graphics software
In computer graphics, graphics software refers to a program or collection of programs that enable a person to manipulate images or models visually on a computer.
Computer graphics can be classified into two distinct categories: raster graphics a ...
that generates images of
fractals
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
. 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
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
.
Fractal generating software creates
mathematical beauty
Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians may express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as ...
through
visualization
Visualization or visualisation may refer to:
*Visualization (graphics), the physical or imagining creation of images, diagrams, or animations to communicate a message
* Data visualization, the graphic representation of data
* Information visualiz ...
. Modern computers may take seconds or minutes to complete a single high
resolution
Resolution(s) may refer to:
Common meanings
* Resolution (debate), the statement which is debated in policy debate
* Resolution (law), a written motion adopted by a deliberative body
* New Year's resolution, a commitment that an individual mak ...
fractal image. Images are generated for both simulation (modeling) and random fractals for art. Fractal generation used for
modeling
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
is part of
realism
Realism, Realistic, or Realists may refer to:
In the arts
*Realism (arts), the general attempt to depict subjects truthfully in different forms of the arts
Arts movements related to realism include:
*Classical Realism
*Literary realism, a move ...
in computer graphics. Fractal generation software can be used to
mimic
MIMIC, known in capitalized form only, is a former simulation computer language developed 1964 by H. E. Petersen, F. J. Sansom and L. M. Warshawsky of Systems Engineering Group within the Air Force Materiel Command at the Wright-Patterson AFB in ...
natural landscapes with
fractal landscape
A fractal landscape is a surface that is 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, ...
s and scenery generation programs. Fractal imagery can be used to introduce irregularity to an otherwise sterile computer generated environment.
Fractals are generated in
music visualization
Music visualization or music visualisation, a feature found in electronic music visualizers and media player software, generates animated imagery based on a piece of music. The imagery is usually generated and rendered in real time and in a way ...
software,
screensaver
A screensaver (or screen saver) is a computer program that blanks the display screen or fills it with moving images or patterns when the computer has been idle for a designated time. The original purpose of screensavers was to prevent phosphor ...
s and
wallpaper
Wallpaper is a material used in interior decoration to decorate the interior walls of domestic and public buildings. It is usually sold in rolls and is applied onto a wall using wallpaper paste. Wallpapers can come plain as "lining paper" (so t ...
generators. This software presents the user with a more limited range of settings and features, sometimes relying a series pre-programmed variables. Because complex images can be generated from simple formula fractals are often used among the
demoscene
The demoscene is an international computer art subculture focused on producing demos: self-contained, sometimes extremely small, computer programs that produce audiovisual presentations. The purpose of a demo is to show off programming, visual ...
. The generation of fractals such as the Mandelbrot set is time-consuming and requires many computations, so it is often used in
benchmarking
Benchmarking is the practice of comparing business processes and performance metrics to industry bests and best practices from other companies. Dimensions typically measured are quality, time and cost.
Benchmarking is used to measure performan ...
devices.
History
The generation of fractals by calculation without computer assistance was undertaken by German mathematician
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
in 1883 to create the
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
Thr ...
. Throughout the following years, mathematicians have postulated the existence of numerous fractals. Some were conceived before the naming of fractals in 1975, for example, the Pythagoras tree by Dutch mathematics teacher Albert E. Bosman in 1942.
The development of the first fractal generating software originated in
Benoit Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) 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 phy ...
's pursuit of a
generalized function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
for a
class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differentl ...
of
shape
A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type.
A pl ...
s known as
Julia set
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wit ...
s. In 1979, Mandelbrot discovered that one image of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
could be created by
iteration
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
. He and programmers working at IBM generated the first rudimentary fractal printouts. This marked the first instance of the generation of fractals by non-linear creations laws or 'escape time fractal'.
Loren Carpenter
Loren C. Carpenter (born February 7, 1947) is a computer graphics researcher and developer.
Biography
He was a co-founder and chief scientist of Pixar Animation Studios. He is the co-inventor of the Reyes rendering algorithm and is one of the ...
created a two-minute color film called Vol Libre for presentation at
SIGGRAPH
SIGGRAPH (Special Interest Group on Computer Graphics and Interactive Techniques) is an annual conference on computer graphics (CG) organized by the ACM SIGGRAPH, starting in 1974. The main conference is held in North America; SIGGRAPH Asia ...
in 1980. The October 1983 issue of
Acorn User
''Acorn User'' magazine was founded by Acorn Computers in 1982, contract-published by Addison-Wesley, to coincide with the launch of the BBC Micro. It covered the range of Acorn home computers, the BBC Micro and Atom at first and later the Elect ...
magazine carried a
BBC BASIC
BBC BASIC is a version of the BASIC programming language released in 1981 as the native programming language for the BBC Micro home/personal computer, providing a standardized language for a UK computer literacy project of the BBC. It was wri ...
listing for generating fractal shapes by Susan Stepney, now Professor of Computer Science at the
University of York
, mottoeng = On the threshold of wisdom
, established =
, type = Public research university
, endowment = £8.0 million
, budget = £403.6 million
, chancellor = Heather Melville
, vice_chancellor = Charlie Jeffery
, students ...
. She followed this up in the March 1984 Acorn User with “Snowflakes and other fractal monsters”. Fractals were rendered in computer games as early as 1984 with the creation of ''
Rescue on Fractalus!
''Rescue On Fractalus!'' is a 1985 first-person shooter computer game created by Lucasfilm Games. It was originally released for the Atari 8-bit family and the Atari 5200 games console. It was also ported to other popular platforms of the day, ...
''. From the early 1980s to about 1995 hundreds of different fractal types were formulated.
The generation of fractal images grew in popularity as the distribution of computers with a maths co-processor or
floating-point unit
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can b ...
in the central processing unit were adopted throughout the 1990s. At this time the rendering of high resolution VGA standard images could take many hours. Fractal generation algorithms display extreme parallelizability. Fractal-generating software was rewritten to make use of multi-threaded processing. Subsequently, the adoption of
graphics processing unit
A graphics processing unit (GPU) is a specialized electronic circuit designed to manipulate and alter memory to accelerate the creation of images in a frame buffer intended for output to a display device. GPUs are used in embedded systems, mobi ...
s in computers has greatly increased the speed of rendering and allowed for real-time changes to parameters that were previously impossible due to render delay. 3D fractal generation emerged around 2009. An early list of fractal-generating software was compiled for the book titled Fractals: The Patterns of Chaos by John Briggs published in 1992. Leading writers in the field include
Dietmar Saupe Dietmar Saupe (born 1954) is a fractal researcher and professor of computer science, Department of Computer and Information Science, University of Konstanz, Germany.
Saupe's book, ''Chaos and Fractals'', won the Association of American Publishers a ...
,
Heinz-Otto Peitgen
Heinz-Otto Peitgen (born April 30, 1945 in Bruch, Nümbrecht near Cologne) is a German mathematician and was President of Jacobs University from January 1, 2013 to December 31, 2013. Peitgen contributed to the study of fractals, chaos theory, an ...
and
Clifford A. Pickover
Clifford Alan Pickover (born August 15, 1957) is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity. For many years, he was employed at the IBM Thomas J. Watson Research ...
.
Methods
There are two major methods of two dimensional fractal generation. One is to apply an iterative process to simple equations by generative
recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
.
Dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s produce a series of values. In fractal software values for a set of points on the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
are calculated and then rendered as
pixel
In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device.
In most digital display devices, pixels are the smal ...
s. This computer-based generation of fractal objects is an endless process. In theory, images can be calculated
infinitely
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
but in practice are approximated to a certain level of detail. Mandelbrot used quadratic formulas described by the French mathematician
Gaston Julia
Gaston Maurice Julia (3 February 1893 – 19 March 1978) was a French Algerian mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoit Mandelbrot; the Julia and Mandelbrot fractals are cl ...
. The maximum
fractal dimension
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 (strictly speaking, a fractal pattern) changes with the scale at which it is meas ...
that can be produced varies according to type and is sometimes limited according to the method implemented. There are numerous coloring methods that can be applied. One of earliest was the escape time algorithm.
Colour banding
Colour banding is a subtle form of posterization in digital images, caused by the colour of each pixel being rounded to the nearest of the digital colour levels. While posterization is often done for artistic effect, colour banding is an unde ...
may appear in images depending on the method of coloring used as well as gradient color density.
Some programs generate
geometric
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
self-similar or deterministic fractals such as the
Koch curve
The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...
. These programs use an initiator followed by a generator that is repeated in a
pattern
A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated l ...
. These simple fractals originate from a technique first proposed in 1904 by Koch.
The other main method is with
Iterated Function Systems
In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were introduced in 1981.
IFS fractal ...
consisting of a number of affine
transformation
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Trans ...
s. In the first method each
pixel
In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device.
In most digital display devices, pixels are the smal ...
in a fractal image is evaluated according to a function and then coloured, before the same process is applied to the next pixel. The former method represents the classical
stochastic
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
approach while the latter implements a linear fractal model. Using recursion allowed programmers to create complex images through
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
direction.
Three dimensional fractals are generated in a variety of ways including by using
quaternion algebra
In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a ma ...
. Fractals emerge from fluid dynamics modelling simulations as turbulence when
contour advection
Contour advection is a Lagrangian method
of simulating the evolution of one or more contours or isolines of
a tracer as it is stirred by a moving fluid.
Consider a blob of dye injected into a river or stream: to first order it could be modelled ...
is used to study
chaotic mixing
In chaos theory and fluid dynamics, chaotic mixing is a process
by which flow tracers develop into complex fractals under the action
of a fluid flow.
The flow is characterized by an exponential growth of fluid filaments.
Even very simple flows, s ...
. The
Buddhabrot
The Buddhabrot is the probability distribution over the trajectories of points that escape the Mandelbrot fractal. Its name reflects its pareidolic resemblance to classical depictions of Gautama Buddha, seated in a meditation pose with a forehea ...
method was introduced in 1993. Programs might use fractal
heightmap
In computer graphics, a heightmap or heightfield is a raster image used mainly as Discrete Global Grid in secondary elevation modeling. Each pixel stores values, such as surface elevation data, for display in 3D computer graphics. A heightm ...
s to generate terrain. Fractals have been generated on computers using the following methods:
Menger sponge
In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Si ...
,
Hypercomplex manifold
In differential geometry, a hypercomplex manifold is a manifold with the tangent bundle
equipped with an action by the algebra of quaternions
in such a way that the quaternions I, J, K
define integrable almost complex structures.
If the almost c ...
,
Brownian tree
In probability theory, the Brownian tree, or Aldous tree, or Continuum Random Tree (CRT) is a special case from random real trees which may be defined from a Brownian excursion. The Brownian tree was defined and studied by David Aldous in three a ...
,
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
,
Decomposition
Decomposition or rot is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and is e ...
,
L-system
An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into some ...
s,
Lyapunov fractal
In mathematics, Lyapunov fractals (also known as Markus–Lyapunov fractals) are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, ''r'', periodically switches between ...
s,
Newton fractal
The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial or transcendental function. It is the Julia set of the meromorphic function which is given by Newton's method. ...
s,
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 discove ...
s and
Strange attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
s.
Features
Many different features are included in fractal-generating software packages. A corresponding diversity in the images produced is therefore possible. Most feature some form of
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
selection, an interactive image zoom, and the ability to save files in
JPEG
JPEG ( ) is a commonly used method of lossy compression for digital images, particularly for those images produced by digital photography. The degree of compression can be adjusted, allowing a selectable tradeoff between storage size and imag ...
,
TIFF
Tag Image File Format, abbreviated TIFF or TIF, is an image file format for storing raster graphics images, popular among graphic artists, the publishing industry, and photographers. TIFF is widely supported by scanning, faxing, word processin ...
, or PNG format, as well as the ability to save parameter files, allowing the user to easily return to previously created images for later modification or exploration. The formula, parameters, variables and coloring algorithms for fractal images can be exchanged between users of the same program. There is no universally adopted standard fractal
file format
A file format is a standard way that information is encoded for storage in a computer file. It specifies how bits are used to encode information in a digital storage medium. File formats may be either proprietary or free.
Some file formats ...
.
One feature of most escape time fractal programs or algebraic-based fractals is a maximum iteration setting. Increasing the iteration count is required if the image is magnified so that fine
detail Detail(s) or The Detail(s) may refer to:
Film and television
* ''Details'' (film), a 2003 Swedish film
* ''The Details'' (film), a 2011 American film
* ''The Detail'', a Canadian television series
* "The Detail" (''The Wire''), a television epis ...
is not lost. Limiting the maximum iterations is important when a device's processing power is low. Coloring options often allow colors to be
random
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. Ind ...
ised. Options for color density are common because some
gradients
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
output hugely variable magnitudes resulting in heavy repetitive banding or large areas of the same color. Because of the convenient ability to add post-processing effects
layering
Layering has evolved as a common means of vegetative propagation of numerous species in natural environments. Layering is also utilized by horticulturists to propagate desirable plants.
Natural layering typically occurs when a branch touches ...
and
alpha compositing
In computer graphics, alpha compositing or alpha blending is the process of combining one image with a background to create the appearance of partial or full transparency. It is often useful to render picture elements (pixels) in separate pas ...
features found in other graphics software have been included. Both 2D and 3D rendering effects such as
plasma effect
The plasma effect is a computer-based visual effect animated in real-time. It uses cycles of changing colours warped in various ways to give an illusion of liquid, organic movement.
Plasma was the name of a VGA graphics demo created by Bret M ...
and
lighting
Lighting or illumination is the deliberate use of light to achieve practical or aesthetic effects. Lighting includes the use of both artificial light sources like lamps and light fixtures, as well as natural illumination by capturing daylig ...
may be included. Many packages also allow the user to input their own formula, to allow for greater control of the fractals, as well as a choice of color rendering, along with the use of filters and other image manipulation techniques. Some fractal software packages allow for the creation of movies from a sequence of fractal images. Others display render time and allow some form of
color cycling
Color cycling, also known as palette shifting, is a technique used in computer graphics in which colors are changed in order to give the impression of animation. This technique was mainly used in early computer games, as storing one image and cha ...
and color palette creation tools.
Standard graphics software (such as
GIMP
GIMP ( ; GNU Image Manipulation Program) is a free and open-source raster graphics editor used for image manipulation (retouching) and image editing, free-form drawing, transcoding between different image file formats, and more specialized task ...
) contains filters or plug-ins which can be used for fractal generation.
Blender
A blender (sometimes called a mixer or liquidiser in British English) is a kitchen appliance, kitchen and laboratory appliance used to mix, crush, purée or emulsion, emulsify food and other substances. A stationary blender consists of a blender ...
contains a fractal (or random) modifier. Many stand-alone fractal-generating programs can be used in conjunction with other graphics programs (such as
Photoshop
Adobe Photoshop is a raster graphics editor developed and published by Adobe Inc. for Windows and macOS. It was originally created in 1988 by Thomas and John Knoll. Since then, the software has become the industry standard not only in raster ...
) to create more complex images.
POV-Ray
The Persistence of Vision Ray Tracer, most commonly acronymed as POV-Ray, is a cross-platform ray-tracing program that generates images from a text-based scene description. It was originally based on DKBTrace, written by David Kirk Buck and Aaro ...
is a ray tracing program which generates images from a text-based scene description that can generate fractals. Scripts on
3ds Max
Autodesk 3ds Max, formerly 3D Studio and 3D Studio Max, is a professional 3D computer graphics program for making 3D animations, models, games and images. It is developed and produced by Autodesk Media and Entertainment. It has modeling capabil ...
and
Autodesk Maya
Autodesk Maya, commonly shortened to just Maya ( ), is a 3D computer graphics application that runs on Windows, macOS and Linux, originally developed by Alias and currently owned and developed by Autodesk. It is used to create assets for interact ...
can be used. A number of web-based interfaces for the fractal generation are freely available including Turtle Graphics Renderer. Fractal Lab can generate both 2D and 3D fractals and is available over the web using
WebGL
WebGL (Short for Web Graphics Library) is a JavaScript API for rendering interactive 2D and 3D graphics within any compatible web browser without the use of plug-ins. WebGL is fully integrated with other web standards, allowing GPU-accelerated ...
. JWildfire is a java-based, open-source
fractal flame
Fractal flames are a member of the iterated function system class of fractals created by Scott Draves in 1992. Draves' open-source code was later ported into Adobe After Effects graphics softwareChris Gehman and Steve Reinke (2005). ''The Sharp ...
generator. Mandelbrot Fractal is a fractal explorer written in
JavaScript
JavaScript (), often abbreviated as JS, is a programming language that is one of the core technologies of the World Wide Web, alongside HTML and CSS. As of 2022, 98% of Website, websites use JavaScript on the Client (computing), client side ...
. Fractal Grower is software written in Java for generating Lindenmayer Substitution Fractals (L-systems).
Programs
Because of the butterfly effect, generating fractals can be difficult to master. A small change in a single variable can have an
unpredictable
Unpredictable may refer to:
Film and TV
* "Unpredictable" (''Eureka''), an episode of ''Eureka''
Music Albums
* ''Unpredictable'' (Classified album), 2000
* ''Unpredictable'' (Jamie Foxx album), 2005
* ''Unpredictable'' (Malik B. album), 2015 ...
effect. Some software presents the user with a
steep learning curve
A learning curve is a graphical representation of the relationship between how Skill, proficient people are at a task and the amount of experience they have. Proficiency (measured on the vertical axis) usually increases with increased experience ...
and an understanding of
chaos theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
is advantageous. This includes the characteristics of
fractal dimension
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 (strictly speaking, a fractal pattern) changes with the scale at which it is meas ...
,
recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
and
self-similarity
__NOTOC__
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
exhibited by all fractals.
There are many fractal generating programs available, both free and commercial. Notable fractal generating programs include:
* Apophysis –
open source
Open source is source code that is made freely available for possible modification and redistribution. Products include permission to use the source code, design documents, or content of the product. The open-source model is a decentralized sof ...
IFS software for
Microsoft Windows
Windows is a group of several proprietary graphical operating system families developed and marketed by Microsoft. Each family caters to a certain sector of the computing industry. For example, Windows NT for consumers, Windows Server for serv ...
-based systems
* Bryce – cross platform commercial software partially developed by
Ken Musgrave
Forest Kenton Musgrave (16 September 1955 – 14 December 2018) 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 Pa ...
Linux
Linux ( or ) is a family of open-source Unix-like operating systems based on the Linux kernel, an operating system kernel first released on September 17, 1991, by Linus Torvalds. Linux is typically packaged as a Linux distribution, which ...
and
Mac OS
Two major famlies of Mac operating systems were developed by Apple Inc.
In 1984, Apple debuted the operating system that is now known as the "Classic" Mac OS with its release of the original Macintosh System Software. The system, rebranded "M ...
. Free for non-commercial use.
*
Electric Sheep
Electric Sheep is a volunteer computing project for animating and evolving fractal flames, which are in turn distributed to the networked computers, which display them as a screensaver.
Process
The process is transparent to the casual user, w ...
– open source distributed screensaver software, developed by
Scott Draves
Scott Draves is the inventor of Fractal Flames and the leader of the distributed computing project Electric Sheep. He also invented patch-based texture synthesis and published the first implementation of this class of algorithms. He is also a v ...
.
*
Fractint
Fractint is a freeware computer program to render and display many kinds of fractals. The program originated on MS-DOS, then ported to the Atari ST, Linux, and Macintosh. During the early 1990s, Fractint was the definitive fractal generating pro ...
–
MS-DOS
MS-DOS ( ; acronym for Microsoft Disk Operating System, also known as Microsoft DOS) is an operating system for x86-based personal computers mostly developed by Microsoft. Collectively, MS-DOS, its rebranding as IBM PC DOS, and a few ope ...
freeware
Freeware is software, most often proprietary, that is distributed at no monetary cost to the end user. There is no agreed-upon set of rights, license, or EULA that defines ''freeware'' unambiguously; every publisher defines its own rules for the f ...
initially released in 1988 with available
source code
In computing, source code, or simply code, is any collection of code, with or without comments, written using a human-readable programming language, usually as plain text. The source code of a program is specially designed to facilitate the wo ...
, later ported to
Linux
Linux ( or ) is a family of open-source Unix-like operating systems based on the Linux kernel, an operating system kernel first released on September 17, 1991, by Linus Torvalds. Linux is typically packaged as a Linux distribution, which ...
and Windows (as WinFract)
* Fyre is a cross-platform open source tool for producing images based on histograms of iterated chaotic functions
* Kalles Fraktaler – Windows based fractal zoomer
* Milkdrop – music visualization plugin distributed with
Winamp
Winamp is a media player for Microsoft Windows originally developed by Justin Frankel and Dmitry Boldyrev by their company Nullsoft, which they later sold to AOL in 1999 for $80 million. It was then acquired by Radionomy in 2014. Sinc ...
*
MojoWorld Generator
MojoWorld was a commercial, fractal-based modelling program for the creation of digital landscapes, and attracted a following among artists who create space art and science fiction scenes. Originally created by Ken Musgrave, it was marketed comme ...
– a defunct landscape generator for Windows
*
openPlaG
openPlaG is a PHP based function graph plotter for use on websites. It was first released in April 2006. In June 2007 its source code was published under the GNU GPL license. PlaG is an abbreviation for Plot a Graph.
The current version 3.2 of ...
– creates fractals by plotting simple functions
*
Picogen
Picogen is a rendering system for the creation and rendering of artificial terrain, based on ray tracing. It is free software.
Overview
While the primary purpose of picogen is to display realistic 3D terrain, both in terms of terrain for ...
- a cross platform open source terrain generator
* Sterling – freeware software for Windows
*
Terragen
Terragen is a scenery generator program for Microsoft Windows and Mac OS X developed and published by Planetside Software. It can be used to create renderings and animations of landscapes.
History
Released in stages (tech preview and beta ...
– a
fractal terrain
A fractal landscape is a surface that is 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, ...
generator that can render animations for Windows and Mac OS X
*
Ultra Fractal
Ultra Fractal is a fractal generation and rendering software application. The program was the first publicly available fractal software which featured layering methods previously only found in image editing software. Because of this, the program h ...