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Algorithmic art

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</style><div role="note" class="hatnote navigation-not-searchable">For the technique of performing basic arithmetic, see <a href="Algorism" title="Algorism">Algorism</a>. For the use of massive diffusion models to produce visuals, see <a href="Artificial_intelligence_art" class="mw-redirect" title="Artificial intelligence art">artificial intelligence art</a>.</div>

Algorithmic art or algorithm art is art, mostly <a href="Visual_art" class="mw-redirect" title="Visual art">visual art</a>, in which the design is generated by an <a href="Algorithm" title="Algorithm">algorithm</a>. Algorithmic artists are sometimes called algorists. Algorithmic art is created in the form of digital paintings and <a href="Sculpture" title="Sculpture">sculptures</a>, <a href="Interactive_installation" class="mw-redirect" title="Interactive installation">interactive installations</a> and <a href="Music_composition" class="mw-redirect" title="Music composition">music compositions</a>.<a href="#cite_note-fctc-2">[2]</a>

Algorithmic art is not a new <a href="Concept" title="Concept">concept</a>. <a href="Islamic_art" title="Islamic art">Islamic art</a> is a good example of the tradition of following a set of rules to create <a href="Pattern" title="Pattern">patterns</a>. The even older practice of <a href="Weaving" title="Weaving">weaving</a> includes elements of algorithmic art.<a href="#cite_note-faa-3">[3]</a>

As <a href="Computer" title="Computer">computers</a> developed so did the art created with them. Algorithmic art encourages <a href="Experimentation" class="mw-redirect" title="Experimentation">experimentation</a> allowing artists to push their <a href="Creativity" title="Creativity">creativity</a> in the <a href="Digital_age" class="mw-redirect" title="Digital age">digital age</a>. Algorithmic art allows creators to devise intricate patterns and designs that would be nearly impossible to achieve by hand.<a href="#cite_note-aawc-4">[4]</a> Creators have a say on what the input criteria is, but not on the outcome.<a href="#cite_note-aid-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></div>

Algorithmic art, also known as computer-generated art, is a subset of <a href="Generative_art" title="Generative art">generative art</a> (generated by an autonomous system) and is related to <a href="Systems_art" title="Systems art">systems art</a> (influenced by systems theory). <a href="Fractal_art" title="Fractal art">Fractal art</a> is an example of algorithmic art.<a href="#cite_note-UCSB-6">[6]</a> Fractal art is both <a href="Abstract_art" title="Abstract art">abstract</a> and mesmerizing.<a href="#cite_note-fctc-2">[2]</a>

For an image of reasonable size, even the simplest <a href="Algorithm" title="Algorithm">algorithms</a> require too much calculation for manual execution to be practical, and they are thus executed on either a single computer or on a cluster of computers. The final output is typically displayed on a <a href="Computer_monitor" title="Computer monitor">computer monitor</a>, printed with a <a href="https://en.wiktionary.org/wiki/raster" class="extiw external" title="wikt:raster">raster</a>-type printer, or drawn using a <a href="Plotter" title="Plotter">plotter</a>. Variability can be introduced by using <a href="Pseudo-random" class="mw-redirect" title="Pseudo-random">pseudo-random</a> numbers. There is no consensus as to whether the product of an algorithm that operates on an existing image (or on any input other than pseudo-random numbers) can still be considered computer-generated art, as opposed to computer-assisted art.<a href="#cite_note-UCSB-6">[6]</a>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<a href="Roman_Verostko" title="Roman Verostko">Roman Verostko</a> argues that <a href="Islamic_geometric_patterns" title="Islamic geometric patterns">Islamic geometric patterns</a> are constructed using algorithms, as are <a href="Italian_Renaissance" title="Italian Renaissance">Italian Renaissance</a> paintings which make use of <a href="Mathematics_and_art" title="Mathematics and art">mathematical techniques</a>, in particular <a href="Linear_perspective" class="mw-redirect" title="Linear perspective">linear perspective</a> and proportion.<a href="#cite_note-Verostko-7">[7]</a>

Some of the earliest known examples of computer-generated algorithmic art were created by <a href="Georg_Nees" title="Georg Nees">Georg Nees</a>, <a href="Frieder_Nake" title="Frieder Nake">Frieder Nake</a>, <a href="A._Michael_Noll" title="A. Michael Noll">A. Michael Noll</a>, <a href="Manfred_Mohr" title="Manfred Mohr">Manfred Mohr</a> and <a href="Vera_Moln%C3%A1r" title="Vera Molnár">Vera Molnár</a> in the early 1960s. These artworks were executed by a plotter controlled by a computer, and were therefore computer-generated art but not <a href="Digital_art" title="Digital art">digital art</a>. The act of creation lay in writing the <a href="Computer_program" title="Computer program">program</a>, which specified the sequence of actions to be performed by the plotter. <a href="Sonia_Landy_Sheridan" class="mw-redirect" title="Sonia Landy Sheridan">Sonia Landy Sheridan</a> established Generative Systems as a program at the <a href="School_of_the_Art_Institute_of_Chicago" title="School of the Art Institute of Chicago">School of the Art Institute of Chicago</a> in 1970 in response to social change brought about in part by the computer-robot communications revolution.<a href="#cite_note-SLS_83-8">[8]</a> Her early work with copier and telematic art focused on the differences between the human hand and the algorithm.<a href="#cite_note-MF1-9">[9]</a>

Aside from the ongoing work of Roman Verostko and his fellow algorists, the next known examples are fractal artworks created in the mid to late 1980s. These are important here because they use a different means of execution. Whereas the earliest algorithmic art was "drawn" by a <a href="Plotter" title="Plotter">plotter</a>, fractal art simply creates an image in <a href="Computer_memory" title="Computer memory">computer memory</a>; it is therefore digital art. The native form of a fractal artwork is an image stored on a computer –this is also true of very nearly all equation art and of most recent algorithmic art in general. However, in a stricter sense "fractal art" is not considered algorithmic art, because the algorithm is not devised by the artist.<a href="#cite_note-UCSB-6">[6]</a>

In light of such ongoing developments, pioneer algorithmic artist <a href="Ernest_Edmonds" title="Ernest Edmonds">Ernest Edmonds</a> has documented the continuing prophetic role of art in human affairs by tracing the early 1960s association between art and the computer up to a present time in which the algorithm is now widely recognized as a key concept for society as a whole.<a href="#cite_note-10">[10]</a>

<div class="mw-heading mw-heading3"><h3 id="Rational_approaches_to_art">Rational approaches to art</h3></div>

While art has strong emotional and psychological ties, it also depends heavily on rational approaches. Artists have to learn how to use various tools, theories and techniques to be able to create impressive artwork. Thus, throughout history, many art techniques were introduced to create various visual effects. For example, <a href="Georges_Seurat" title="Georges Seurat">Georges-Pierre Seurat</a> invented <a href="Pointillism" title="Pointillism">pointillism</a>, a painting technique that involves placing dots of complementary colors adjacent to each other.<a href="#cite_note-:0-11">[11]</a> <a href="Cubism" title="Cubism">Cubism</a> and <a href="Color_theory" title="Color theory">Color Theory</a> also helped revolutionize visual arts. <a href="Cubism" title="Cubism">Cubism</a> involved taking various reference points for the object and creating a 2-Dimensional rendering. <a href="Color_theory" title="Color theory">Color Theory</a>, stating that all colors are a combination of the three primary colors (Red, Green and Blue), also helped facilitate the use of colors in visual arts and in the creation of distinct colorful effects.<a href="#cite_note-:0-11">[11]</a> In other words, humans have always found algorithmic ways and discovered patterns to create art. Such tools allowed humans to create more visually appealing artworks efficiently. In such ways, art adapted to become more methodological.

<div class="mw-heading mw-heading3"><h3 id="Creating_perspective_through_algorithms">Creating perspective through algorithms</h3></div>

Another important aspect that allowed art to evolve into its current form is <a href="Perspectivity" title="Perspectivity">perspective</a>. Perspective allows the artist to create a 2-Dimensional projection of a 3-Dimensional object. Muslim artists during the <a href="Islamic_Golden_Age" title="Islamic Golden Age">Islamic Golden Age</a> employed <a href="Linear_perspective" class="mw-redirect" title="Linear perspective">linear perspective</a> in most of their designs. The notion of perspective was rediscovered by Italian artists during the Renaissance. The <a href="Golden_ratio" title="Golden ratio">Golden Ratio</a>, a famous mathematical ratio, was utilized by many <a href="Renaissance" title="Renaissance">Renaissance</a> artists in their drawings.<a href="#cite_note-:0-11">[11]</a> Most famously, <a href="Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo DaVinci</a> employed that technique in his <a href="Mona_Lisa" title="Mona Lisa">Mona Lisa</a>, and many other paintings, such as <a href="Salvator_Mundi_(Leonardo)" title="Salvator Mundi (Leonardo)">Salvator Mundi</a>.<a href="#cite_note-12">[12]</a> This is a form of using algorithms in art. By examining the works of artists in the past, from the Renaissance and Islamic Golden Age, a pattern of mathematical patterns, geometric principles and natural numbers emerges.

<div class="mw-heading mw-heading2"><h2 id="Role_of_the_algorithm">Role of the algorithm</h2></div>

From one point of view, for a work of art to be considered algorithmic art, its creation must include a process based on an <a href="Algorithm" title="Algorithm">algorithm</a> devised by the artist. An artists may also select parameters and interact as the composition is generated. Here, an algorithm is simply a detailed recipe for the design and possibly execution of an artwork, which may include <a href="Source_code" title="Source code">computer code</a>, <a href="Function_(mathematics)" title="Function (mathematics)">functions</a>, <a href="Expression_(mathematics)" title="Expression (mathematics)">expressions</a>, or other input which ultimately determines the form the art will take.<a href="#cite_note-Verostko-7">[7]</a> This input may be <a href="Mathematical" class="mw-redirect" title="Mathematical">mathematical</a>, <a href="Computation" title="Computation">computational</a>, or generative in nature. Inasmuch as algorithms tend to be <a href="Deterministic" class="mw-redirect" title="Deterministic">deterministic</a>, meaning that their repeated execution would always result in the production of identical artworks, some external factor is usually introduced. This can either be a random number generator of some sort, or an external body of data (which can range from recorded heartbeats to frames of a movie.) Some artists also work with organically based gestural input which is then modified by an algorithm. By this definition, <a href="Fractal" title="Fractal">fractals</a> made by a fractal program are not art, as humans are not involved. However, defined differently, algorithmic art can be seen to include fractal art, as well as other varieties such as those using <a href="Genetic_algorithm" title="Genetic algorithm">genetic algorithms</a>. The artist <a href="Kerry_Mitchell" title="Kerry Mitchell">Kerry Mitchell</a> stated in his 1999 Fractal Art Manifesto:<a href="#cite_note-Mitchell2009-13">[13]</a><a href="#cite_note-UCSB-6">[6]</a><a href="#cite_note-14">[14]</a>

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</style><blockquote class="templatequote">Fractal Art is not..Computer(ized) Art, in the sense that the computer does all the work. The work is executed on a computer, but only at the direction of the artist. Turn a computer on and leave it alone for an hour. When you come back, no art will have been generated.<a href="#cite_note-Mitchell2009-13">[13]</a></blockquote>

<div class="mw-heading mw-heading2"><h2 id="Algorists">Algorists</h2></div>

"Algorist" is a term used for <a href="Digital_artist" class="mw-redirect" title="Digital artist">digital artists</a> who create algorithmic art.<a href="#cite_note-Verostko-7">[7]</a> Pioneering algorists include <a href="Vera_Moln%C3%A1r" title="Vera Molnár">Vera Molnár</a>, <a href="D%C3%B3ra_Maurer" title="Dóra Maurer">Dóra Maurer</a> and Gizella Rákóczy.<a href="#cite_note-thp-15">[15]</a>

Algorists formally began correspondence and establishing their identity as artists following a panel titled "Art and Algorithms" at <a href="SIGGRAPH" title="SIGGRAPH">SIGGRAPH</a> in 1995. The co-founders were <a href="Jean-Pierre_H%C3%A9bert" title="Jean-Pierre Hébert">Jean-Pierre Hébert</a> and <a href="Roman_Verostko" title="Roman Verostko">Roman Verostko</a>. Hébert is credited with coining the term and its definition, which is in the form of his own algorithm:<a href="#cite_note-Verostko-7">[7]</a>

<pre>if (creation && object of art && algorithm && one's own algorithm) {

return * an algorist *

} else {

return * not an algorist *

}

</pre>

<div class="mw-heading mw-heading3"><h3 id="Types">Types</h3></div>

Artists can write code that creates <a href="Complexity" title="Complexity">complex</a> and dynamic visual compositions.<a href="#cite_note-fctc-2">[2]</a>

<a href="Cellular_automata" class="mw-redirect" title="Cellular automata">Cellular automata</a> can be used to generate artistic patterns with an appearance of randomness, or to modify images such as photographs by applying a transformation such as the stepping stone rule (to give an impressionist style) repeatedly until the desired artistic effect is achieved.<a href="#cite_note-16">[16]</a> Their use has also been explored in music.<a href="#cite_note-Burraston_Edmonds_2005_pp._165–185-17">[17]</a>

Fractal art consists of varieties of computer-generated <a href="Fractals" class="mw-redirect" title="Fractals">fractals</a> with colouring chosen to give an attractive effect.<a href="#cite_note-fgad-18">[18]</a> Especially in the western world, it is not drawn or painted by hand. It is usually created indirectly with the assistance of <a href="Fractal-generating_software" title="Fractal-generating software">fractal-generating software</a>, <a href="Iteration" title="Iteration">iterating</a> through three phases: setting <a href="Parameter" title="Parameter">parameters</a> of appropriate fractal software; executing the possibly lengthy calculation; and evaluating the product. In some cases, other <a href="Graphics_software" title="Graphics software">graphics programs</a> are used to further modify the images produced. This is called post-processing. Non-fractal imagery may also be integrated into the artwork.<a href="#cite_note-mra-19">[19]</a>

Genetic or evolutionary art makes use of <a href="Genetic_algorithm" title="Genetic algorithm">genetic algorithms</a> to develop images iteratively, selecting at each "generation" according to a rule defined by the artist.<a href="#cite_note-20">[20]</a><a href="#cite_note-21">[21]</a>

Algorithmic art is not only produced by computers. Wendy Chun explains:<a href="#cite_note-Chun2011-22">[22]</a>

<blockquote class="templatequote">Software is unique in its status as metaphor for metaphor itself. As A universal imitator/machine, it encapsulates a logic of general substitutability; a logic of ordering and creative, animating disordering. Joseph Weizenbaum has argued that computers have become metaphors for "effective procedures," that is, for anything that can be solved in a prescribed number of steps, such as gene expression and clerical work.<a href="#cite_note-Chun2011-22">[22]</a></blockquote>

The American artist, <a href="Jack_Ox" title="Jack Ox">Jack Ox</a>, has used algorithms to produce paintings that are <a href="Music_visualization" title="Music visualization">visualizations of music</a> without using a computer. Two examples are visual performances of extant scores, such as <a href="Anton_Bruckner" title="Anton Bruckner">Anton Bruckner</a>'s <a href="Symphony_No._8_(Bruckner)" title="Symphony No. 8 (Bruckner)">Eighth Symphony</a><a href="#cite_note-23">[23]</a><a href="#cite_note-24">[24]</a> and <a href="Kurt_Schwitters" title="Kurt Schwitters">Kurt Schwitters</a>' Ursonate.<a href="#cite_note-25">[25]</a><a href="#cite_note-26">[26]</a> Later, she and her collaborator, Dave Britton, created the 21st Century Virtual Color Organ that does use computer coding and algorithms.<a href="#cite_note-27">[27]</a>

Since 1996 there have been <a href="Ambigram#Ambigram_generators" title="Ambigram">ambigram generators</a> that auto generate ambigrams.<a href="#cite_note-28">[28]</a><a href="#cite_note-29">[29]</a><a href="#cite_note-30">[30]</a>

<div class="mw-heading mw-heading2"><h2 id="Contemporary_views_on_algorithmic_art">Contemporary views on algorithmic art</h2></div>

<div class="mw-heading mw-heading3"><h3 id="The_necessity_of_algorithmic_art">The necessity of algorithmic art</h3></div>

In modern times, humans have witnessed a drastic change in their lives. One such glaring difference is the need for more comfortable and <a href="Aesthetic" class="mw-redirect" title="Aesthetic">aesthetic</a> environment. People have started to show particular interest towards decorating their environment with paintings. While it is not uncommon to see renowned, famous <a href="Oil_painting" title="Oil painting">oil paintings</a> in certain environments, it is still unusual to find such paintings in an ordinary family house. Oil paintings can be costly, even if its a copy of the painting. Thus, many people prefer simulating such paintings.<a href="#cite_note-:1-31">[31]</a> With the emergence of Artificial Intelligence, such simulations have become possible. Artificial intelligence image processors utilize an algorithm and machine learning to produce the images for the user.<a href="#cite_note-:1-31">[31]</a>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Artificial_intelligence_art" class="mw-redirect" title="Artificial intelligence art">Artificial intelligence art</a></div>

<div class="mw-heading mw-heading3"><h3 id="Studies_on_algorithmic_and_generative_art">Studies on algorithmic and generative art</h3></div>

Recent studies and experiments have shown that <a href="Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>, using algorithms and <a href="Machine_learning" title="Machine learning">machine learning</a>, is able to replicate oil paintings. The image look relatively accurate and identical to the original image.<a href="#cite_note-:1-31">[31]</a> Such improvements in algorithmic art and artificial intelligence can make it possible for many people to own renowned paintings, at little to no cost. This could prove to be revolutionary for various environments, especially with the rapid rise in demand for improved aesthetic. Using the algorithm, the simulator can create images with an accuracy of 48.13% to 64.21%, which would be imperceptible to most humans. However, the simulations are not perfect and are bound to error. They can sometimes give inaccurate, extraneous images. Other times, they can completely malfunction and produce a physically impossible image. However, with the emergence of newer technologies and finer algorithms, research are confident that simulations could witness a massive improvement.<a href="#cite_note-:1-31">[31]</a> Other contemporary outlooks on art have focused heavily on making art more interactive. Based on the environment or audience <a href="Feedback" title="Feedback">feedback</a>, the algorithm is fine-tuned to create a more appropriate and appealing output. However, such approaches have been criticized since the artist is not responsible for every detail in the painting. Merely, the artist facilitates the interaction between the algorithm and its environment and adjusts it based on the desired outcome.<a href="#cite_note-32">[32]</a>

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<ul><li><a href="Algorithmic_composition" title="Algorithmic composition">Algorithmic composition</a></li>

<li><a href="Computer-aided_design" title="Computer-aided design">Computer-aided design</a></li>

<li><a href="DeepDream" title="DeepDream">DeepDream</a></li>

<li><a href="Demoscene" title="Demoscene">Demoscene</a></li>

<li><a href="Display_hack" class="mw-redirect" title="Display hack">Display hack</a></li>

<li><a href="Low-complexity_art" title="Low-complexity art">Low-complexity art</a></li>

<li><a href="Infinite_compositions_of_analytic_functions" title="Infinite compositions of analytic functions">Infinite compositions of analytic functions</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><a href="Oliver_Grau" title="Oliver Grau">Oliver Grau</a> (2003). <a rel="nofollow" class="external text" href="http://leonardo.info/isast/leobooks/books/grau.html">Virtual Art: From Illusion to Immersion</a> (MIT Press/Leonardo Book Series). Cambridge, Massachusetts: The MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-262-07241-6</bdi>.</li>

<li>Wands, Bruce (2006). Art of the Digital Age, London: Thames & Hudson. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-500-23817-0</bdi>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Algorithmic_art" class="extiw external" title="commons:Category:Algorithmic art">Algorithmic art</a>.</div></div>

</div>

<li><a rel="nofollow" class="external text" href="http://www.verostko.com/algorithm.html">Algorithmic Art: Composing the Score for Visual Art</a> - Roman Verostko</li>

<li><a rel="nofollow" class="external text" href="http://dada.compart-bremen.de/">Compart - Database of Digital and Algorithmic Art</a></li>

<li><a rel="nofollow" class="external text" href="http://phillipmfeldman.org/art/computer%20art/index.html">Fun with Computer-Generated Art</a></li>

<li><a rel="nofollow" class="external text" href="http://iasl.uni-muenchen.de/links/NAKSe.html">Thomas Dreher: Conceptual Art and Software Art: Notations, Algorithms and Codes</a></li>

<li><a rel="nofollow" class="external text" href="https://makeprojects.com/project/real-time-computer-generated-digital-painting">Real-Time Computer Generated Digital Painting</a></li></ul>

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</style><div id="Mathematics_and_art295" style="font-size:114%;margin:0 4em"><a href="Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algorithm" title="Algorithm">Algorithm</a></li>

<li><a href="Catenary" title="Catenary">Catenary</a></li>

<li><a href="Fractal" title="Fractal">Fractal</a></li>

<li><a href="Golden_ratio" title="Golden ratio">Golden ratio</a></li>

<li><a href="Hyperboloid_structure" title="Hyperboloid structure">Hyperboloid structure</a></li>

<li><a href="Minimal_surface" title="Minimal surface">Minimal surface</a></li>

<li><a href="Paraboloid" title="Paraboloid">Paraboloid</a></li>

<li><a href="Perspective_(graphical)" title="Perspective (graphical)">Perspective</a>

<ul><li><a href="Camera_lucida" title="Camera lucida">Camera lucida</a></li>

<li><a href="Camera_obscura" title="Camera obscura">Camera obscura</a></li></ul></li>

<li><a href="Plastic_ratio" title="Plastic ratio">Plastic ratio</a></li>

<li><a href="Projective_geometry" title="Projective geometry">Projective geometry</a></li>

<li>Proportion

<ul><li><a href="Proportion_(architecture)" title="Proportion (architecture)">Architecture</a></li>

<li><a href="Body_proportions" title="Body proportions">Human</a></li></ul></li>

<li><a href="Symmetry" title="Symmetry">Symmetry</a></li>

<li><a href="Tessellation" title="Tessellation">Tessellation</a></li>

<li><a href="Wallpaper_group" title="Wallpaper group">Wallpaper group</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="9" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Forms</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul>

<li><a href="Anamorphosis" title="Anamorphosis">Anamorphic art</a></li>

<li><a href="Mathematics_and_architecture" title="Mathematics and architecture">Architecture</a>

<ul><li><a href="Geodesic_dome" title="Geodesic dome">Geodesic dome</a></li>

<li><a href="Pyramid" title="Pyramid">Pyramid</a></li>

<li><a href="Vastu_shastra" title="Vastu shastra">Vastu shastra</a></li></ul></li>

<li><a href="Computer_art" title="Computer art">Computer art</a></li>

<li><a href="Mathematics_and_fiber_arts" title="Mathematics and fiber arts">Fiber arts</a></li>

<li><a href="Fractal_art" title="Fractal art">Fractal art</a></li>

<li><a href="Islamic_geometric_patterns" title="Islamic geometric patterns">Islamic geometric patterns</a>

<ul><li><a href="Girih" title="Girih">Girih</a></li>

<li><a href="Muqarnas" title="Muqarnas">Muqarnas</a></li>

<li><a href="Zellij" title="Zellij">Zellij</a></li></ul></li>

<li><a href="Knot" title="Knot">Knotting</a>

<ul><li><a href="Celtic_knot" title="Celtic knot">Celtic knot</a></li>

<li><a href="Croatian_interlace" title="Croatian interlace">Croatian interlace</a></li>

<li><a href="Interlace_(art)" title="Interlace (art)">Interlace</a></li></ul></li>

<li><a href="Music_and_mathematics" title="Music and mathematics">Music</a></li>

<li><a href="Origami" title="Origami">Origami</a>

<ul><li><a href="Mathematics_of_paper_folding" title="Mathematics of paper folding">Mathematics</a></li></ul></li>

<li><a href="Mathematical_sculpture" title="Mathematical sculpture">Sculpture</a></li>

<li><a href="String_art" title="String art">String art</a></li>

<li><a href="String_figure" title="String figure">String figure</a></li>

<li><a href="Tessellation" title="Tessellation">Tiling</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Artworks</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_works_designed_with_the_golden_ratio" title="List of works designed with the golden ratio">List of works designed with the golden ratio</a></li>

<li><a href="Continuum_(sculpture)" title="Continuum (sculpture)">Continuum</a></li>

<li><a href="Mathemalchemy" title="Mathemalchemy">Mathemalchemy</a></li>

<li><a href="Mathematica%3A_A_World_of_Numbers..._and_Beyond" title="Mathematica: A World of Numbers... and Beyond">Mathematica: A World of Numbers... and Beyond</a></li>

<li><a href="Octacube_(sculpture)" title="Octacube (sculpture)">Octacube</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematics_and_architecture" title="Mathematics and architecture">Buildings</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cathedral_of_Saint_Mary_of_the_Assumption_(San_Francisco)" title="Cathedral of Saint Mary of the Assumption (San Francisco)">Cathedral of Saint Mary of the Assumption</a></li>

<li><a href="Hagia_Sophia" title="Hagia Sophia">Hagia Sophia</a></li>

<li><a href="Kresge_Auditorium" title="Kresge Auditorium">Kresge Auditorium</a></li>

<li><a href="Pantheon%2C_Rome" title="Pantheon, Rome">Pantheon</a></li>

<li><a href="Parthenon" title="Parthenon">Parthenon</a></li>

<li><a href="Great_Pyramid_of_Giza" title="Great Pyramid of Giza">Pyramid of Khufu</a></li>

<li><a href="Sagrada_Fam%C3%ADlia" title="Sagrada Família">Sagrada Família</a></li>

<li><a href="Sydney_Opera_House" title="Sydney Opera House">Sydney Opera House</a></li>

<li><a href="Taj_Mahal" title="Taj Mahal">Taj Mahal</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_mathematical_artists" title="List of mathematical artists">Artists</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Renaissance</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Paolo_Uccello" title="Paolo Uccello">Paolo Uccello</a></li>

<li><a href="Piero_della_Francesca" title="Piero della Francesca">Piero della Francesca</a></li>

<li><a href="Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a>

<ul><li><a href="Vitruvian_Man" title="Vitruvian Man">Vitruvian Man</a></li></ul></li>

<li><a href="Albrecht_D%C3%BCrer" title="Albrecht Dürer">Albrecht Dürer</a></li>

<li><a href="Parmigianino" title="Parmigianino">Parmigianino</a>

<ul><li><a href="Self-portrait_in_a_Convex_Mirror" class="mw-redirect" title="Self-portrait in a Convex Mirror">Self-portrait in a Convex Mirror</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">19th–20th Century</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="William_Blake" title="William Blake">William Blake</a>

<ul><li><a href="The_Ancient_of_Days" title="The Ancient of Days">The Ancient of Days</a></li>

<li><a href="Newton_(Blake)" title="Newton (Blake)">Newton</a></li></ul></li>

<li><a href="Jean_Metzinger" title="Jean Metzinger">Jean Metzinger</a>

<ul><li><a href="Dancer_in_a_Caf%C3%A9" title="Dancer in a Café">Danseuse au café</a></li>

<li><a href="L'Oiseau_bleu_(Metzinger)" class="mw-redirect" title="L'Oiseau bleu (Metzinger)">L'Oiseau bleu</a></li></ul></li>

<li><a href="Giorgio_de_Chirico" title="Giorgio de Chirico">Giorgio de Chirico</a></li>

<li><a href="M._C._Escher" title="M. C. Escher">M. C. Escher</a>

<ul><li><a href="Circle_Limit_III" title="Circle Limit III">Circle Limit III</a></li>

<li><a href="Print_Gallery_(M._C._Escher)" title="Print Gallery (M. C. Escher)">Print Gallery</a></li>

<li><a href="Relativity_(M._C._Escher)" title="Relativity (M. C. Escher)">Relativity</a></li>

<li><a href="Reptiles_(M._C._Escher)" title="Reptiles (M. C. Escher)">Reptiles</a></li>

<li><a href="Waterfall_(M._C._Escher)" title="Waterfall (M. C. Escher)">Waterfall</a></li></ul></li>

<li><a href="Ren%C3%A9_Magritte" title="René Magritte">René Magritte</a>

<ul><li><a href="The_Human_Condition_(Magritte)" title="The Human Condition (Magritte)">La condition humaine</a></li></ul></li>

<li><a href="Salvador_Dal%C3%AD" title="Salvador Dalí">Salvador Dalí</a>

<ul><li><a href="Crucifixion_(Corpus_Hypercubus)" title="Crucifixion (Corpus Hypercubus)">Crucifixion</a></li>

<li><a href="The_Swallow's_Tail" title="The Swallow's Tail">The Swallow's Tail</a></li></ul></li>

<li><a href="Crockett_Johnson" title="Crockett Johnson">Crockett Johnson</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Contemporary</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Martin_Demaine" title="Martin Demaine">Martin</a> and <a href="Erik_Demaine" title="Erik Demaine">Erik Demaine</a></li>

<li><a href="Scott_Draves" title="Scott Draves">Scott Draves</a></li>

<li><a href="Jan_Dibbets" title="Jan Dibbets">Jan Dibbets</a></li>

<li><a href="John_Ernest" title="John Ernest">John Ernest</a></li>

<li><a href="Helaman_Ferguson" title="Helaman Ferguson">Helaman Ferguson</a></li>

<li><a href="Peter_Forakis" title="Peter Forakis">Peter Forakis</a></li>

<li><a href="Susan_Goldstine" title="Susan Goldstine">Susan Goldstine</a></li>

<li><a href="Bathsheba_Grossman" title="Bathsheba Grossman">Bathsheba Grossman</a></li>

<li><a href="George_W._Hart" title="George W. Hart">George W. Hart</a></li>

<li><a href="Desmond_Paul_Henry" title="Desmond Paul Henry">Desmond Paul Henry</a></li>

<li><a href="Anthony_Hill_(artist)" title="Anthony Hill (artist)">Anthony Hill</a></li>

<li><a href="Charles_Jencks" title="Charles Jencks">Charles Jencks</a>

<ul><li><a href="Garden_of_Cosmic_Speculation" title="Garden of Cosmic Speculation">Garden of Cosmic Speculation</a></li></ul></li>

<li><a href="Andy_Lomas" title="Andy Lomas">Andy Lomas</a></li>

<li><a href="Robert_Longhurst" title="Robert Longhurst">Robert Longhurst</a></li>

<li><a href="Jeanette_McLeod" title="Jeanette McLeod">Jeanette McLeod</a></li>

<li><a href="Hamid_Naderi_Yeganeh" title="Hamid Naderi Yeganeh">Hamid Naderi Yeganeh</a></li>

<li><a href="Istv%C3%A1n_Orosz" title="István Orosz">István Orosz</a></li>

<li><a href="Hinke_Osinga" title="Hinke Osinga">Hinke Osinga</a></li>

<li><a href="Antoine_Pevsner" title="Antoine Pevsner">Antoine Pevsner</a></li>

<li><a href="Tony_Robbin" title="Tony Robbin">Tony Robbin</a></li>

<li><a href="Reza_Sarhangi" class="mw-redirect" title="Reza Sarhangi">Reza Sarhangi</a></li>

<li><a href="Oliver_Sin" title="Oliver Sin">Oliver Sin</a></li>

<li><a href="Hiroshi_Sugimoto" title="Hiroshi Sugimoto">Hiroshi Sugimoto</a></li>

<li><a href="Daina_Taimi%C5%86a" title="Daina Taimiņa">Daina Taimiņa</a></li>

<li><a href="Roman_Verostko" title="Roman Verostko">Roman Verostko</a></li>

<li><a href="Margaret_Wertheim" title="Margaret Wertheim">Margaret Wertheim</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorists</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Ancient</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Polykleitos" title="Polykleitos">Polykleitos</a>

<ul><li>Canon</li></ul></li>

<li><a href="Vitruvius" title="Vitruvius">Vitruvius</a>

<ul><li><a href="De_architectura" title="De architectura">De architectura</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Renaissance</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Filippo_Brunelleschi" title="Filippo Brunelleschi">Filippo Brunelleschi</a></li>

<li><a href="Leon_Battista_Alberti" title="Leon Battista Alberti">Leon Battista Alberti</a>

<ul><li><a href="De_pictura" title="De pictura">De pictura</a></li>

<li><a href="De_re_aedificatoria" title="De re aedificatoria">De re aedificatoria</a></li></ul></li>

<li><a href="Piero_della_Francesca" title="Piero della Francesca">Piero della Francesca</a>

<ul><li><a href="De_prospectiva_pingendi" title="De prospectiva pingendi">De prospectiva pingendi</a></li></ul></li>

<li><a href="Luca_Pacioli" title="Luca Pacioli">Luca Pacioli</a>

<ul><li><a href="Divina_proportione" title="Divina proportione">De divina proportione</a></li></ul></li>

<li><a href="Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a>

<ul><li><a href="A_Treatise_on_Painting" title="A Treatise on Painting">A Treatise on Painting</a></li></ul></li>

<li><a href="Albrecht_D%C3%BCrer" title="Albrecht Dürer">Albrecht Dürer</a>

<ul><li>Vier Bücher von Menschlicher Proportion</li></ul></li>

<li><a href="Sebastiano_Serlio" title="Sebastiano Serlio">Sebastiano Serlio</a>

<ul><li>Regole generali d'architettura</li></ul></li>

<li><a href="Andrea_Palladio" title="Andrea Palladio">Andrea Palladio</a>

<ul><li><a href="I_quattro_libri_dell'architettura" title="I quattro libri dell'architettura">I quattro libri dell'architettura</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Romantic</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Samuel_Colman" title="Samuel Colman">Samuel Colman</a>

<ul><li>Nature's Harmonic Unity</li></ul></li>

<li><a href="Frederik_Macody_Lund" title="Frederik Macody Lund">Frederik Macody Lund</a>

<ul><li>Ad Quadratum</li></ul></li>

<li><a href="Jay_Hambidge" title="Jay Hambidge">Jay Hambidge</a>

<ul><li>The Greek Vase</li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modern</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Owen_Jones_(architect)" title="Owen Jones (architect)">Owen Jones</a>

<ul><li><a href="Owen_Jones_(architect)#The_Grammar_of_Ornament" title="Owen Jones (architect)">The Grammar of Ornament</a></li></ul></li>

<li><a href="Ernest_Hanbury_Hankin" title="Ernest Hanbury Hankin">Ernest Hanbury Hankin</a>

<ul><li>The Drawing of Geometric Patterns in Saracenic Art</li></ul></li>

<li><a href="G._H._Hardy" title="G. H. Hardy">G. H. Hardy</a>

<ul><li><a href="A_Mathematician's_Apology" title="A Mathematician's Apology">A Mathematician's Apology</a></li></ul></li>

<li><a href="George_David_Birkhoff" title="George David Birkhoff">George David Birkhoff</a>

<ul><li>Aesthetic Measure</li></ul></li>

<li><a href="Douglas_Hofstadter" title="Douglas Hofstadter">Douglas Hofstadter</a>

<ul><li><a href="G%C3%B6del%2C_Escher%2C_Bach" title="Gödel, Escher, Bach">Gödel, Escher, Bach</a></li></ul></li>

<li><a href="Nikos_Salingaros" title="Nikos Salingaros">Nikos Salingaros</a>

<ul><li>The 'Life' of a Carpet</li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Publications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Journal_of_Mathematics_and_the_Arts" title="Journal of Mathematics and the Arts">Journal of Mathematics and the Arts</a></li>

<li><a href="Lumen_Naturae" title="Lumen Naturae">Lumen Naturae</a></li>

<li><a href="Making_Mathematics_with_Needlework" title="Making Mathematics with Needlework">Making Mathematics with Needlework</a></li>

<li><a href="Rhythm_of_Structure" title="Rhythm of Structure">Rhythm of Structure</a></li>

<li><a href="Viewpoints%3A_Mathematical_Perspective_and_Fractal_Geometry_in_Art" title="Viewpoints: Mathematical Perspective and Fractal Geometry in Art">Viewpoints: Mathematical Perspective and Fractal Geometry in Art</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Organizations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ars_Mathematica_(organization)" title="Ars Mathematica (organization)">Ars Mathematica</a></li>

<li><a href="The_Bridges_Organization" title="The Bridges Organization">The Bridges Organization</a></li>

<li><a href="European_Society_for_Mathematics_and_the_Arts" title="European Society for Mathematics and the Arts">European Society for Mathematics and the Arts</a></li>

<li><a href="Goudreau_Museum_of_Mathematics_in_Art_and_Science" title="Goudreau Museum of Mathematics in Art and Science">Goudreau Museum of Mathematics in Art and Science</a></li>

<li><a href="Institute_For_Figuring" title="Institute For Figuring">Institute For Figuring</a></li>

<li><a href="Mathemalchemy" title="Mathemalchemy">Mathemalchemy</a></li>

<li><a href="National_Museum_of_Mathematics" title="National Museum of Mathematics">National Museum of Mathematics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Droste_effect" title="Droste effect">Droste effect</a></li>

<li><a href="Mathematical_beauty" title="Mathematical beauty">Mathematical beauty</a></li>

<li><a href="Patterns_in_nature" title="Patterns in nature">Patterns in nature</a></li>

<li><a href="Sacred_geometry" title="Sacred geometry">Sacred geometry</a></li></ul>

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<ul><li> Category</li></ul>

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