Fractal landscape

Last updated
Animated fractal mountain.gif

A fractal landscape is a surface generated using a stochastic algorithm designed to produce fractal behavior that mimics the appearance of natural terrain. In other words, the result of the procedure is not a deterministic fractal surface, but rather a random surface that exhibits fractal behavior. [1]

Contents

Many natural phenomena exhibit some form of statistical self-similarity that can be modeled by fractal surfaces. [2] Moreover, variations in surface texture 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. [3] The modeling of the Earth's rough surfaces via fractional Brownian motion was first proposed by Benoit Mandelbrot. [4]

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 stationarity and even the overall fractal behavior of such a surface, in the interests of producing a more convincing landscape.

According to R. R. Shearer, the generation of natural looking surfaces and landscapes was a major turning point in art history, where the distinction between geometric, computer generated images and natural, man made art became blurred. [5] The first use of a fractal-generated landscape in a film was in 1982 for the movie Star Trek II: The Wrath of Khan . [6] Loren Carpenter refined the techniques of Mandelbrot to create an alien landscape. [7]

Behavior of natural landscapes

An example of a computer-generated fractal landscape FractalLandscape.jpg
An example of a computer-generated fractal landscape
Computer generated fractal terrain Fractal terrain texture.jpg
Computer generated fractal terrain
Computer-generated fractal wooded hills BlueRidgePastures.jpg
Computer-generated fractal wooded hills

Whether or not natural landscapes behave in a generally fractal manner has been the subject of some research. Technically speaking, any surface in three-dimensional space has a topological dimension of 2, and therefore any fractal surface in three-dimensional space has a Hausdorff dimension between 2 and 3. [8] 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 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. [9] In general, there is no reason to suppose that the geological processes that shape terrain on large scales (for example plate tectonics) exhibit the same mathematical behavior as those that shape terrain on smaller scales (for instance, soil creep).

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 fractal function, 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. [10]

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 of real landscapes [11]

Generation of fractal landscapes

A way to make such a landscape is to employ the random midpoint displacement algorithm, in which a square 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 level of detail is reached. There are many fractal procedures (such as combining multiple octaves of Simplex noise) capable of creating terrain data, however, the term "fractal landscape" has become more generic over time.

Fractal plants

Fractal plants can be procedurally generated using L-systems in computer-generated scenes. [12]

See also

Notes

  1. "The Fractal Geometry of Nature".
  2. Advances in multimedia modeling: 13th International Multimedia Modeling by Tat-Jen Cham 2007 ISBN   3-540-69428-5 page
  3. Human symmetry perception and its computational analysis by Christopher W. Tyler 2002 ISBN   0-8058-4395-7 pages 173–177
  4. Dynamics of Fractal Surfaces by Fereydoon Family and Tamas Vicsek 1991 ISBN   981-02-0720-4 page 45
  5. Rhonda Roland Shearer "Rethinking Images and Metaphors" in The languages of the brain by Albert M. Galaburda 2002 ISBN   0-674-00772-7 pages 351–359
  6. "The First Completely Computer-Generated (CGI) Cinematic Image Sequence in a Feature Film (1982)". HistoryofInformation.com. Jeremy Norman & Co. Retrieved 15 June 2014.
  7. Briggs, John (1992). Fractals: The Patterns of Chaos : a New Aesthetic of Art, Science, and Nature. Simon and Schuster. p. 84. ISBN   978-0671742171 . Retrieved 15 June 2014.
  8. Lewis
  9. Richardson
  10. Ken Musgrave, 1993
  11. Joost van Lawick van Pabst et al.
  12. de la Re, Armando; Abad, Francisco; Camahort, Emilio; Juan, M. C. (2009). "Tools for Procedural Generation of Plants in Virtual Scenes" (PDF). Computational Science – ICCS 2009. Lecture Notes in Computer Science. 5545. pp. 801–810. doi:10.1007/978-3-642-01973-9_89. ISBN   978-3-642-01972-2.

Related Research Articles

Benoit Mandelbrot French-American mathematician

Benoit B.Mandelbrot was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.

Fractal Self similar mathematical structures

In mathematics, a fractal is a subset of Euclidean space with a fractal dimension that strictly exceeds its topological dimension. Fractals appear the same at different scales, as illustrated in successive magnifications of the Mandelbrot set. Fractals exhibit similar patterns at increasingly smaller scales, a property called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.

Mandelbrot set Fractal named after mathematician Benoit Mandelbrot

The Mandelbrot set is the set of complex numbers for which the function does not diverge when iterated from , i.e., for which the sequence , , etc., remains bounded in absolute value. Its definition is credited to Adrien Douady who named it in tribute to the mathematician Benoit Mandelbrot, a pioneer of fractal geometry.

Self-similarity The whole of an object being mathematically similar to part of itself

In mathematics, a self-similar object is exactly or approximately similar to a part of itself. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.

Fractal art

Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still digital images, animations, and media. Fractal art developed from the mid-1980s onwards. It is a genre of computer art and digital art which are part of new media art. The mathematical beauty of fractals lies at the intersection of generative art and computer art. They combine to produce a type of abstract art.

In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer.

A Lévy flight, named for French mathematician Paul Lévy, is a random walk in which the step-lengths have a Lévy distribution, a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions.

How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension Paper by Benoît Mandelbrot discussing the nature of fractals (without using the term)

"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoît Mandelbrot, first published in Science in 5 May 1967. In this paper, Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2. These curves are examples of fractals, although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first publications on the topic of fractals.

Diffusion-limited aggregation

Diffusion-limited aggregation (DLA) is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by T.A. Witten Jr. and L.M. Sander in 1981, is applicable to aggregation in any system where diffusion is the primary means of transport in the system. DLA can be observed in many systems such as electrodeposition, Hele-Shaw flow, mineral deposits, and dielectric breakdown.

Pickover stalk

Pickover stalks are certain kinds of details to be found empirically in the Mandelbrot set, in the study of fractal geometry. They are so named after the researcher Clifford Pickover, whose "epsilon cross" method was instrumental in their discovery. An "epsilon cross" is a cross-shaped orbit trap.

Forest Kenton Musgrave was a professor at The George Washington University in the USA. A computer artist who worked with fractal images, he worked on the Bryce landscape software and later as CEO/CTO of Pandromeda, Inc. developed and designed the innovative MojoWorld software.

Fractal curve Mathematical curve whose shape is a fractal, pathological irregularity, regardless of magnification. Each non-zero arc has infinite length

A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite length — and every subarc longer than a single point has infinite length.

Coastline paradox Counterintuitive observation that the coastline of a landmass does not have a well-defined length

The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal curve-like properties of coastlines, i.e., the fact that a coastline typically has a fractal dimension. The first recorded observation of this phenomenon was by Lewis Fry Richardson and it was expanded upon by Benoit Mandelbrot.

Scenery generator Type of software

A scenery generator is software used to create landscape images, 3D models, and animations. These programs often use procedural generation to generate the landscapes. If not using procedural generation to create the landscapes, then normally a 3D artist would render and create the landscapes. These programs are often used in video games or movies. Basic elements of landscapes created by scenery generators include terrain, water, foliage, and clouds. The process for basic random generation uses a diamond square algorithm.

Fractal-generating software

Fractal-generating software is any type of graphics software that generates images of fractals. There are many fractal generating programs available, both free and commercial. Mobile apps are available to play or tinker with fractals. Some programmers create fractal software for themselves because of the novelty and because of the challenge in understanding the related mathematics. The generation of fractals has led to some very large problems for pure mathematics.

Lateral computing is a lateral thinking approach to solving computing problems. Lateral thinking has been made popular by Edward de Bono. This thinking technique is applied to generate creative ideas and solve problems. Similarly, by applying lateral-computing techniques to a problem, it can become much easier to arrive at a computationally inexpensive, easy to implement, efficient, innovative or unconventional solution.

Procedural texture

In computer graphics, a procedural texture is a texture created using a mathematical description rather than directly stored data. The advantage of this approach is low storage cost, unlimited texture resolution and easy texture mapping. These kinds of textures are often used to model surface or volumetric representations of natural elements such as wood, marble, granite, metal, stone, and others.

Computer-generated imagery Application of computer graphics to create or contribute to images

Computer-generated imagery (CGI) is the application of computer graphics to create or contribute to images in art, printed media, video games, simulators, computer animation and VFX in films, television programs, shorts, commercials, and videos. The images may be dynamic or static, and may be two-dimensional (2D), although the term "CGI" is most commonly used to refer to the 3-D computer graphics used for creating characters, scenes and special effects in films and television, which is described as 'CGI animation'. It was first used in the 1986 film Flight of the Navigator.

Box counting Fractal analysis technique

Box counting is a method of gathering data for analyzing complex patterns by breaking a dataset, object, image, etc. into smaller and smaller pieces, typically "box"-shaped, and analyzing the pieces at each smaller scale. The essence of the process has been compared to zooming in or out using optical or computer based methods to examine how observations of detail change with scale. In box counting, however, rather than changing the magnification or resolution of a lens, the investigator changes the size of the element used to inspect the object or pattern. Computer based box counting algorithms have been applied to patterns in 1-, 2-, and 3-dimensional spaces. The technique is usually implemented in software for use on patterns extracted from digital media, although the fundamental method can be used to investigate some patterns physically. The technique arose out of and is used in fractal analysis. It also has application in related fields such as lacunarity and multifractal analysis.

References