194:, 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.
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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
197:
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
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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
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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
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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
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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
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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
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behave in a generally fractal manner has been the subject of some research. Technically speaking, any surface in three-dimensional space has a
100:, the generation of natural looking surfaces and landscapes was a major turning point in art history, where the distinction between geometric,
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and natural, man made art became blurred. The first use of a fractal-generated landscape in a film was in 1982 for the movie
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resulting from the procedure is not a deterministic, but rather a random surface that exhibits fractal behavior.
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is reached. There are many fractal procedures (such as combining multiple octaves of
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Fractals: The
Patterns of Chaos : a New Aesthetic of Art, Science, and Nature
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by Ken Perlin, 1998; a Java applet showing a sphere with a generated landscape.
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de la Re, Armando; Abad, Francisco; Camahort, Emilio; Juan, M. C. (2009).
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522:. Lecture Notes in Computer Science. Vol. 5545. pp. 801–810.
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Advances in multimedia modeling: 13th
International Multimedia Modeling
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refined the techniques of
Mandelbrot to create an alien landscape.
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586:"Dynamic Terrain Generation Based on Multifractal Techniques"
513:"Tools for Procedural Generation of Plants in Virtual Scenes"
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93:, in the interests of producing a more convincing landscape.
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405:
Rhonda Roland
Shearer "Rethinking Images and Metaphors" in
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Human symmetry perception and its computational analysis
61:
Many natural phenomena exhibit some form of statistical
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Richardson, L.F. (1961). "The
Problem of Continuity".
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584:van Lawick van Pabst, Joost; Jense, Hans (2001).
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565:"Is the Fractal Model Appropriate for Terrain?"
50:behavior that mimics the appearance of natural
154:Computer-generated fractal wooded hills using
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16:Stochastically generated naturalistic terrain
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386:by Fereydoon Family and Tamas Vicsek 1991
605:"Methods for Realistic Landscape Imaging"
138:Computer generated fractal terrain using
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1556:List of fractals by Hausdorff dimension
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218:random midpoint displacement algorithm
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243:can be procedurally generated using
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439:. Simon and Schuster. p. 84.
91:fractal behavior of such a surface
14:
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1538:How Long Is the Coast of Britain?
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520:Computational Science – ICCS 2009
500:Joost van Lawick van Pabst et al.
235:
174:in three-dimensional space has a
327:"The Fractal Geometry of Nature"
202:Generation of fractal landscapes
126:A fractal landscape rendered in
30:to create a mountainous terrain.
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107:Star Trek II: The Wrath of Khan
1562:The Fractal Geometry of Nature
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247:in computer-generated scenes.
118:Behavior of natural landscapes
46:algorithm designed to produce
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364:by Christopher W. Tyler 2002
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528:10.1007/978-3-642-01973-9_89
384:Dynamics of Fractal Surfaces
7:
1578:Chaos: Making a New Science
577:General Systems Yearbook. 6
272:Fractal-generating software
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10:
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951:Mathematical visualization
926:Computer-generated imagery
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407:The languages of the brain
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76:fractional Brownian motion
69:. Moreover, variations in
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102:computer generated images
946:Iterated function system
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267:Diamond-square algorithm
170:of 2, and therefore any
65:that can be modeled by
1570:The Beauty of Fractals
603:Musgrave, Ken (1993).
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78:was first proposed by
54:. In other words, the
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433:Briggs, John (1992).
342:by Tat-Jen Cham 2007
212:A way to make such a
168:topological dimension
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89:and even the overall
42:is generated using a
22:
1516:Lewis Fry Richardson
1511:Hamid Naderi Yeganeh
1301:Burning Ship fractal
1233:Weierstrass function
885:Burning Ship fractal
156:Visual Nature Studio
1274:Space-filling curve
1251:Multifractal system
1134:Space-filling curve
1119:Sierpinski triangle
931:Fractal compression
854:MojoWorld Generator
816:Wolfram Mathematica
411:Albert M. Galaburda
198:of real landscapes
176:Hausdorff dimension
1501:Aleksandr Lyapunov
1481:Desmond Paul Henry
1445:Self-avoiding walk
1440:Percolation theory
1084:Iterated function
1025:Fractal dimensions
489:Ken Musgrave, 1993
164:natural landscapes
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1544:Coastline paradox
1521:Wacław Sierpiński
1506:Benoit Mandelbrot
1430:Fractal landscape
1338:Misiurewicz point
1243:Strange attractor
1124:Apollonian gasket
1114:Sierpinski carpet
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936:Fractal landscape
846:Scenery generator
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675:Graphics software
537:978-3-642-01972-2
292:Scenery generator
216:is to employ the
208:Scenery generator
80:Benoit Mandelbrot
36:fractal landscape
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1461:Michael Barnsley
1328:Lyapunov fractal
1186:Sierpiński curve
1139:Blancmange curve
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622:A Web-Wide World
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591:. Archived from
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181:plate tectonics
172:fractal surface
162:Whether or not
144:Adobe Photoshop
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112:Loren Carpenter
71:surface texture
63:self-similarity
40:fractal surface
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598:on 2011-07-24.
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1590:Chaos theory
1585:Kaleidoscope
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1486:Gaston Julia
1466:Georg Cantor
1429:
1291:Escape-time
1223:Gosper curve
1171:Lévy C curve
1156:Dragon curve
1035:Box-counting
935:
825:Windows only
593:the original
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563:Lewis, J.P.
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473:
462:
450:. Retrieved
435:
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406:
401:
383:
379:
361:
357:
339:
335:
321:
240:
239:
211:
196:
189:
161:
140:Perlin noise
105:
95:
87:stationarity
84:
60:
39:
35:
33:
1581:(1987 book)
1573:(1986 book)
1565:(1982 book)
1551:Fractal art
1471:Bill Gosper
1435:Lévy flight
1181:Peano curve
1176:Moore curve
1062:Topological
1047:Correlation
689:Open-source
680:Fractal art
670:Digital art
1605:Categories
1389:Orbit trap
1384:Buddhabrot
1377:techniques
1365:Mandelbulb
1166:Koch curve
1099:Cantor set
956:Orbit trap
910:Mandelbulb
579:: 139–187.
557:References
478:Richardson
185:soil creep
44:stochastic
25:triangular
1496:Paul Lévy
1375:Rendering
1360:Mandelbox
1306:Julia set
1218:Hexaflake
1149:Minkowski
1069:Recursion
1052:Hausdorff
900:Mandelbox
895:Julia set
696:Apophysis
282:Heightmap
245:L-systems
214:landscape
1616:Surfaces
1611:Fractals
1406:fractals
1293:fractals
1261:L-system
1203:T-square
1011:Fractals
969:Category
869:VistaPro
864:Terragen
801:Chaotica
768:Fractint
761:Freeware
747:openPlaG
721:Sterling
716:MilkDrop
546:33892094
394:page 45
307:Quadtree
297:Terragen
251:See also
128:Terragen
28:fractals
1355:Tricorn
1208:n-flake
1057:Packing
1040:Higuchi
1030:Assouad
919:Related
859:Picogen
773:IFStile
701:Blender
452:15 June
287:Outerra
56:surface
52:terrain
48:fractal
23:Use of
1454:People
1404:Random
1311:Filled
1279:H tree
1198:String
1086:system
832:VisSim
782:Retail
544:
534:
443:
417:
390:
368:
346:
302:Octree
222:square
1530:Other
806:Maple
796:Bryce
608:(PDF)
596:(PDF)
589:(PDF)
568:(PDF)
542:S2CID
516:(PDF)
467:Lewis
413:2002
350:page
313:Notes
277:Grome
262:Bryce
142:with
752:XaoS
742:GIMP
706:Fyre
532:ISBN
454:2014
441:ISBN
415:ISBN
388:ISBN
366:ISBN
344:ISBN
730:GNU
524:doi
409:by
187:).
110:.
38:or
1607::
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540:.
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82:.
34:A
1536:"
1003:e
996:t
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329:.
158:.
130:.
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