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"One, two, three, . . . ." – the row of these sounds (spoken ordinal numbers) we know by heart as a series without end, i.e., which continues forever according to a law which is known as fixed.
Besides this row of auditory images we possess other sequences of representations which proceed according to a fixed law, such as the row of visual symbols (written ordinal numbers) 1, 2, 3, . . . .
These things are intuitively clear.

L.E.J. Brouwer: The Foundations of Mathematics, Ph. D. Dissertation, University of Amsterdam, 1907. [Beginnning of Chapter 1: The Construction of Mathematics.]

Piero Manzoni

Alfabeto, 1958


1-30 Settembre, 1959

Konrad Bayer & Gerhard Rühm
10 Lebensregeln für Vierzigjährige
 1.) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
 2.) 2, 3, 4, 5, 6, 7, 8, 9, 10.
 3.) 3, 4, 5, 6, 7, 8, 9, 10.
 4.) 4, 5, 6, 7, 8, 9, 10.
 5.) 5, 6, 7, 8, 9, 10.
 6.) 6, 7, 8, 9, 10.
 7.) 7, 8, 9, 10.
 8.) 8, 9, 10.
 9.) 9, 10.
10.) 10.

In: Gerhard Rühm (ed.): Die Wiener Gruppe. Achleitner, Artmann, Bayer, Rühm, Wiener.
Reinbek: Rowohlt Verlag, 1967, p. 318.

Ellsworth Kelly

Blue Green Yellow Orange Red, 1966

Spectrum II, 1966/1967

Spectrum V, 1969



One of the most basic algorithms is the iterative loop: repeat the same calculation forever, merely changing the value of one parameter: i = 0, 1, 2, 3, . . . The very form of this algorithm shows that its output sequence is isomorphic with the series of natural numbers. No matter what the output looks like, its abstract form embodies the Ur-intuition of counting. 


The Universal Library

A schema allowing finitely many instantiations is succinctly represented by an algorithm enumerating these instantiations, one by one, in some systematic order. An alternative (extensional) representation is the output sequence of such an algorithm.

The first artwork instantiating this format is an early piece of "concept art": the "Universal Library" described by Kurd Laßwitz in his dialogue "Die Universalbibliothek" (1901). The Universal Library is the set of all possible books: the set of all sequences of typographical characters with a certain maximum length. (Laßwitz notes that the maximum length does not introduce an essential limitation: longer texts merely consist of multiple volumes from the library. W.V.O. Quine ("Universal Library") observed that a small pocket library of two one-bit volumes may thus be used to represent the Universal Library in a very efficient way.) The history of this idea is beautifully summarized by Jorge Luis Borges in his essay "La Biblioteca Total" (1939). Borges mentions Lewis Carroll and Gustav Theodor Fechner as direct antecedents of Laßwitz. The Universal Library re-appears in The Race with the Tortoise by Theodor Wolff (1929) and "The Library of Babel" by Jorge Luis Borges (1944).


Interviewer: "Can you tell us something about your new opera?"
Giuseppe Verdi: "It's exactly the same notes as the previous one, I merely changed the order a little bit."

[To do: Find source and exact formulation!]

When things around me seemed once more to be real, Arthur was saying “I’m afraid there’s no help for it: they must be finite in number.”
  “I should be sorry to have to believe it,” said Lady Muriel. “Yet, when one comes to think of it, there are no new melodies, now-a-days. What people talk of as ‘the last new song’ always recalls to me some tune I’ve known as a child!”
  “The day must come – if the world lasts long enough –” said Arthur, “when every possible tune will have been composed and every possible pun perpetrated –” (Lady Muriel wrung her hands, like a tragedy-queen) "and worse than that, every possible book written! For the number of words is finite."
   “It’ll make very little difference to the authors,” I suggested. “Instead of saying ’what book shall I write?’ an author will ask himself ’which book shall I write?’ A mere verbal distinction!”
   Lady Muriel gave me an approving smile. “But lunatics would always write new books, surely?” she went on. They couldn’t write the same books over again!”
   “True,” said Arthur. “But their books would come to an end, also. The number of lunatic books is as finite as the number of lunatics.”

Lewis Carroll: Sylvie and Bruno Concluded, 1893. Chapter IX: The Farewell-Party.

The caprice or fancy or utopia of the Total Library contains certain traits that could be confused with virtues. Actually, it is astonishing how long it took mankind to dream up the idea. Certain examples Aristotle attributes to Democritus and to Leucippus clearly prefigure it, but its late inventor is Gustav Theodor Fechner and its first expounder is Kurd Laßwitz. Its connections are illustrious and multiple: it is related to atomism and combinatory analysis, to typography and to chance. In The Race with the Tortoise (Berlin, 1929), Dr. Theodor Wolff suggests that it is either a derivation from or a parody of Raymond Lull's mental machine; I would add that it is a typographical avatar of the doctrine of the eternal return which, adopted by the Stoics or by Blanqui, by the Pythagoreans or by Nietzsche, eternally returns.

Jorge Luis Borges: "La Biblioteca Total", 1939.

Chaque fois que l'on discute de tel ou tel produit de l'esprit sous le rapport de l'art, il faut s'en tenir à ce dilemme: ou c'est une intuition lyrique, ou ce sera n'importe quoi d'autre, même éminemment respectable, mais non pas de l'art. Si la peinture était, comme pour certaines théories, une imitation ou une reproduction des objets donnés, elle ne serait pas de l'art mais quelque chose de mécanique et de pratique; si les peintres étaient, comme selon d'autres théories, des assembleurs de lignes, de lumière et de couleurs au moyen de nouveautés ingénieuses, de trouvailles efficaces, ils ne seraient pas des artistes mes des inventeurs de techniques; si la musique consistait en de semblables combinaisons de tons il serait possible de composer des partitions sans connaître la musique, en réalisant le paradoxe de Leibniz et du Père Kircher, ou il serait à craindre, avec Proudhon pour la poésie et Stuart Mill pour la musique, que, une fois épuisé le nombre des combinaisons possibles de mots et de notes, poésie et musique disparaissent du monde.

Benedetto Croce: "Aesthetica in nuce." (Laterza, 1935) French translation by Gilles A. Tiberghien
in: Benedetto Croce: Essais d'esthétique (Paris: Gallimard, 1991), p. 47.


Louis Auguste Blanqui: L'éternité par les astres (1872).

Jorge Luis Borges: "La Biblioteca Total." Sur (August 1939).

Jorge Luis Borges: "The Library of Babel." In: Ficciones (1944).

Lewis Carroll: Sylvie and Bruno Concluded, 1893.

Kurd Laßwitz: "Die Universalbibliothek." In: Traumkristalle, 1901.

W.V.O. Quine: "Universal Library"

Theodor Wolff: The Race with the Tortoise. Berlin, 1929.

Permutation Poetry

In the 1960's, poets such as Brion Gysin and Dick Higgins produced enumerative literary works by means of digital computers. Their "permutation poems" consisted simply of all possible permutations of the words of a simple phrase.


Brion Gysin: Permutation poems. In: William S. Burroughs: The Third Mind. Viking, New York, 1978.

Dick Higgins: Computers for the Arts. Somerville, Mass.: Abyss Publications, 1968/1970.
[Including "Hank and Mary" by Dick Higgins, and "Proposition No. 2 for Emmett Williams" by Alison Knowles.]


Conceptual Art

An algorithm enumerating a relatively small finite sequence may be carried out by a human person, without the aid of a digital computer. This approach was used in many "conceptual art" pieces by Sol LeWitt.

"The artist's will is secondary to the process he initiates from idea to completion. [...] The process is mechanical and should not be tampered with. It should run its course."

Sol LeWitt: "Sentences on Conceptual Art." Art-Language 1,1 (May 1969).


Sol LeWitt: Arcs circles and grids. Arcs, from corners & sides, circles, & grids and all their combinations. Bern: Kunsthalle Bern & Paul Bianchini, 1972.

Sol LeWitt: Arcs and Lines. All combinations of arcs from four corners, arcs from four sides, straight lines, not-straight lines, and broken lines. Lausanne: Éditions des Massons, 1974.

Sol LeWitt: Geometric Figures & Color. Circle, Square, Triangle, Trapezoid and Parallelogram in Red, Yellow and Blue on Red, Yellow and Blue. New York: Harry N. Abrams, 1979.



Ryszard Winiarski:
Game 7x7 – logical course – the elements
appear keeping the diagonal direction,




Performance Art

Some of the minimalist art of the 1960's shows an obsession with repetition which is not primarily concerned with visual strategy but just as much with compulsive behaviour. (Andy Warhol, Yayoi Kusama, Jan Schoonhoven.) This attitude was developed into a curiously formal version of performance art in the oeuvres of On Kawara, Roman Opalka and Hanne Darboven, who all (in different ways) construed their lives as mere enumerations of  temporal marks.


Wo Mechanismus ist, ist keine Geschichte, und umgekehrt, wo Geschichte ist, ist kein Mechanismus. Können wir uns z.B. die Geschichte einer Uhr denken, die immer regelmäßig (der Einheit ihres Princips gemäß) geht? Aus doppeltem Grund nicht: einmal, weil in ihr keine Freiheit des Princips, und dann, weil in ihr keine Mannichfaltigkeit der Handlung ist, denn es ist eine und dieselbe immer wiederholte Begebenheit, die wir an ihr sehen. Daher ist auch der Mensch nach der Uhr – der selbst Maschine geworden ist (er aß, trank, nahm ein Weib und starb) – kein Objekt – nicht einmal der Erzählung.

Friedrich Wilhelm Joseph Schelling: Ist eine Philosophie der Geschichte möglich? 1797/1798

The things I want to show are mechanical. Machines have less problems. I'd like to be a machine, wouldn't you?

Andy Warhol. (In: David Bourdon: Warhol. New York: Harry N. Abrams, 1989, p. 140.)


Maurice Fréchuret:  La Machine à peindre, Jacqueline Chambon, 1994. [Discusses the artist as machine and the machine as artist: Alberto Giacometti, Andy Warhol, Piero Manzoni, Jean Tinguely, Giuseppe Pinot-Gallizio, Roman Opalka, On Kawara, Hanne Darboven, Buren/Mosset/Parmentier/Toroni.]

On Kawara's date paintings.

Barthelémy Schwartz: Review of: Maurice Fréchuret: La Machine à Peindre.

Arthur Elsenaar and Remko Scha: The Varieties of Human Facial Expression (1997).

All pixel grids

The visual equivalent of the Universal Library is the set of all images with a certain resolution and a certain color depth. In the 1990's, several artists implemented algorithms doing this.

  • Lars Eijssen & Boele Klopman: "The Wishing Well" (Enschede, 1991). Program printing out 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 picture when it will be produced. (Pascal program running on IBM-compatible PC's under MSDOS.) This piece was displayed as an installation at the 1991 TART Festival, University of Twente, the Netherlands; it was mentioned in: Remko Scha: "The Artificial Artist." Natuur en Techniek 60, 7 (1992), pp. 526-539.

  • Jochem van der Spek: "Borges" (Amsterdam, 1993).

  • John F. Simon Jr.: "Every Icon" (New York, 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). [This program has been accessible online, and may become accessible again. See .]

Some related issues:
  •  An aesthetic experience evoked by these pieces: Even if the number of 71 x 71 black-and-white images is finite, it is extremely large. The attempt at mental representation of such an image set is experienced as a sublime vertigo. (Cf. Kant: "Das physikalische Erhabene.") [Computer Science correlary: Theory of randomness: Almost all of these images lack structure. Regularity is rare.]
  •  A more difficult challenge is to design and implement a theory describing all possible images not as pixel grids but as visual Gestalt structures, taking into account the way in which images are perceived by human observers. This is the goal of the IAAA project Artificial, initiated in 1988.

Non-enumerable sets

black line

Each point on this line is a composition.

Henry Flynt, January 1961

To add: Pieces by Christer Hennix which deal with the real numbers.



Fodor's problem.

Risa Horowitz: Generic Form

Yasuka Matsui: Algorithms for Combinatorial Enumeration Problems


© Remko Scha, 2001-2006