**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.^{ [2] } 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.

Fractal art (especially in the western world) is rarely drawn or painted by hand. It is usually created indirectly with the assistance of fractal-generating software, iterating through three phases: setting parameters of appropriate fractal software; executing the possibly lengthy calculation; and evaluating the product. In some cases, other graphics programs are used to further modify the images produced. This is called post-processing. Non-fractal imagery may also be integrated into the artwork.^{ [3] } The Julia set and Mandelbrot sets can be considered as icons of fractal art.^{ [4] }

It was assumed that fractal art could not have developed without computers because of the calculative capabilities they provide.^{ [5] } Fractals are generated by applying iterative methods to solving non-linear equations or polynomial equations. Fractals are any of various extremely irregular curves or shapes for which any suitably chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size.^{ [6] }

There are many different kinds of fractal images and can be subdivided into several groups.

- Fractals derived from standard geometry by using iterative transformations on an initial common figure like a straight line (the Cantor dust or the von Koch curve), a triangle (the Sierpinski triangle), or a cube (the Menger sponge). The first fractal figures invented near the end of the 19th and early 20th centuries belong to this group.
- IFS (iterated function systems)
- Strange attractors
- Fractal flame
- L-system fractals
- Fractals created by the iteration of complex polynomials: perhaps the most famous fractals.
- Newton fractals, including Nova fractals
- Quaternionic and (recently) hypernionic
^{[ clarification needed ]}fractals^{ [7] } - Fractal terrains generated by random fractal processes
^{ [8] } - Mandelbulbs are a kind of three dimensional fractal.

Fractal Expressionism is a term used to differentiate traditional visual art that incorporates fractal elements such as self-similarity for example. Perhaps the best example of fractal expressionism is found in Jackson Pollock's dripped patterns. They have been analysed and found to contain a fractal dimension which has been attributed to his technique.^{ [9] }

Fractals of all kinds have been used as the basis for digital art and animation. High resolution color graphics became increasingly available at scientific research labs in the mid-1980s. Scientific forms of art, including fractal art, have developed separately from mainstream culture.^{ [10] } Starting with 2-dimensional details of fractals, such as the Mandelbrot Set, fractals have found artistic application in fields as varied as texture generation, plant growth simulation, and landscape generation.

Fractals are sometimes combined with evolutionary algorithms, either by iteratively choosing good-looking specimens in a set of random variations of a fractal artwork and producing new variations, to avoid dealing with cumbersome or unpredictable parameters, or collectively, as in the Electric Sheep project, where people use fractal flames rendered with distributed computing as their screensaver and "rate" the flame they are viewing, influencing the server, which reduces the traits of the undesirables, and increases those of the desirables to produce a computer-generated, community-created piece of art.

Many fractal images are admired because of their perceived harmony. This is typically achieved by the patterns which emerge from the balance of order and chaos. Similar qualities have been described in Chinese painting and miniature trees and rockeries.^{ [11] }

The first fractal image that was intended to be a work of art was probably the famous one on the cover of * Scientific American *, August 1985. This image showed a landscape formed from the potential function on the domain outside the (usual) Mandelbrot set. However, as the potential function grows fast near the boundary of the Mandelbrot set, it was necessary for the creator to let the landscape grow downwards, so that it looked as if the Mandelbrot set was a plateau atop a mountain with steep sides. The same technique was used a year after in some images in * The Beauty of Fractals * by Heinz-Otto Peitgen and Michael M. Richter. They provide a formula to estimate the distance from a point outside the Mandelbrot set to the boundary of the Mandelbrot set (and a similar formula for the Julia sets). Landscapes can, for example, be formed from the distance function for a family of iterations of the form .

Notable fractal artists include Desmond Paul Henry, Hamid Naderi Yeganeh, and musician Bruno Degazio. The British artist William Latham, has used fractal geometry and other computer graphics techniques in his works.^{ [12] } Greg Sams has used fractal designs in postcards, T-shirts, and textiles. American Vicky Brago-Mitchell has created fractal art which has appeared in exhibitions and on magazine covers. Scott Draves is credited with inventing flame fractals. Carlos Ginzburg has explored fractal art and developed a concept called "homo fractalus" which is based around the idea that the human is the ultimate fractal.^{ [13] } Merrin Parkers from New Zealand specialises in fractal art.^{ [14] } Kerry Mitchell wrote a "Fractal Art Manifesto", claiming that^{ [15] }

Fractal Art is a subclass of two-dimensional visual art, and is in many respects similar to photography—another art form that was greeted by skepticism upon its arrival. Fractal images typically are manifested as prints, bringing fractal artists into the company of painters, photographers, and printmakers. Fractals exist natively as electronic images. This is a format that traditional visual artists are quickly embracing, bringing them into Fractal Art's digital realm. Generating fractals can be an artistic endeavor, a mathematical pursuit, or just a soothing diversion. However, Fractal Art is clearly distinguished from other digital activities by what it is, and by what it is not.

^{ [15] }

According to Mitchell, fractal art is not computerized art, lacking in rules, unpredictable, nor something that any person with access to a computer can do well. Instead, fractal art is expressive, creative, and requires input, effort, and intelligence. Most importantly, "fractal art is simply that which is created by Fractal Artists: ART."^{ [15] }

More recently, American artist Hal Tenny was hired to design environment in Guardians of the Galaxy Vol. 2.

Fractal art has been exhibited at major international art galleries.^{ [16] } One of the first exhibitions of fractal art was "Map Art", a travelling exhibition of works from researchers at the University of Bremen.^{ [17] } Mathematicians Heinz-Otto Peitgen and Michael M. Richter discovered that the public not only found the images aesthetically pleasing but that they also wanted to understand the scientific background to the images.^{ [18] }

In 1989, fractals were part of the subject matter for an art show called *Strange Attractors: Signs of Chaos* at the New Museum of Contemporary Art.^{ [10] } The show consisted of photographs, installations and sculptures designed to provide greater scientific discourse to the field which had already captured the public's attention through colourful and intricate computer imagery.

In 2014, emerging British fractal artist Vienna Forrester^{ [19] } created an exhibition held at the I-node of the Planetary Collegium ^{ [20] }^{[ circular reference ]}, Kefalonia, entitled "IO. Fragmented Myths and Memories: A Fractal Exploration of Kefalonia",^{ [21] } part of the 2013–14 international arts festival "Stone Kingdom Kefalonia" commemorating the devastating 1953 Ionian earthquake.^{ [22] }^{[ circular reference ]} Her works were created by using geographical coordinates and photographs from parts of the island which still bear the scars.

**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.

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.

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.

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.

The **Sierpiński triangle**, also called the **Sierpiński gasket** or **Sierpiński sieve**, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński.

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.

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.

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.

**Heinz-Otto Peitgen** 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, and medical image computing, as well as helping to introduce fractals to the broader public.

**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.

**Algorithmic art** or **algorithm art** is art, mostly visual art, in which the design is generated by an algorithm. Algorithmic artists are sometimes called *algorists*.

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.

* The Beauty of Fractals* is a 1986 book by Heinz-Otto Peitgen and Peter Richter which publicises the fields of complex dynamics, chaos theory and the concept of fractals. It is lavishly illustrated and as a mathematics book became an unusual success.

**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.

In mathematics, an **orbit trap** is a method of colouring fractal images based upon how close an iterative function, used to create the fractal, approaches a geometric shape, called a "trap". Typical traps are points, lines, circles, flower shapes and even raster images. Orbit traps are typically used to colour two dimensional fractals representing the complex plane.

* The Fractal Geometry of Nature* is a 1982 book by the Franco-American mathematician Benoît Mandelbrot.

**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*.

**Kerry Mitchell** is an American artist known for his algorithmic and fractal art, which has been exhibited at the Nature in Art Museum, The Bridges Conference, and the Los Angeles Center for Digital Art, and for his "Fractal Art Manifesto".

- ↑ Trivedi, K. (1989). "Hindu Temples: Models of a Fractal Universe".
*The Visual Computer*.**5**(4): 243–258. doi:10.1007/BF02153753. S2CID 29185088. - ↑ Bovill, Carl (1996).
*Fractal geometry in architecture and design*. Boston: Birkhauser. p. 153. ISBN 0-8176-3795-8 . Retrieved October 28, 2011. - ↑ Elysia Conner (February 25, 2009). "Meet Reginald Atkins, mathematical artist". CasperJournal.com. Archived from the original on April 20, 2012. Retrieved October 28, 2011.
- ↑ Burger, Edward B.; Michael P. Starbird (2005).
*The heart of mathematics: an invitation to effective thinking*. Springer. p. 475. ISBN 1-931914-41-9 . Retrieved October 30, 2011. - ↑ Steven R., Holtzman (1995).
*Digital Mantras: The languages of abstract and virtual worlds*. MIT Press. p. 241. ISBN 0-262-58143-4 . Retrieved October 28, 2011. - ↑ Fractal – Definition. Free Merriam-Webster Dictionary.
- ↑ Quaternion Julia Fractals
- ↑ Fractal Art FAQ
- ↑ Érdi, Péter (2008).
*Complexity explained*. Springer. p. 214. ISBN 978-3-540-35777-3 . Retrieved October 29, 2011. - 1 2 Penny, Simon (1995).
*Critical issues in electronic media*. State University of New York Press. pp. 81–82. ISBN 0-7914-2317-4 . Retrieved October 29, 2011. - ↑ Wang, Hongyu (2005). "Chinese aesthetics, Fractals and the Tao of Curriculum". In Doll, Jr, William E.; Fleener, Jayne; Trueit, Donna; et al. (eds.).
*Chaos, complexity, curriculum and culture*. New York: Peter Lang Publishing. p. 301. ISBN 978-0-8204-6780-1 . Retrieved October 28, 2011. - ↑ Briggs, John (1992).
*Fractals: The Patterns of Chaos*. London: Thames and Hudson. p. 169. ISBN 0-500-27693-5. - ↑ "Carlos Ginzburg".
*Leonardo*. Leonardo/ISAST, the International Society for the Arts, Sciences and Technology. 2001. Retrieved October 29, 2011. - ↑ Jackson, William Joseph (2004).
*Heaven's fractal net: retrieving lost visions in the humanities, Volume 1*. Indiana University Press. p. 116. ISBN 0-253-21620-6 . Retrieved October 30, 2011. - 1 2 3 Mitchell, Kerry. "The Fractal Art Manifesto" . Retrieved December 28, 2015.
- ↑ Barrow, John D. (1995).
*The artful universe expanded*. Oxford University Press. p. 69. ISBN 0-19-280569-X . Retrieved October 28, 2011. - ↑ Robertson, George (1996).
*FutureNatural*. London: Routledge. pp. 220–221. ISBN 0-415-07013-9 . Retrieved October 28, 2011. - ↑ Richard Wright. "Art and Science in Chaos: Contesting Readings of Scientific Visualisation".
*ISEA'94 Proceedings – The Next Generation*. University of Iowa. Archived from the original on October 1, 2011. Retrieved October 28, 2011. - ↑ https://viennaforrester.art/fractal
- ↑ Planetary Collegium
- ↑ https://ionionartscenter.gr/2014/10/25/1383/
- ↑ 1953 Ionian earthquake

- Duarte, German A. (2014).
*Fractal Narrative. About the Relationship Between Geometries and Technology and Its Impact on Narrative Spaces*. Transcript-Verlag. ISBN 9783837628296. - Pickover, Clifford (1990).
*Computers, Pattern, Chaos and Beauty*. St. Martin's Press. ISBN 0-486-41709-3. - Schroeder, Manfred (1991).
*Fractals, Chaos, Power Laws*. Freeman. ISBN 0-7167-2357-3.

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