205:, however, is statistically stationary, meaning that its bulk statistical properties are the same everywhere. Thus, any real approach to modeling landscapes requires the ability to modulate fractal behavior spatially. Additionally, real landscapes have very few natural minima (most of these are lakes), whereas a fractal function has as many minima as maxima, on average. Real landscapes also have features originating with the flow of water and ice over their surface, which simple fractals cannot model.
146:
190:
natural phenomena, even those commonly thought to exhibit fractal behavior, do not do so over more than a few orders of magnitude. For instance, Richardson's examination of the western coastline of
Britain showed fractal behavior of the coastline over only two orders of magnitude. In general, there is no reason to suppose that the geological processes that shape terrain on large scales (for example
31:
162:
134:
208:
It is because of these considerations that the simple fractal functions are often inappropriate for modeling landscapes. More sophisticated techniques (known as 'multi-fractal' techniques) use different fractal dimensions for different scales, and thus can better model the frequency spectrum behavior
189:
between 2 and 3. Real landscapes however, have varying behavior at different scales. This means that an attempt to calculate the 'overall' fractal dimension of a real landscape can result in measures of negative fractal dimension, or of fractal dimension above 3. In particular, many studies of
1548:
84:
provide important visual cues to the orientation and slopes of surfaces, and the use of almost self-similar fractal patterns can help create natural looking visual effects. The modeling of the Earth's rough surfaces via
235:
is subdivided into four smaller equal squares and the center point is vertically offset by some random amount. The process is repeated on the four new squares, and so on, until the desired
96:
Because the intended result of the process is to produce a landscape, rather than a mathematical function, processes are frequently applied to such landscapes that may affect the
201:
Real landscapes also have varying statistical behavior from place to place, so for example sandy beaches don't exhibit the same fractal properties as mountain ranges. A
660:
1566:
177:
behave in a generally fractal manner has been the subject of some research. Technically speaking, any surface in three-dimensional space has a
111:, the generation of natural looking surfaces and landscapes was a major turning point in art history, where the distinction between geometric,
546:
236:
1012:
455:
1045:
429:
402:
380:
358:
117:
115:
and natural, man made art became blurred. The first use of a fractal-generated landscape in a film was in 1982 for the movie
1572:
653:
433:
406:
596:
384:
362:
1428:
1385:
1506:
1588:
1072:
669:
282:
961:
936:
112:
86:
1238:
979:
646:
69:
resulting from the procedure is not a deterministic, but rather a random surface that exhibits fractal behavior.
1094:
956:
277:
228:
243:) capable of creating terrain data, however, the term "fractal landscape" has become more generic over time.
1531:
1580:
1139:
1005:
523:
1365:
1057:
178:
108:
194:) exhibit the same mathematical behavior as those that shape terrain on smaller scales (for instance,
1526:
1521:
1311:
1243:
895:
706:
166:
1284:
1261:
1196:
1144:
1129:
1062:
941:
864:
826:
811:
421:
186:
1511:
1491:
1455:
1450:
1213:
731:
711:
552:
97:
1626:
1621:
1554:
1516:
1348:
1253:
1159:
1134:
1124:
1067:
1050:
1040:
1035:
998:
856:
685:
542:
451:
425:
398:
376:
354:
302:
232:
218:
182:
174:
101:
90:
77:
783:
1471:
1338:
1321:
1149:
816:
806:
721:
534:
272:
267:
202:
632:
30:
1486:
1423:
1084:
716:
191:
154:
122:
81:
73:
1181:
575:
145:
1501:
1445:
1433:
1404:
1360:
1343:
1326:
1279:
1223:
1208:
1176:
1114:
915:
900:
747:
1615:
1355:
1331:
1201:
1171:
1154:
1119:
1104:
951:
821:
240:
239:
is reached. There are many fractal procedures (such as combining multiple octaves of
224:
556:
1600:
1595:
1496:
1476:
1233:
1166:
150:
447:
Fractals: The
Patterns of Chaos : a New Aesthetic of Art, Science, and Nature
538:
445:
337:
1561:
1481:
1191:
1186:
690:
680:
603:
161:
635:
by Ken Perlin, 1998; a Java applet showing a sphere with a generated landscape.
133:
17:
1414:
1399:
1394:
1375:
1109:
966:
920:
615:
195:
54:
1370:
1316:
1228:
1079:
910:
905:
292:
255:
522:
de la Re, Armando; Abad, Francisco; Camahort, Emilio; Juan, M. C. (2009).
1271:
879:
874:
778:
757:
726:
317:
307:
138:
35:
1301:
1218:
1021:
869:
533:. Lecture Notes in Computer Science. Vol. 5545. pp. 801–810.
351:
Advances in multimedia modeling: 13th
International Multimedia Modeling
297:
66:
62:
58:
38:
1289:
842:
312:
638:
125:
refined the techniques of
Mandelbrot to create an alien landscape.
287:
160:
144:
132:
762:
752:
994:
642:
597:"Dynamic Terrain Generation Based on Multifractal Techniques"
524:"Tools for Procedural Generation of Plants in Virtual Scenes"
104:, in the interests of producing a more convincing landscape.
990:
416:
Rhonda Roland
Shearer "Rethinking Images and Metaphors" in
373:
Human symmetry perception and its computational analysis
72:
Many natural phenomena exhibit some form of statistical
586:
Richardson, L.F. (1961). "The
Problem of Continuity".
1540:
1464:
1413:
1384:
1300:
1270:
1252:
1093:
1028:
929:
888:
855:
835:
799:
792:
771:
740:
699:
595:van Lawick van Pabst, Joost; Jense, Hans (2001).
576:"Is the Fractal Model Appropriate for Terrain?"
61:behavior that mimics the appearance of natural
165:Computer-generated fractal wooded hills using
1006:
654:
27:Stochastically generated naturalistic terrain
8:
1013:
999:
991:
796:
661:
647:
639:
397:by Fereydoon Family and Tamas Vicsek 1991
616:"Methods for Realistic Landscape Imaging"
149:Computer generated fractal terrain using
29:
1567:List of fractals by Hausdorff dimension
329:
229:random midpoint displacement algorithm
7:
254:can be procedurally generated using
450:. Simon and Schuster. p. 84.
102:fractal behavior of such a surface
25:
1549:How Long Is the Coast of Britain?
531:Computational Science – ICCS 2009
511:Joost van Lawick van Pabst et al.
185:in three-dimensional space has a
338:"The Fractal Geometry of Nature"
213:Generation of fractal landscapes
137:A fractal landscape rendered in
41:to create a mountainous terrain.
488:
118:Star Trek II: The Wrath of Khan
1573:The Fractal Geometry of Nature
499:
258:in computer-generated scenes.
129:Behavior of natural landscapes
57:algorithm designed to produce
1:
375:by Christopher W. Tyler 2002
539:10.1007/978-3-642-01973-9_89
395:Dynamics of Fractal Surfaces
1589:Chaos: Making a New Science
588:General Systems Yearbook. 6
283:Fractal-generating software
1643:
962:Mathematical visualization
937:Computer-generated imagery
510:
477:
418:The languages of the brain
216:
87:fractional Brownian motion
80:. Moreover, variations in
975:
676:
113:computer generated images
957:Iterated function system
278:Diamond-square algorithm
181:of 2, and therefore any
76:that can be modeled by
1581:The Beauty of Fractals
614:Musgrave, Ken (1993).
170:
158:
142:
89:was first proposed by
65:. In other words, the
42:
444:Briggs, John (1992).
353:by Tat-Jen Cham 2007
223:A way to make such a
179:topological dimension
164:
148:
136:
100:and even the overall
53:is generated using a
33:
1527:Lewis Fry Richardson
1522:Hamid Naderi Yeganeh
1312:Burning Ship fractal
1244:Weierstrass function
896:Burning Ship fractal
167:Visual Nature Studio
1285:Space-filling curve
1262:Multifractal system
1145:Space-filling curve
1130:Sierpinski triangle
942:Fractal compression
865:MojoWorld Generator
827:Wolfram Mathematica
422:Albert M. Galaburda
209:of real landscapes
187:Hausdorff dimension
1512:Aleksandr Lyapunov
1492:Desmond Paul Henry
1456:Self-avoiding walk
1451:Percolation theory
1095:Iterated function
1036:Fractal dimensions
500:Ken Musgrave, 1993
175:natural landscapes
171:
159:
143:
43:
1609:
1608:
1555:Coastline paradox
1532:Wacław Sierpiński
1517:Benoit Mandelbrot
1441:Fractal landscape
1349:Misiurewicz point
1254:Strange attractor
1135:Apollonian gasket
1125:Sierpinski carpet
988:
987:
947:Fractal landscape
857:Scenery generator
851:
850:
686:Graphics software
548:978-3-642-01972-2
303:Scenery generator
227:is to employ the
219:Scenery generator
91:Benoit Mandelbrot
47:fractal landscape
16:(Redirected from
1634:
1472:Michael Barnsley
1339:Lyapunov fractal
1197:Sierpiński curve
1150:Blancmange curve
1015:
1008:
1001:
992:
797:
722:Kalles Fraktaler
670:Fractal software
663:
656:
649:
640:
633:A Web-Wide World
622:
620:
610:
608:
602:. Archived from
601:
591:
582:
580:
561:
560:
528:
519:
513:
508:
502:
497:
491:
486:
480:
475:
469:
468:
466:
464:
441:
435:
414:
408:
392:
386:
370:
364:
348:
342:
341:
334:
268:Brownian surface
203:fractal function
78:fractal surfaces
21:
1642:
1641:
1637:
1636:
1635:
1633:
1632:
1631:
1612:
1611:
1610:
1605:
1536:
1487:Felix Hausdorff
1460:
1424:Brownian motion
1409:
1380:
1303:
1296:
1266:
1248:
1239:Pythagoras tree
1096:
1089:
1085:Self-similarity
1029:Characteristics
1024:
1019:
989:
984:
971:
925:
884:
847:
831:
788:
767:
736:
695:
672:
667:
629:
618:
613:
606:
599:
594:
585:
578:
573:
570:
565:
564:
549:
526:
521:
520:
516:
509:
505:
498:
494:
487:
483:
476:
472:
462:
460:
458:
443:
442:
438:
415:
411:
393:
389:
371:
367:
349:
345:
336:
335:
331:
326:
264:
249:
237:level of detail
221:
215:
192:plate tectonics
183:fractal surface
173:Whether or not
155:Adobe Photoshop
131:
123:Loren Carpenter
82:surface texture
74:self-similarity
51:fractal surface
28:
23:
22:
18:Fractal surface
15:
12:
11:
5:
1640:
1638:
1630:
1629:
1624:
1614:
1613:
1607:
1606:
1604:
1603:
1598:
1593:
1585:
1577:
1569:
1564:
1559:
1558:
1557:
1544:
1542:
1538:
1537:
1535:
1534:
1529:
1524:
1519:
1514:
1509:
1504:
1502:Helge von Koch
1499:
1494:
1489:
1484:
1479:
1474:
1468:
1466:
1462:
1461:
1459:
1458:
1453:
1448:
1443:
1438:
1437:
1436:
1434:Brownian motor
1431:
1420:
1418:
1411:
1410:
1408:
1407:
1405:Pickover stalk
1402:
1397:
1391:
1389:
1382:
1381:
1379:
1378:
1373:
1368:
1363:
1361:Newton fractal
1358:
1353:
1352:
1351:
1344:Mandelbrot set
1341:
1336:
1335:
1334:
1329:
1327:Newton fractal
1324:
1314:
1308:
1306:
1298:
1297:
1295:
1294:
1293:
1292:
1282:
1280:Fractal canopy
1276:
1274:
1268:
1267:
1265:
1264:
1258:
1256:
1250:
1249:
1247:
1246:
1241:
1236:
1231:
1226:
1224:Vicsek fractal
1221:
1216:
1211:
1206:
1205:
1204:
1199:
1194:
1189:
1184:
1179:
1174:
1169:
1164:
1163:
1162:
1152:
1142:
1140:Fibonacci word
1137:
1132:
1127:
1122:
1117:
1115:Koch snowflake
1112:
1107:
1101:
1099:
1091:
1090:
1088:
1087:
1082:
1077:
1076:
1075:
1070:
1065:
1060:
1055:
1054:
1053:
1043:
1032:
1030:
1026:
1025:
1020:
1018:
1017:
1010:
1003:
995:
986:
985:
983:
982:
976:
973:
972:
970:
969:
964:
959:
954:
949:
944:
939:
933:
931:
927:
926:
924:
923:
918:
916:Mandelbrot set
913:
908:
903:
901:Jerusalem cube
898:
892:
890:
886:
885:
883:
882:
877:
872:
867:
861:
859:
853:
852:
849:
848:
846:
845:
839:
837:
833:
832:
830:
829:
824:
819:
814:
809:
803:
801:
800:Cross-platform
794:
790:
789:
787:
786:
781:
775:
773:
769:
768:
766:
765:
760:
755:
750:
748:Electric Sheep
744:
742:
738:
737:
735:
734:
729:
724:
719:
714:
709:
703:
701:
697:
696:
694:
693:
688:
683:
677:
674:
673:
668:
666:
665:
658:
651:
643:
637:
636:
628:
627:External links
625:
624:
623:
611:
609:on 2011-07-24.
592:
583:
569:
566:
563:
562:
547:
514:
503:
492:
481:
470:
457:978-0671742171
456:
436:
432:pages 351–359
409:
387:
383:pages 173–177
365:
343:
328:
327:
325:
322:
321:
320:
315:
310:
305:
300:
295:
290:
285:
280:
275:
270:
263:
260:
252:Fractal plants
248:
247:Fractal plants
245:
217:Main article:
214:
211:
130:
127:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1639:
1628:
1625:
1623:
1620:
1619:
1617:
1602:
1599:
1597:
1594:
1591:
1590:
1586:
1583:
1582:
1578:
1575:
1574:
1570:
1568:
1565:
1563:
1560:
1556:
1553:
1552:
1550:
1546:
1545:
1543:
1539:
1533:
1530:
1528:
1525:
1523:
1520:
1518:
1515:
1513:
1510:
1508:
1505:
1503:
1500:
1498:
1495:
1493:
1490:
1488:
1485:
1483:
1480:
1478:
1475:
1473:
1470:
1469:
1467:
1463:
1457:
1454:
1452:
1449:
1447:
1444:
1442:
1439:
1435:
1432:
1430:
1429:Brownian tree
1427:
1426:
1425:
1422:
1421:
1419:
1416:
1412:
1406:
1403:
1401:
1398:
1396:
1393:
1392:
1390:
1387:
1383:
1377:
1374:
1372:
1369:
1367:
1364:
1362:
1359:
1357:
1356:Multibrot set
1354:
1350:
1347:
1346:
1345:
1342:
1340:
1337:
1333:
1332:Douady rabbit
1330:
1328:
1325:
1323:
1320:
1319:
1318:
1315:
1313:
1310:
1309:
1307:
1305:
1299:
1291:
1288:
1287:
1286:
1283:
1281:
1278:
1277:
1275:
1273:
1269:
1263:
1260:
1259:
1257:
1255:
1251:
1245:
1242:
1240:
1237:
1235:
1232:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1203:
1202:Z-order curve
1200:
1198:
1195:
1193:
1190:
1188:
1185:
1183:
1180:
1178:
1175:
1173:
1172:Hilbert curve
1170:
1168:
1165:
1161:
1158:
1157:
1156:
1155:De Rham curve
1153:
1151:
1148:
1147:
1146:
1143:
1141:
1138:
1136:
1133:
1131:
1128:
1126:
1123:
1121:
1120:Menger sponge
1118:
1116:
1113:
1111:
1108:
1106:
1105:Barnsley fern
1103:
1102:
1100:
1098:
1092:
1086:
1083:
1081:
1078:
1074:
1071:
1069:
1066:
1064:
1061:
1059:
1056:
1052:
1049:
1048:
1047:
1044:
1042:
1039:
1038:
1037:
1034:
1033:
1031:
1027:
1023:
1016:
1011:
1009:
1004:
1002:
997:
996:
993:
981:
978:
977:
974:
968:
965:
963:
960:
958:
955:
953:
952:Fractal flame
950:
948:
945:
943:
940:
938:
935:
934:
932:
928:
922:
919:
917:
914:
912:
909:
907:
904:
902:
899:
897:
894:
893:
891:
889:Found objects
887:
881:
878:
876:
873:
871:
868:
866:
863:
862:
860:
858:
854:
844:
841:
840:
838:
834:
828:
825:
823:
822:Ultra Fractal
820:
818:
815:
813:
810:
808:
805:
804:
802:
798:
795:
791:
785:
782:
780:
777:
776:
774:
770:
764:
761:
759:
756:
754:
751:
749:
746:
745:
743:
739:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
704:
702:
698:
692:
689:
687:
684:
682:
679:
678:
675:
671:
664:
659:
657:
652:
650:
645:
644:
641:
634:
631:
630:
626:
617:
612:
605:
598:
593:
589:
584:
577:
572:
571:
567:
558:
554:
550:
544:
540:
536:
532:
525:
518:
515:
512:
507:
504:
501:
496:
493:
490:
485:
482:
479:
474:
471:
459:
453:
449:
448:
440:
437:
434:
431:
430:0-674-00772-7
427:
423:
419:
413:
410:
407:
404:
403:981-02-0720-4
400:
396:
391:
388:
385:
382:
381:0-8058-4395-7
378:
374:
369:
366:
363:
360:
359:3-540-69428-5
356:
352:
347:
344:
339:
333:
330:
323:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
279:
276:
274:
271:
269:
266:
265:
261:
259:
257:
253:
246:
244:
242:
241:Simplex noise
238:
234:
231:, in which a
230:
226:
220:
212:
210:
206:
204:
199:
197:
193:
188:
184:
180:
176:
168:
163:
157:and Terragen.
156:
152:
147:
140:
135:
128:
126:
124:
120:
119:
114:
110:
109:R. R. Shearer
107:According to
105:
103:
99:
94:
92:
88:
83:
79:
75:
70:
68:
64:
60:
56:
52:
48:
40:
37:
32:
19:
1601:Chaos theory
1596:Kaleidoscope
1587:
1579:
1571:
1497:Gaston Julia
1477:Georg Cantor
1440:
1302:Escape-time
1234:Gosper curve
1182:Lévy C curve
1167:Dragon curve
1046:Box-counting
946:
836:Windows only
604:the original
587:
574:Lewis, J.P.
530:
517:
506:
495:
484:
473:
461:. Retrieved
446:
439:
417:
412:
394:
390:
372:
368:
350:
346:
332:
251:
250:
222:
207:
200:
172:
151:Perlin noise
116:
106:
98:stationarity
95:
71:
50:
46:
44:
1592:(1987 book)
1584:(1986 book)
1576:(1982 book)
1562:Fractal art
1482:Bill Gosper
1446:Lévy flight
1192:Peano curve
1187:Moore curve
1073:Topological
1058:Correlation
700:Open-source
691:Fractal art
681:Digital art
1616:Categories
1400:Orbit trap
1395:Buddhabrot
1388:techniques
1376:Mandelbulb
1177:Koch curve
1110:Cantor set
967:Orbit trap
921:Mandelbulb
590:: 139–187.
568:References
489:Richardson
196:soil creep
55:stochastic
36:triangular
1507:Paul Lévy
1386:Rendering
1371:Mandelbox
1317:Julia set
1229:Hexaflake
1160:Minkowski
1080:Recursion
1063:Hausdorff
911:Mandelbox
906:Julia set
707:Apophysis
293:Heightmap
256:L-systems
225:landscape
1627:Surfaces
1622:Fractals
1417:fractals
1304:fractals
1272:L-system
1214:T-square
1022:Fractals
980:Category
880:VistaPro
875:Terragen
812:Chaotica
779:Fractint
772:Freeware
758:openPlaG
732:Sterling
727:MilkDrop
557:33892094
405:page 45
318:Quadtree
308:Terragen
262:See also
139:Terragen
39:fractals
1366:Tricorn
1219:n-flake
1068:Packing
1051:Higuchi
1041:Assouad
930:Related
870:Picogen
784:IFStile
712:Blender
463:15 June
298:Outerra
67:surface
63:terrain
59:fractal
34:Use of
1465:People
1415:Random
1322:Filled
1290:H tree
1209:String
1097:system
843:VisSim
793:Retail
555:
545:
454:
428:
401:
379:
357:
313:Octree
233:square
1541:Other
817:Maple
807:Bryce
619:(PDF)
607:(PDF)
600:(PDF)
579:(PDF)
553:S2CID
527:(PDF)
478:Lewis
424:2002
361:page
324:Notes
288:Grome
273:Bryce
153:with
763:XaoS
753:GIMP
717:Fyre
543:ISBN
465:2014
452:ISBN
426:ISBN
399:ISBN
377:ISBN
355:ISBN
741:GNU
535:doi
420:by
198:).
121:.
49:or
1618::
1551:"
551:.
541:.
529:.
93:.
45:A
1547:"
1014:e
1007:t
1000:v
662:e
655:t
648:v
621:.
581:.
559:.
537::
467:.
340:.
169:.
141:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
