THE Algoristsby Roman Verostko © 2012

THE ALGORISTS http://www.algorists.org/algorist.html

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Above: Jean Pierre Hebert, a distinguished founding member of the algorists, with one of his recent "Bright Wavelet" digital drawings, 2008 (38" by 77"). www.jeanpierrehebert.com photo by roman, 2012

Often I am asked "Who are the algorists?" Simply put, algorists are artists who create art using algorithmic procedures that include their own algorithms. This page presents an account of the origin of this usage and the algorithm serving as the "algorist manifesto".

In the latter half of the 20th Century, with the growth in information science and digital technologies the use of algorithmic procedure spread far beyond the dreams of its earliest practitioners. We have, among the living in 2012, a number of pioneer algorists whose vision and work with programming pointed the way that permeates the world of art today. Jean Pierre Hebert, who wrote the algorithm serving as a manifesto of the "algorists", is shown above with one of his drawings that speaks eloquently of algorithmic form-generation. Pictured below are four algorist pioneers who were present in the context of a display of digital pioneer works at the Victoria & Albert in London, 2010.

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Left to Right: Mark Wilson, Manfred Mohr, Roman Verostko & Frieder Nake: Four algorist pioneers at the Victoria & Albert, Museum, Jan 5, 2010. Photo by Douglas Dodds in the context of a V & A display of their work.

While not present for the photo above, celebrated pioneers like Herbert Franke, Vera Molnar, Charles Csuri, and Harold Cohen (shown below), were well represented in the Digital Pioneers (Note 8.) .

Herbert Franke Vera Molnar, 1985, archives of Charles Csuri on occasion of Harold Cohen From the film: Age of photo by Andreas Hübner Frieder Nake's comp-art data base. his SIGGRAPH Lifetime Achieve- the Intelligent Machines ment Award, 2011,.Vancouver, Canada.

Historically the term "algorithm" refers to the step by step mathematical procedure for carrying out a specific calculating task (see note 3) . The detailed instruction directing a drawing machine on how to draw a visual form is also an algorithm. Today, more broadly, a composer's score for musical form and a choreographer's notations for dance may also be viewed as algorithms. Such notations, similar to computer software formats, are detailed procedures for carrying out a task.

This Seventeenth Century keyboard music, is thought to be in the hand of the great keyboard composer, Girolamo Frescobaldi (1583-1643). The Vatican folios shown (23 & 24) display the opening from his Fourth Toccata. The score may be viewed as an algorithm, expressive both in structure and in the writing hand of the cvomposer. © Vatican Library MS: Chig. Q. IV 29 fols. 23 verso-24 recto music17 NB.40. Image source: http://www.ibiblio.org/expo/vatican.exhibit

Algorithmic art may be found throughout history, from prehistoric basket weaving to geometric and conceptual art in the 20th Century. For more on usage of the term algorithm see my notes on algorithmic art. With the advent of computers the possibilities for generating visual form with more complex algorithmic procedure attracted some visual and sound artists to explore these avenues for creating their art.

In the last quarter of the 20th Century I was one of a like minded group of artists who undertook to write instructions for executing our art. On occasion I referred to my art as "writing the score for drawing". "Computer art" was the common term applied to all art associated with computers We felt a need to identify the nature of algorithmic art. Some of us had been working with algorithmic

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procedure for about a quarter century before our 1995 declaration as "algorists". The 1995 manifesto of the "algorists" was not a declaration of something new; rather, it was the declaration of an artistic practice that had already brought radical change and would continue to change the way we would create art in the 21st Century. This practice also had a valuable presence in sculpture and music. Helaman Ferguson would be a sculptor whose algorist practice celebrated an aesthetic rooted in mathematics. His "Four Canoes", shown below, is an excellent example with links to information on the procedure for achieving the form of the tori, the pedestal, and the platform.

Helaman Ferguson

"Four Canoes", , 1997, University of St Thomas Science and Engineering Center, St. Paul, MN, USA. This impressive algorithmic presence features two interlocked six foot granite tori mounted on a platform created with hexagonal tiling. Note 6

Photo Credit: RV

Yet algorist art should not be confused with mathematics. Algorithmic drawings, like my "Green Cloud" shown below, evolved from my passion, as a painter, for the marriage of spontaneous brushwork and studied arrangement. With elementary programming abilities I explored the same goals I had set for myself as a painter. Clearly programming and mathematics do not create art. Programming is a tool that serves the the vision and passion of the artist who creates the procedure.

*Green Cloud (the drawing)" by Roman Verostko. Pen & ink plotter drawing. Three Story Drawing Machine, north wall, MCAD, Nothern Spark, "White Night", ca. Drawing process presented as a Three Story Drawing Machine, June 2011 at the 4:30 AM, June 5, 2011, nearing the end of an 8 hour, all night drawing session. Minneapolis College of Art & Design (MCAD).

LOOKING BACK

Before looking back let me note that several pioneer algorists have carried their form explorations

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well into the 21st Century. Of these Manfred Mohr and Harold Cohen stand out as two of those master pioneers who were the first to achieve mature algorithmic styles. Manfred's "Klangfarben" series demonstrates the power of algorithmic procedure in the hands of a master. Manfred has labored for over 40 years creating visual tension fields as an art of pure visual form. In this series we experience algorithmic form generation in real time.

Manfred Mohr, "Klangfarben" series (2006-2007) Harold Cohen, Starting Over, 2011, oil painting, 50 by 80 inches from Bitforms Gallery on Vimeo. © Manfred Mohr

"Aaron", a personification of Harold Cohen's software, represents 40 years of work on creating an intelligent artist. Aaron's drawings and paintings, grew from Harold's earlier experience as a painter. Consequently "Aaron's" code appears to yield a mysterious relationship to Harold as the artist-painter in his earlier years.

Historically how did we get here?

In the earlier days of computing there were no software tools for artists. In the 1960's a few artists like Manfred Mohr saw the "form-generating" power of computers as an opportunity for their art. Software and technical procedures for visualization grew hand in hand with hardware. Artists engaging new computing and visualizing technologies had to either collaborate with engineers for programming their ideas or else create their own programs (algorithms).

Manfred Mohr, New York

P-021/A, "band-structure", ink/paper, 1969, 50cm x 50cm Early plotter drawing in a series achieved with algorithmic procedures that included rules he viewed as "aesthetical-filters"

P702F, Endurachrome, 2000. canvas, 76cm x 100cm. This work belongs to the artist's space/color work phase that followed about 30 years of his work in monochrome. Mohr's work with coded procedure, steadfast for over a period of 43 years, demonstrates the power of algorithmic procedure. Yet the term algorist was not introduced until 26 years after his first algorithmic work.

http://www.emohr.com/

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Those artists who first experimented writing coded artistic procedures for computers included Herbert Frank, Manfred Mohr, Frieder Nake, Georg Nees, Vera Molnar, and Edward Zajec. Artists who used computers in the art-making process were often called "computer artists". Frieder Nake, a pioneer algorist, has, in recent years, served as the chief investigator documenting first generation "Digital Art" at the University of Bremen in Germany. This "CompArt" data bank provides an excellent source for perusing the breadth and depth of early algorithmic art. From the 1970's up to the early 1990's this work was generally referred to as "computer art", a term that became the umbrella for any kind of art associated with computers.

A.Michael Noll, USA, Computer composition with lines, 1964. This drawing was meant to mime procedures similar to those employed by Mondrian for his "Composition with lines".

For sources see Note 7

Hardware: IBM 7094 computer and a General Dynamics SC-4020 microfilm plotter.

Frieder Nake, Germany

Nr. 2 Homage à Paul Klee Dated: September 13, 1965 Program: COMPART ER 56. Drawn with Zuse-Graphomat Z 64 Size: 50 × 50 cm Signed lower left: Nake/ER56/Z64

Produced at the Technische Hochschule Stuttgart

Artwork Type: serigraphic print edition after plotter drawing. Ink on paper. Collections include: Museum of Contemporary Art, Zagreb V&A Collection, London. ©

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Vera Molnar, Quadrilaterals, pen & ink, 1988 Herbert Franke Electronic Graphics,1961-62.. , 30 by 35 cm., Coll Szollosi-Nagy-Nemes

Harold Cohen, 1999 Aaron's Garden, pen and ink drawing plotted by Aaron in 1989. "Aaron" is a "personal expert system" that Harold Cohen developed for generating art. "Aaron" consists of Harold's coded program of generative drawing procedures and a flat bed pen plotter. Aaron's drawings of two figures in a garden recall the tradition of representing Adam and Eve in the Garden of Eden.

Above: Edward Zajec ram2/6 plotter drawing, 1969

left: Georg Nees, «Schotter». " a computer graphic from the 1960s, produced by a structured operation by random generators that lead to the discovery of new images. This graphic visually displays the relationship between order and disorder, and the effects of change." Georg Nees

Venues for exhibiting "computer art" emerged in the 1980's when various symposia and conferences, both national and international, began including exhibitions and papers related to the use of computing procedures in the arts. Leonardo, the quarterly on art science and technology, played an important role for emerging algorithmic art by including essays and documentation of

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exhibitions for ISEA, the Digital Salon, and SIGGRAPH . Ars Electronica, with its annual symposium, catalogues and awards provided us with papers and examples with a growing community of practitioners. (See note 1)

By the mid to later 1980's a fair number of artists using original algorithms had achieved distinctive algorithmic styles, each with a body of mature work. Working independently of each other, we found, when we did meet, that we shared many similar experiences. Furthermore we could be helpful to each other by establishing our unique identity. While I had little contact with Manfred Mohr, Frieder Nake, and Vera Molnar I was aware of their work and shared some of the same exhibiting venues.

ALGORITHMIC SEDUCTION

Detail: Flowers of Learning, Madame Curie. Algorithmic pen ink drawing, one of seven in a 25 foot permanent installation, Spalding University, Academic Learning Center, Louiville, Kentucky, USA, Roman Verostko, 2006. PDF Catalogue: Flowers...

Jean Pierre Hebert, 1999 Santa Barbara, CA

Artist's coded procedure from 100 views of a megagon.

print 8" by 8" image on paper, Somerset Book, measuring 12.75" by 19".

Peter Beyls, 1988 Ghent, Belgium

Untitled algorithmic pen plotter drawing tinted with watercolor.

11.5" by 16.5" Artist's software Hardware: Symbolics 3600 & HP plotter

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Stephen Bell, 1977-79. Detail of one of a drawing created with his ranstack algorithm.

The ranstak programs were the first programs developed by Bell as a postgraduate student at the Slade School of Art in 1977-79 in the Department of Experimental & Electronic Art.

http://stephenbell.org.uk/

In Minneapolis in 1993 Peter Beyls and I discussed forming a panel on algorithmic art for the 1994 Helsinki ISEA. Peter chaired the Helsinki panel and planned a similar panel on "Artists and Algorithms" for the 1995 SIGGRAPH conference in Los Angeles, USA. This 1995 panel, chaired by Peter Beyls (Belgium), included Jean Pierre Hebert (US), Brian Evans (US), Steve Bell (UK), Ken Musgrave (US) and myself.

In an informal casual meeting sometime after the panel, Jean Pierre, Ken Musgrave and I briefly discussed forming an informal group of artists who shared similar interests in algorithmic art. Jean Pierre, Peter Beyls and I had already been sharing experiences for several years.

Following the conference we corresponded in search of a term and possible ways to share views. During that period we invited Helaman Ferguson to join in this exchange. It was JPH who proposed using the term "algorist". Noting that algorists "ought to be defined with an algorithm", he also wrote an algorithm identifying an algorist as one who uses one's own algorithms for creating art objects. Jean Pierre's name for the group and his algorithm were both brilliant contributions serving as an identity and a manifesto. The Hebert algorithm, as quoted below, dates from correspondence in September 1995:

if (creation && object of art && algorithm && one's own algorithm) { include * an algorist * } elseif (!creation || !object of art || !algorithm || !one's own algorithm) { exclude * not an algorist * }

So artists who create an object of art with a process that includes their own algorithms are identified as algorists.

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Above: Brian Evans, A visualization (a mapping) of Ken Musgrave, USA, Inkjet on rag paper . Beethoven’s Ode to Joy, date unknown. see note 5 4" of 5 ', ca. 1995 Slickrock III, 11.5" by 20.5" see: www.brianevans.net © Brian Evans © k.musgrave

Hans Dehlinger, Kassel, Germany

D162 "Turm", 1993 Paper: 11.75" by 16.5"

Pen & ink. algorithmic pen plotted drawing.

Private Collection

As artists employing algorithmic procedures the term algorist fit our interests well. In general we are fairly agreed that algorists are artists who include original algorithmic procedures in the course of creating their work. The use of algorithms in and of itself does not constitute algorist work. It is the inclusion of one's own algorithms that makes the difference.

As algorists with or without computers, we all employ algorithms created by colleagues and predecessors. The algorithms for geometry and perspective come to mind. We depend on algorithms others have created for everything from the circuit logic and operating system of our computers to the computer languages and the editors that we use.

The algorist goes one step further by introducing original algorithms for generating "art". The jewel of algorist art lies in the artist's own "form-generating algorithms", the artist's unique procedures for creating the form. From this perspective Mozart's score for a Sonata, the architect's plan for the building, and Hans Dehlinger's code for a drawing are all "form-generating" algorithms.

Algorithms have been used variously in the arts for centuries. And the procedures were carried out manually without the use of machines. Clearly an algorist may also practice her art without the use of computers. That has been the case in the work of Channa Horwitz whose algorist work is executed entirely by hand. Channa is one of the four algorists in the first show that Jean Pierre Hebert curated at the Kavli Institute for Theoretical Physics in Santa Barbara, a show noted at the end of this essay.

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Chanah Horwitz, "Canon # 10", 1983, Plaka on Mylar, 23-1/2 x 26-1/2 inch. © C Horwitz

Finally, one aspect of the algorist definition remains open to interpretation. An algorist, by definition, creates "art". The definition does not identify what constitutes "art". One might employ original algorithms and create work that one person may experience as art while another may not experience it as art. For example a mathematician may use original algorithms in creating a scientific visualization that some may view as "art". Yet the visualization may not fit another's conception of "art". While we may have our various preferences the definition cannot and, in my view, should not attempt to define what kind of work belongs in the world of the art we choose to cultivate.

BREADTH OF ALGORIST ART

This account has outlined the introduction of the term "algorist" by only a few of us in 1995 for a procedure being practiced by ourselves and others for many years. We were only a few artists within a limited circle identifying a practice employed by a much broader group of artists. I came to view our algorist identity as embracing algorists rather widely with a history yet to be written. By 1995 algorithmic procedure, at one level or another, had already had an impact on the way we did animated film, architecture, sculpture, music, dance, environmental art and a wide range of new experimental forms of art. Note 4.

Paul Brown, 1997, UK, Australia Gymnasts, Print, 60 X 80 cm

This essay can only suggest the breadth of this practice. For example let me point to the important

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work of Paul Brown, an algorist whose writings and newsletters kept us well informed for years, even before we had the web. Dividing his living between London and Australia he identifies with those circles in the UK involved with the theory and practice of generative art. This includes Ernest Edmonds whom I first met in 1996 at his annual conference on "Creativity and Cognition" at Loughborough. As an algorist, deeply involved with the nature and practice of generative art, his writing, teaching and art work has been felt widely by colleagues and students.

Ernest Edmonds, Four Shaping Forms, 2009, Acrylic and digital print on canvas, 50x50 cm each. © E Edmonds

Yoshiyuki Abe, Japan

Stochastica Crossmodulation 2000, prints

< Image 75

Image 97 ►

Algorists who had been active since the 1970's had no common identity and therefore no web site or historical reference to their work specifically. In 1996 J.P. Hebert introduced several pages dedicated to the algorists. These pages spelled out essential features of "algorist" art, historical precedents, and links to related topics and issues. These initial pages identified an informal group of active algorists including Yoshiyuke Abbe, Harold Cohen, Charles Csuri, Hans Dehlinger, Helaman Ferguson, Manfred Mohr, Vera Molnar, Mark Wilson, and others.

Mark Wilson USA, "12D90", 1990 plotter drawing

36 x 48 inches (91 x 121 cm.), pen & ink on paper. Private collection.

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Vlatko Čerić, 2005, Zagreb, Croatia Evolution #5 of 10, digital print, 6.7" by 6.7" Hahnemuhle paper measuring 11.7" by 16.5" .

Vlatko Čerić is one of many algorists, who employ generative procedures and demonstrate the breadth and depth of algorist artistic achievement.

©Vlatco Ceric

Many first generation algorists worked with pen plotters as their first graphic interface. Some shared a common history that grew out of conceptual art and an interest in the emerging information processing technologies of the 1960's. Put in perspective the work of first generation algorists marked a turning point in the history of western culture. Algorithmic procedures have changed everything from the way we create film to the way we run wall street, the military and our communications systems.

"Paul", a robotic arm drawing my portrait on Sept 28, 2011, at the University of London, Goldsmiths. This robotic arm, "Paul", is the work of Patrick Tresset in "The Aikon Project". His research uses computational and robotic technologies to explore drawing activity with a focus on face sketching.

See: http://www.aikon-gold.com/

The numerous sites and groups involved with algorist art at the turn of the century testify to a remarkable growth of coded procedures in virtually all the arts. Other terms may be used but they all share one thing in common - algorithmic procedure. This holds for generative art, genetic art, bio arts and a great deal of the art mounted in cyberspace, as well as a wide range of interactive and robotic work. As this 21st Century unfolds algorist art emerges as a major force in the creative arts.

left to right: Hans Dehlinger, Jean Pierre Hebert, Channa Horwitz, Roman Verostko

Above: work shown in " The Algorists: four visual artists in the land of Newton", curated by Hebert at the Kavli Institute for Theoretical Physics, University of California, Santa Barbara, USA. March 13-June 30, 2006. Click here for catalogue

While algorists themselves presented their own shows and participated in group shows it was Jean

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Pierre Hebert who curated the first exhibition formally identified as "algorist". This exhibition was presented at the Kavli Institute for Theoretical Physics in 2006 (University of California at Santa Barbara). The catalogue for this show provides an excellent introduction to the nature of algorithmic practice with specific examples of algorithmic drawings by four artists: The Algorists: four visual artists in the land of Newton".

Artists in first algorist show: Roman Verostko, Channa Horwitz, Jean Pierre Hebert, Hans Dehlinger This picture was taken in 2005 in Channa Horwtiz' studio, Santa Monica CA., USA.

Note 1a. algorism & algorithm

algorithm: Early usage in the English language appears variously as augrime, augryme, algorisme, algarism and variations with latinizing influence as in algorismus. In English the term algorism appears in 17th Century writing as algorithm which is commonly used today. The transformation to algorithm may have been influenced by classical learning since the Greek term for number is ’arithmós (’αριθμός) the root for our English term arithmetic.

algorism: The term algorism most probably descended from the name of an Arabian mathematician active in Bagdad around 820 AD. This mathematician, Abu Ja'far Mohammed Ben Musa, a native of Khwarasm, surnamed al-Khowarazmi, wrote treatises on Hindu arithmetic and algebra. The title of one of his works, al-jabrawa al-maqàbala, is taken as the source for the term algebra. It is also believed that his name, al-Khowarazmi is the source for the term algorism.

Note 1. By the late 1980's the established symposia and exhibition venues that were known to me included: The Inter-Society for Electronic Art (ISEA), SIGGRAPH , Ars Electronica and the semi-annual Komputerkunst and Golden Plotter Awards, in Germany (Gladbeck). The annual Small Computers in the Arts conference (Philadelphia, 1980 ff) was also an important venue in the U.S. recognizing the impact of the PC for individual artists. At all of these conferences artists could see the work of others and share mutual concerns. The exhibitions, papers, panels, and publications of these venues provided an overview of what was generally called "computer art" but there was no single venue for specifically "algorist" work. There were many others. Let me identify several that had a broad impact in the early years:

SIGGRAPH , the Special Interest group for GRAPHics of the Association for Computer Machinery, ACM, played an important role both nationally and internationally. Its international exhibitions of art and technology with its workshops, lectures and awards provided both learning and exhibition opportunities that served us well. Many local SIGGRAPH chapter groups, including Minneapolis, met on a regular basis and held regional media exhibitions. http://www.siggraph.org

ISEA The Internationational Symposium on Electronic Art. This symposium grew out of the special interest of artists and composers from a broad spectrum of new media that embraced emerging digital technologies. More specialized than SIGGRAPH for those immersed in the arts it also provided workshops, lectures and exhibition venues. First held in Utrecht in 1988 it has enjoyed a healthy growth and remains a premier venue for those involved in electronic arts. www.isea-web.org (For my role as founding member (1988), active board member in the 1990's and Director in 1993 see: http://www.verostko.com/isea/roman-isea.html

ARS ELECTRONICA. Founded in 1979 as a Festival for Arts ,Technology and Scociety, it has served as a premier center of the electronic arts with its exhibitions, publications and prestigious award, the Prix Ars Elecrtronica. Located in Linz, Austria the AQrs Electronia Center in cludes a Museum of the Future and the Ars Electronica Futurelab. I especially appreciated their well documented publications and the invitation to exhibit in "Code the Language of Our Time".

Komputerkunst, Gladbeck, Germany. (more notes to be added later)

Symposium: Art & Algorithm - Mind & Machine, 1991

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Roman Verostko, The Magic Hand of Chance, 1982. Photo of monitor, frame from an algorithmic sequence,

To address algorithmic procedure in the arts I organized a small symposium at the Minneapolis College of Art and Design in 1991 (February 23/24). This symposium Art & Algorithm - Mind & Machine, included an audio visual show, "Images of the Unseen From the Worlds of Art & Science". The presentation of video, sound tracks and slides included the work of 23 artists and scientists from 6 countries.

Algorithmic drawing Vera Molnar 74.338/14.29.00 c.1990

Courtesy of the artist for the 1991 symposium on Art & algorithm... .

©v.molnar

Artists and scientists whose work was shown included: Stephen C.G.Bell (UK), Donna Cox (US), Charlotte Davies (Canada), Hans Dehlinger (Germany), Helamen Ferguson (US), Samia A. Halaby (US), Bruce Hamilton (US), Jean Pierre Hebert (US), Yoichiro Kawaguchi (Japan), William Latham (UK), Vera Molnar (France), Jim Otis (US), Clifford Pickover (US), Jeffrey Ventrella (US), Mark Wilson (US), Toshifumi Kawahara (Japan).<

Note 2. My paper on algorithmic art included examples of work by several algorists with a note pointing to the JPH work on display. Our works were shown on opposite walls in the same Sidney gallery and I was especially impressed with the new work JPH had sent. In another gallery Brian Evans had presented a very impressive show of algorithmic work that included sound and light works. The ISEA show traveled to Melbourne and the Hebert's large triptych was never returned. His triptych, as I recall now, was an outstanding work excelling anything I had ever seen in algorithmic pen and ink work. All efforts to recover the work have failed and I view this as a loss to the history of 20th Century algorist art. <

Note 3. The term algorism most probably descended from the name of an Arabian mathematician who was active around 820 AD in Baghdad. It is believed that his surname, al-Khowarazmi is the source for the term algorism. The use of the term algorism appeared with various spellings in several languages and often with latinizing influence as in algorismus. In English the term algorism came to be replaced with the term algorithm which is more commonly used today. So one who employs algorithms (algorisms) would be an algorist. <

Note 4. The use of the term "algorist" to identify a specific group must be understood in the context of its use. Historically it applied to those who used algorithms for solving mathematical problems. In the late 20th Century we used it to identify the growing number of artists who write algorithms in computer languages for implementing their art.

The term itself has undergone interesting usage in the history of mathematics. In the late Middle Ages controversies arose between the "algorists" and "abacists". The algorists, having adapted algebra and Arabic numbering, found resistance among the abacists with their classical numbering system and Euclidean Geometry. So the "algorists" were mathematicians who embraced calculating procedures with the arabic numbering system.

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Patric Prince (Art Historian) poses with 3 algorists in San Diego, August, 2003. Patric Prince collected works by many early digital artists.

Left to Right: Hans Dehlinger, Patric Prince, Jean Pierre Hebert, Roman Verostko

Algorist art, in the most proper sense, belongs to our own unique time in history. We have, in the view of Peter Weibel, undergone an algorithmic revolution, a revolution that has transformed the worlds of commerce and politics as well as arts & culture. And the first generation of artist algorists were pioneers in applying procedures that are now changing world culture. <

Note 5. Quote here is from Brian Evan's paper Numbers to Neurons: Digital Synaesthesi from the PDF version at: http://www.brianevans.net/

Evan Brian presents this pseudo code excerpt (the first two bars) of Beethoven's Ode to Joy for his visualization.

calc_tune(){ int xcnt, ycnt; float radx, rady;

for (xcnt = 0; xcnt < XDIM; ++xcnt) { radx = TWO_PI*((xcnt*1.0)/(XDIM*1.0)); curr_x = (1.0)*cos(radx*(1.25)+(1.0/64.0)*TWO_PI) +

(1.0)*cos(radx*(1.25)+(2.0/64.0)*TWO_PI) + (1.0)*cos(radx*(1.33333)+(3.0/64.0)*TWO_PI) + (1.0)*cos(radx*(1.5)+(4.0/64.0)*TWO_PI) + (1.0)*cos(radx*(1.5)+(5.0/64.0)*TWO_PI) + (1.0)*cos(radx*(1.33333)+(6.0/64.0)*TWO_PI) + (1.0)*cos(radx*(1.25)+(7.0/64.0)*TWO_PI) + (1.0)*cos(radx*(1.125)+(8.0/64.0)*TWO_PI) + . . . etc... <

Note 6. See: http://www.stthomas.edu/mathematics/fourcanoes.html Return to Illustration

Note 7. See Frank Dietrich's account of this period: "Visual Intelligence: The First Decade of Computer Art (1965-1975)", in IEEE Computer Graphics and Applications, Vol.5, No.7, pp 34-35, July 1985, Reprinted in Leonardo Vol.19, No.2, pp.159-69, 1986. (PDF copy at: https://design.osu.edu/carlson/history/PDFs/dietrich-leonardo.pdf)

For an early account published with a landmark exhibition of computer art see Cynthia Goodman, Digital Visions, Computers & Art, Everson Museum of Art, Syracuse, Harry N. Abrams, Inc., NY, 1987. <

Note 8. V&A Pattern: Digital Pioneers, V&A Publishing, Victoria & Albert Museum, 2009. ISBN 9781851775873. <

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