670 1740 1810 1880 1950 2020 2090 2160 2230 2300 2370 2440 2510 2580 2650 2720 2790 2860 2930 3000 3070 3140 3210 3280 3350 0 2270 Rep 1 5500 5810 6120 6430 6740 7050 7360 7670 7980 8290 8600 8910 9220 9530 9840 10150 10460 10770 11080 11390 11700 12010 12320 12630 12940 4790 4580 4370 4160 3950 3740 100 600 m, rep 3 ln. W** 6890 7300 7710 8120 8530 8940 9350 9760 10170 10580 10990 11400 11810 12220 12630 13040 13450 13860 14270 14680 15090 15500 15910 12660 12350 12040 11730 11420 11110 10800 10490 3530 3320 3110 2900 2690 2480 600 m, rep 2, ln. W** 8730 9040 9350 9660 9970 10280 10590 10900 11210 11520 11830 12140 12450 12760 13070 13380 13690 14000 14310 14620 14930 15240 15550 15860 16170 9870 9560 9250 8940 8630 8320 8010 10180 -4000 2270 200 7450 7960 8470 8980 9490 10000 10510 11020 11530 12040 12550 13060 13570 14080 14590 15100 15610 16120 16630 17140 17650 18160 18670 -2000 2060 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775 800 825 850 875 900 300 9460 9870 10280 10690 11100 11510 11920 12330 12740 13150 13560 13970 14380 14790 15200 15610 16020 16430 16840 17250 17660 18070 18480 18890 19300 0 7700 -500 7390 0 7080 -500 1430 1510 1590 1670 1750 1830 1910 1990 2070 2150 2230 2310 2390 2470 2550 2630 2710 2790 2870 2950 3030 3110 3190 3270 3350 3430 3510 3590 0 1850 -100 6770 0 5520 5930 6340 6750 7160 7570 7980 8390 8800 9210 9620 10030 10440 10850 11260 11670 12080 12490 12900 13310 13720 14130 14540 14950 15360 600 m, rep 1 ln. W** cs 600, Rep 5 ln. W** 150 5 50 -100 1200 m, Rep 5 ln. W** 300 200 100 -200 Rep 5 -300 -400 cs 1800, Rep 5 ln. W** -2000 -3000 cs 2400, Rep 5 ln. W** 1000 -4000 -6000 Rep 5 -7000 -8000 -9000 cs 3000, Rep 5 ln. W** -6000 5 -12000 -14000 Figure S 3. Configurational entropy curves for 15 real landscapes varying in size in the San Francisco Peaks region of Arizona. The x-axis is total edge length. They y-axis is configurational entropy (ln. W** where W** is the number of permutations (microstates) of the lattice that produce a given total edge length (macrostate). The arrow indicates the configurational entropy of the real landscape along the curve of possible configurational entropies of landscapes with That same landscape composition (number of pixels, patch richness, and proportion in each landscape. The first row is five replicate landscapes with 600 m Size (30 m pixels, 20 x 20 dimensionality). Second row – landscapes 40 x 40 pixels, Third row – 60 x 60. Fourth row – 80 x 80. Fifth Row – 100 x 100.