Grids – Institute of Artificial Art Amsterdam

What is an image?

This course is concerned with image-generation algorithms, in particular with image generation algorithms carried out by digital computers. Such algorithms must be based on some formal conception of what an image is — they must be inscribed within a mathematical definition of the set of all possible images.


Shapes. Mainstream mathematics assumes a reductionist ontology about geometry: it treats lines and planes as sets of points –– where a point is a position in a given n-dimensional space. A point may thus be represented as an n-tuple of numbers: the Cartesian coordinates which specify its position. A point in a two-dimensional plane, for instance, is described by a pair, consisting of its x-coordinate and its y-coordinate.

Colors. In black & white graphics, color is a boolean: yes/no, on/off, 1/0, black/white. The scale of grey-tones is a one-dimensional continuum (for instance: the segment of the real numbers between 0 and 1). The space of all colors is a three-dimensional continuum. One can make different choices about which dimensions to use. E.g.: RGB, CYM, HSB.

An image is thus a mapping from pairs of real numbers (in a given interval) to colors (e.g., booleans, real numbers, or triples of real numbers).


This mathematical point of view does not translate into a practical convention for finite computational representations, however. The mathematical correlate of a visible shape or line-segment is a set of uncountably many points, and the coordinates of most of these points are not integers, but irrational numbers which cannot be represented in a finite way.

In the design of image-generation algorithms, the representations to be used can thus not be derived from mathematical first principles. No image-generation algorithm can explore the space of all possible images if we understand that notion in its mathematical sense. Any algorithm must be based on a definition of a space of finitely representable images; specifying this definition is in some sense an arbitrary (in other words, artistic) decision.

Different “schools” in algorithmic art make different decisions about this. We discuss these different approaches in separate chapters of this course.

We distinguish:

  •          Grid-based algorithms, which work with a two-dimensional array of cells.
  •          Scatter-algorithms, which map bags of objects onto positions on the plane.
  •          Partition-algorithms, which split a 2D object into parts.
  •          Turtle-graphics algorithms, which draw sequences of line-segments in terms of a “local” coordinate-system.
  •          Movement simulation algorithms, which draw lines according to a continuous dynamics.
  •          Synthetic systems, which combine one or more of these approaches

We first discuss algorithms based on grids.


The Grid

The conception of an image as a uniform m x n grid of colored squares is derived from the mathematical image definition in a simple, direct way. It is the discrete version of the mathematical model, which makes the image representations finite, and makes the set of image representations enumerable (if we also assume a similar discretization of the colour space). The grid therefore inherits the feel of “objectivity” of the mathematical research tradition. It is also technologically convenient, since monitors and printers are based on the same idea: m x n “pixels”.

Correlarium 1: Hoe bouw ik een rechthoekig vlak op uit deelvlakjes van dezelfde vorm?
Correlarium 2: Wat is de dichtste pakking van gelijke rechthoekige elementen?

Note, however, that the grid does not inherit the mathematical properties of the mathematical image definition. Discretization is not an innocent operation. The maxim that “all directions have equal rights”, which is characteristic for Euclidean space, is no longer valid: two orthogonal directions (“horizontal” and “vertical”) get preferential treatment. As a result, the notions of distance and neighbourhood change beyond recognition, and the Euclidean notions of translation and rotation get largely lost.

The grid does not instantiate the mathematical image conception –– it denotes it.

Cf. Lev Manovich: The Engineering of Vision from Constructivism to Computers, Chapter I (“Visual Atomism”).


Piet Mondriaan: Compositie
‘dambord’ met lichte kleuren, 1919.

The grid in early twentieth-century art

Sophie Taeuber: Sans titre, ca. 1916-1918; Composition verticale horizontale,ca. 1916-1918.
Jean Arp & Sophie Taeuber: Sans titre, 1918.
Jean Arp: Papiers coupés au massicot, 1918.

Arp 1886-1966. Exhibition Catalogue Württembergischer Kunstverein, Stuttgart, 1986, pp. 52-55.

Vilmos Huszár: Compositie, 1918.

Sjarel Ex: “Vilmos Huszár.” In: De beginjaren van De Stijl 1917-1922. Utrecht: Reflex, 1982, pp. 83-124.

Piet Mondriaan: Losengique met grijze lijnen, 1918; Losengique met vlakken in oker en grijs, 1919; Losengique met kleurvlakken, 1919; Compositie ‘dambord’ met lichte kleuren, 1919; Compositie ‘dambord’ met donkere kleuren, 1919.

Els Hoek: “Mondriaan”. In: De beginjaren van De Stijl 1917-1922. Utrecht: Reflex, 1982, pp. 47-82.

Georges Vantongerloo: Étude, 1920.

Georges Vantongerloo 1886-1965. Exhibition Catalogue Musées Royaux des Beaux-Arts de Belgique, Brussels, 1981, p. 44.

“Hij zet nu met zijn Belgisch intellect een hulpstelsel op touw dat volgens mij op de natuur gebaseerd is. (…) Met zijn gewone bewustzijn zit hij alles uit te rekenen. (…) Hij gaat te werk als een (gewone) theosoof.” [Mondriaan writing to Van Doesburg about Vantongerloo, September 5, 1920.] Cf. Nicolette Gast: “Georges Vantongerloo.” In: De beginjaren van De Stijl 1917-1922. Utrecht: Reflex, 1982, pp. 233-261.

Paul Klee: Einst dem Grau der Nacht enttaucht …, 1918; Alter Klang, 1925; Farbtafel auf maiorem Grau, 1930; Individualisierte Höhenmessung der Lagen, 1930; Paneelschildering QU I, 1930; Polyphonie, 1932.

Jürg Spiller: Paul Klee. Utrecht: Bruna, 1962, pp. 61/83.
Norbert Lynton: Klee. London: Hamlyn, 1964.

Ellsworth Kelly: Linear Color Sequences (1952-1955)

Yve-Alain Bois: Ellsworth Kelly: The Early Drawings, 1948-1955.Cambridge, Mass.: Harvard University Art Museums, 1999, # 122-158, 168-177.

Ellsworth Kelly: 3 X 3 Color Grids with Borders (1953-1954)

Yve-Alain Bois: Ellsworth Kelly: The Early Drawings, 1948-1955.Cambridge, Mass.: Harvard University Art Museums, 1999, # 159-167.


“In the early part of this century there began to appear, first in France and then in Russia and in Holland, a structure that has remained emblematic of the modernist ambition in the visual arts ever since. Surfacing in pre-War cubist painting and subsequently becoming ever more stringent and manifest, the grid announces, among other things, modern art’s will to silence, its hostility to literature, to narrative, to discourse.”

“It is not just the sheer number of careers that have been devoted to the exploration of the grid that is impressive, but the fact that never could exploration have chosen less fertile ground. As the experience of Mondrian amply demonstrates, development is precisely what the grid resists. But no one seems to have been deterred by that example, and modernist practice continues to generate ever more instances of grids.”

“In the spatial sense, the grid states the autonomy of the realm of art. Flattened, geometricized, ordered, it is antinatural, antimimetic, antireal. It is what art looks like when it turns its back to nature. In the flatness that results from its coordinates, the grid is the means of crowding out the dimensions of the real and replacing them with the lateral result not of imitation, but of aesthetic decree. Insofar as its order is that of pure relationship, the grid is a way of abrogating the claims of natural objects to have an order particular to themselves; the relationships in the aesthetic field are shown by the grid to be in a world apart and, with respect to natural objects, to be both prior and final.”

“. . . the bottom line of the grid is a naked and determined materialism. But (. . .) that is not the way that artists have ever discussed it. (. . .) Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit.”

“The grid’s mythic power is that it makes us able to think we are dealing with materialism (or sometimes science, or logic) while at the same time it provides us with a release into belief (or illusion, or fiction).”

From: Rosalind Krauss: “Grids” October 9, Summer 1979. [Reprinted in: The Originality of the Avant-Garde and Other Modernist Myths. Cambridge, MA: The MIT Press, 1985, pp. 9-22. The quotes above are from pages 9-12; Eduardo Navas put a cut-up version of this essay online in his project“Grids”.]

“It’s supposed to be indexical of all that is rational, but I think it’s as mad as many logical things turn out to be – artificial, hysterical, subsuming its own version of chaos. It’s rigid but flexible, a measure of scale but scaleless, it’s flat with imitations of depth, democratic about space but really absolutist, stamped with rigidity but alert with permutational virtuosity. It’s a container that contains itself, that is both form and content.”

Patrick Ireland, 1998.

The Centrifugal Grid versus the Centripetal Grid.

David Sylvester: “Mondrian in London,” Studio International, December 1966.
John Elderfield: “Grids,” Artforum 10 (May 1972), pp. 52-59.
Rosalind Krauss: “Grids,” October 9 (Summer 1979).

Tony Smith: Mondrian’s early grids from a mathematical perspective.

“De Machine” in de opvatting van b.v. Piet Mondriaan en Andy Warhol. Mondriaan was daar tegen, en Warhol niet.

De machine houdt niet de nieuwe cultuur in. (. . .) De machine veralgemeent, zoals het militarisme veralgemeent. Het kan het individueele verminderen — het kan het ook dooden.

Piet Mondriaan: “De Jazz en de Neo-plastiek.” In: Yve-Alain Bois: Arthur Lehning en Mondriaan.
Amsterdam: Van Gennep, 1984, pp. 89/90.

Order can serve as a metaphor for order.

Gombrich: The Sense of Order, p. 247. (In deze passage van Gombrich heeft de gedenoteerde ordelijkheid een positieve lading: Harmonie, het gevoel van ontspanning als de “puzzelstukjes op hun plaats vallen”.)


Interactive programs: PixelPaint

     In 2002, Aaron Neugebauer submitted the JavaScript-programPixelPaint to the <5k internet-software-contest. PixelPaint is an interactive program which enables its user to specify an image by choosing the colours of the individual cells in a grid.
A pixel-paint program is obviously not the ideal image-construction tool for most people: it is not particularly efficient or convenient, nor does it provide a very interesting “interactive experience”. And for the rare occasions that one actually does want to manipulate individual pixels, that functionality is readily offered by many of the most common graphic tools, from MacPaint to PhotoShop. Thus, PixelPaint is probably best viewed as a primarily conceptual piece.
A 3D version of the same idea is one of the modes of operation of the program Blocks, written by Wouter De Jong and Roy Tanck in 2000.

If we do not try to use Neugebauer’s PixelPaint to construct specific preconceived images, but reflect on the set of its potential outcomes, the program itself becomes a symbol –– representing a seemingly limitless set of possibilities, while paradoxically emphasizing the finite constructability of each element and of the sequence of all elements.


Random Grids

A different strategy was popular in the 1960’s: the arbitrary pixel configuration, chosen at random from the set of all pixel configurations. Such a configuration also functions as a symbol, representative of the whole set. But, unlike the interactive tool which can be used to make anything, it does not make a mock-positive gesture, inviting the end-user to decide, choose, play and construct. Instead, it posits the equivalence of all possible images, and demonstrates that equivalence: the image displayed by the randomly sampled pixel configuration is noise.

Jean Arp – Arrangement according to Laws of Chance (Collage with Squares) – 1916-17.
Motherwell: “putting on a piece of cardboard pieces of paper that he had cut out at random and then colored: he placed the scraps colored side down and then shook the cardboard; finally he would paste them to the cardboard just as they had fallen.” (The Dada Painters and Poets)

Ellsworth Kelly: Spectrum Colors Arranged by Chance VI (1951)

Ellsworth Kelly: Spectrum Colors Arranged by ChanceVII (1951)

Ellsworth Kelly: Sixty-Four Panels: Colors for a Large Wall (1951).

Cowart, J., “Method and motif: Ellsworth Kelly’s ‘Chance’ grids and his development of color panel painting, 1948-1951,” pages 37-45 of Ellsworth Kelly: The Years in France. 1948-1954, National Gallery of Art, Washington, 1992.

Yve-Alain Bois: Ellsworth Kelly: The Early Drawings, 1948-1955. Cambridge, Mass.: Harvard University Art Museums, 1999, # 110-116.

Paul C. Vitz and Arnold B. Glimcher: Modern art and modern science : the parallel analysis of vision. New York: Praeger, 1984.

François Morellet: Répartition aléatoire de
40 000 carrés. 50 % noir, 50 % blanc. 1961.

herman de vries: Terre Provençale, 1991. (Rubbings with earth from different locations in the Provence.)

Zie ook:

herman de vries: Random Objectivation (Chance Collage), 1970. (33 x 33 vierkantjes met kleuren at random gekozen uit een verzameling van 8.)

Rul Gunzenhäuser: Maß und Information als ästhetische Kategorien. Einführung in die ästhetische Theorie G.D. Birkhoffs und die Informationsästhetik. Baden-Baden: Agis Verlag, 1962.

Jim Hodges. “As close as I can get”, 1998, a collage of small patches of color in a checkerboard pattern. The piece comprises nine panels, each made up of hundreds of Pantone color chips. Arrangement (partially?) based on chance.

George Korsmit.

image 1: “Colourhealingscreen, Chance & Choice”, Acrylic on canvas, 230*420cm, 2002
image 2: Left: “I’m not your Brother 2”, Acrylic on canvas, 190*260cm, 2002. Right: “Sssh”, acrylic on canvas, 240*300cm, 2000.
Non-computational chance procedures (using dice and blindfolding), resulting in random colors on a randomly irregular grid. (Cf. Michael Byron: Korsmit and Chance.)

Online algorithms generating random pixel grids

Andreas Fünderich: Squares, 2000.
3D version: Wouter De Jong and Roy Tanck: Blocks, 2000.


Every Grid (enumerating all patterns)

Minos: In predicate (2), you speak of ‘any transversal’: a little while ago, you spoke of ‘every exterior angle.’ Do you make any distinction between ‘any’ and ‘every’?
Euclid: Where the things spoken of are limited in number, I use ‘every’; where infinite, I use ‘any’ in order to bring the idea within the grasp of our finite intellects. For instance, you may talk of ‘everygrain of sand in the world’: there are, no doubt, what country-folk would call ‘a good few’ of them, but still the number is limited, and the mind can just grasp the idea. But if you tell me that ‘every cubic inch of Space contains eight cubic half-inches,’ my mind is unable to form a distinct conception of the subject of your Proposition: you would convey the same truth, and in a form I could grasp, by saying ‘any cubic inch.’

Charles L. Dodgson [= Lewis Carroll]: Euclid and his modern rivals. London: Macmillan, 1879. [Second edition, 1885, p. 25.]

  •  Lars Eijssen & Boele Klopman: “De Wensput” (1991). [“The Wishing Well.”] Enumeration of all configurations of a 71 by 71 black & white pixel grid. With a graphical interface for looking into the future, and an inverse mapping which calculates for any input picture when it will be produced. (Pascal program running on IBM-compatible PC’s under MSDOS.)
    [Discussed in: Remko Scha: “De Kunstkunstenaar” (“The Artificial Artist”). Natuur en Techniek60, 7 (1992), pp. 526-539.]
  •   Jochem van der Spek: “Borges” (1993).
  •   John F. Simon Jr.: “Every Icon” (1996). Enumeration of all configurations of a 32 by 32 black & white pixel grid. (Platform-independent Java-applet.)
  •  Leander Seige: “Imagen” (2000). Enumeration of all configurations of various pixel grids. E.g.: up to 150 by 150 black & white; up to 64 by 64 grey or RGB (up to 16 bit). Server-side application accessible through web-interface. [This program has been accessible online, and may become accessible again. See .]


herman de vries:
toevallige puntrastertekening
v 72-145, 1972.

herman de vries:

fields of randomly chosen numbers in random distributions.

[in: “on language”. subvers 8, august 1972, p.14.]


Grid-based processes: Cellular Automata.

Wolfram, Geurts & Meertens, Conway, Toffoli, Struycken, Driessens & Verstappen.

          See separate page.

Other formal operations on the grid: Vasarély, Richard Paul Lohse.


The grid is often used as a format for enumerating a bag of identical or distinct images (possibly using the axes to represent 1 or 2 of their parameters). Artists who used this format fairly consistently include Jennifer Bartlett, Gerhard von Graevenitz, Jan Henderikse, Sol LeWitt, herman de vries, Andy Warhol and Don Judd.

Andy Warhol: Marilyn Monroe, 1967.

It seems that the salient metaphysical question lately is: “Why does Andy Warhol paint Campbell Soup cans?” The only available answer is “Why not?” (…) Actually it is not very interesting to think about the reasons, since it is easy to imagine Warhol’s paintings without such subject matter, simply as “overall” paintings of repated elements. The novelty and the absurdity of the repeated images of Marilyn Monroe, Troy Donahue, and Coca-Cola bottles is not great. Although Warhol thought of using these subjects, he certainly did not think of the format. (…) The gist of this is that Warhol’s work is able but general. It certainly has possiblities, but it is so far not exceptional.

Donald Judd: “Andy Warhol,” Arts Magazine 37 (January 1963), p. 49. [Reprinted in: Alan R. Pratt (ed.): The Critical Response to Andy Warhol. Westport, CT: Greenwood Press, 1997. Pp. 2-3.]


Ellsworth Kelly: “Brushstrokes Cut into Twenty-Seven Squares and Arranged by Chance,” 1951. (Left)
” ‘Cité’: Brushstrokes Cut into Twenty Squares and Arranged by Chance,” 1951. (Above)

Zelfde principe: Mended Skylight serie (1949), Pages from a Magazine (1950), Cut Up Drawing Rearranged by Chance (1950), Drawing Cut into Strips and rearranged by Chance (1950), etc.

[Yve-Alain Bois: Ellsworth Kelly: The Early Drawings, 1948-1955. Cambridge, Mass.: Harvard University Art Museums, 1999, # 30-32, 40-42, 95-103.]

Discretisering van fotografische beelden.

Herbert W. Franke: Serie Einstein, 1972.

Leon Harmon: “The Recognition of Faces.” Scientific American, November 1973.

Chuck Close.

Remko Scha: “Computer/Art/Photography.” Perspektief 37 (1989), pp. 4-10.

Artboy: The History of the Pixel.

The Arcimboldo Effect:

Arthur Mole

Roy Lichtenstein (cf. Chuck Close.)


Salvador Dali: “Gala Contemplating the Mediterranean Sea which at Twenty Metres Becomes the Portrait of Abraham Lincoln,” 1976.

Het grid als thema. Distortions & mappings.

H. Philip Peterson: Digital Mona Lisa, 1965

Christoph Brachmann and Romana Walter: “Rastersturz. Interaktive Auseinandersetzung mit einer Grafik von Georg Nees.” In: Lutz Dickmann, Lars Fehr, Susanne Grabowski, Philipp Kehl, Frieder Nake, Romana Walter (eds.): Der Bericht zum Projekt macS. (Mediating Art in Computational Spaces.) UniversitŠt Bremen Medieninformatik (B.Sc.) January 2004.

Jochem van der Spek No Noise 2001

Bit-101 Laboratory: 01 sep 01; 01 oct 23/29; 01 nov 04/06-08; 01 dec 24; 02 jan 30/31; 02 oct 04; 02 mar 02/12; 02 may 11; 02 sep 12/13.

Projecting the grid onto the “real world”.

Man Ray photo.

François Morellet:

“2 trames 0°, 90°”, 1971.
“2 doubles trames 30°, 60°”, 1971.
“1 double trames 0°”, 1973.

François Morellet: Exhibition Catalogue Nationalgalerie Berlin, 1977, pp. 138-144.

Michael McDonough:

Grid House. Boston, Massachusetts, 1978. [Shown to the left.]

No image: The constant function: Empty grids.

Manzoni, Schoonhoven, Henderikse, Andre.

Ellsworth Kelly: White Panels on Green (1950), Bathroom Tiles (1951), Two Yellows (1952), Red and White (1952), White and Black (1952)

[Yve-Alain Bois: Ellsworth Kelly: The Early Drawings, 1948-1955. Cambridge, Mass.: Harvard University Art Museums, 1999, # 37, 117-121.]


Cellular Modules

Sebastien Truchet (1704).

Dominique Douat: Méthode pour faire une infinité de desseins différents avec des carreaux mi-partis de deux couleurs par une Ligne diagonale. Paris, 1722.

E.H. Gombrich: The Sense of Order. A Study in the Psychology of Decorative Art.
Oxford: Phaidon Press, 1979, pp. 70-72.

Eric W. Weisstein: Truchet Tiling.

Naomi Pitcairn: The Rotator Engine.

Philippe Esperet & Denis Girou: “Coloriage du pavage dit ‘de Truchet'”, Cahiers GUTenberg, n¡ 31 (December 1998), pp. 5-18.

Marjory Pratt (1940).

Marjory B. Pratt: Formal Designs from Ten Shakespeare Sonnets. Privately printed, 1940.

Susanne B. Langer: Mind. An Essay on Human Feeling. Baltimore: Johns Hopkins University Press, 1967, pp. 183-185.

François Morellet: Répartition aléatoire de triangles suivant les chiffres pairs et impairs d’un annuaire de téléphone, 1958.

François Morellet: Exhibition Catalogue Nationalgalerie Berlin, 1977, pp. 147-150.

K.O. Götz: Statistisch-metrische Modulationen (Rasterbilder), 1959-1961.

Heinz Ohff: Kunst ist Utopie. Gütersloh: Bertelsmann, 1972, pp. 132-134, 217.
K.O. Gštz: Erinnerungen und Werk, Vol. II. DŸsseldorf: Concept-Verlag, 1983.

Rul Gunzenhaüser (1962)

Rul Gunzenhaüser: Maß und Information als ästhetische Kategorien. Einführung in die ästhetische Theorie G.D. Birkhoffs und die Informationsästhetik. Baden-Baden: Agis, 1975, pp. 126-128.

Remko Scha and Rens Bod: “Computationele Esthetica.” Informatie en Informatiebeleid 11, 1 (1993), pp. 54-63. [English translation: “Computational Esthetics.”]

Peter Struycken (1969-1980): Computer Structures / OSTRE / METRO

Jean Leering: Programmi Sistematici. (Ricerca Contemporanea 4.) Milano: Vanni Scheiwiller, 1975.
Peter Struycken: Beelden en Projecten. Otterlo: Rijksmuseum Kröller-Müller, 1977, pp. 37-39.
Peter Struycken: Structuur Elementen 1969 1980. Rotterdam: Museum Boymans-van Beuningen, 1980, pp. 8-13.

Manuel Barbadillo & Michael Thompson (1970-1973).

Manuel Barbadillo: “Modules/Structures/Relationships: Ideograms of Universal Rapport.” PAGE 12. Bulletin of the Computer Arts Society, November 1970.

Michael Thompson: “Computer Art: A Visual Model for the Modular Pictures of Manuel Barbadillo.” Leonardo 5, 3 (Summer 1972).

Michael Thompson: “All done by graphs (undirected).” PAGE 31. Bulletin of the Computer Arts Society, October 1973.

Martin Schwencke (1973)

Martin Schwencke: Denken in de Ruimte. Uitgave in eigen beheer: Martin J. Schwencke, Zeist, 1973. [Tentoonstelling: Ploegh-huis, Amersfoort.]

Zdenek Sykora (1974): Tegelvloer, Gorcum.

Antoinette Hilgemann & Rien Robijns (eds.): Symposium Gorinchem 1974.

Groupe Belfort (G.F. Kammerer-Luka, Jean-Baptiste Kempf, Marcel Kibler, Claude Noll, Yves Normand, Anne-Marie Quemar), 1975.

IBM Deutschland: Computerkunst, 1975, pp. 72-75.

Paul Panhuysen & Johan Goedhart, 1977.

Remko J. H. Scha: “Visuele patronen volgens formele procedees.” In: Paul Panhuysen. Schilderijen, situasies, ordeningssystemen en omgevingsontwerpen. Exhibition Catalogue. Eindhoven: Van Abbemuseum, 1978, pp. 38-59. [English translation by Lucas Van Beeck: “Visual patterns according to formal procedures.” In: Paul Panhuysen (ed.): Number Made Visible. The story of the magic square of 8 by Benjamin Franklin. Eindhoven: Het Apollohuis / Breda: Tijdschrift # 11, 1997.]

Paul Bos: Padded Tile Piece, 1979. (J. Fields Gallery, New York.)

Brian P. Hoke: Cellular Automata and Art (Section VI: Creating Cellular Automata Art), 1996.