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Mathematics and systems in art

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Schillinger’s system also points to the role played by mathematical figures in art. Until the advent of computer-generated photorealism the most readily recognisable “computerised” images were mathematical surfaces and shapes. Mathematical abstract forms whose linear qualities lent themselves to computer generated images have their roots in the systems of proportion and decoration used in many traditional systems of art.

Most geometrical art systems are not mathematical in the sense of being produced by equations; rather they consist of a repertoire of forms within a grid guideline. They do not “generate” the image so much as define its parameters. However, within Islamic patterns, geometrical forms arise from the system itself and could be said to be generated within it. Islamic art is also the result of religious restrictions on figurative imagery forcing artists to invent a tightly defined system of decorative forms. This is important, as an image constructed within a set of rules could then conceivably be assembled – even “created” – by a machine.

El-Said and Parman assert that the transformation of basic geometric figures has a very ancient history. Titus Burckhardt calls this manipulation of form “geometric speculation”, which is a good term for the experimentation of many early (and some later) computer artists with repetitive geometric shapes in many different attitudes and transformations. Whether overtly influenced by Islamic art or not, it would seem that they have a strong internal notion of the fascination of repeat geometric forms.

Despite the non-numerical basis of the Islamic geometrical system (if Parman and El-Said are correct) it does seem to allow for form generation in a way analogous to certain types of computer graphics program. “The creation of patterns by infinite combinations of the geometric forms providing these systems of proportioning becomes possible.”[1]

There are strong similarities between this conception of geometric art as a system of generative forms based on an underlying structure determined by the circle and its derivatives, and the use of algorithms by many computer artists (such as Latham and the Algorists) to generate infinite but structured abstract artforms. John Whitney connected the technological form to the ancient in his animation Arabesque, which made a further link with Arabic music as well. Larry Cuba explained to me how this work was inspired by Whitney’s visit to the city of Isfahan and his observation of Islamic architecture.[2]

An analogy can be drawn with the refinement and codification of ideas of proportion amongst the Byzantines, which involved mathematics very closely in art. This was in part a result of the rational basis of Byzantine aesthetics and the separation of the world of the mind – noetos -- from the material world, aisthetos.[3] Mathematics provided a way of realising ideas in material reality, as pure geometry and number theory were closely linked:

The harmonies of number could be translated into geometric terms […] Though both belonged to the world of noetos, both moulded subordinate arts in aisthetos and the material world. It was this that led Michael Psellos in the 11th century to note the value of mathematics for philosophy since it linked abstract thought and material things.[4]

Computer graphics also provides a bridge between one’s visual imagination (including one’s internal conception of the world), and the external manifestation of reality. The distinction between noetos and aisthetos remains current. Through art, the power of imagination can directly influence the external world, either through direct execution, such as drawing and sculpting; or by conceiving of plans that can be executed by others. In the latter case, the internal idea has to be expressed in intelligible form, which can take the form of instructions – verbal, illustrated, or mathematical. As Gervase Mathew notes, Byzantine mathematics was divided along lines of practical application:

For the Byzantine mathematician the theory of numbers and pure geometry belong to the world of noetos; the art of calculation, applied geometry, of optics and mechanics to that of aisthetos. Material was moulded inevitably by the laws of mind.[5]

Once systematised as a scale or a system, a previously freeform art could potentially by produced mechanically. Even though a mechanical realisation might not arise immediately, nor would it be the intention or motivation behind codification, it remains a possibility once music or the visual arts exist within a definite conceptual framework. In particular, when mathematical or geometrical rules are involved, then the art becomes much more amenable to mechanical production. It could be that the first stage of this is simply reproduction – witness the music box and camera – before moving onto the much more complex issue of generation, which requires a machine capable of independent action, such as a digital or analogue computer. Additionally, mathematical correlation makes possible a degree of connection between artforms: witness the “colour keyboards” of Scarlatti and Klein (see below). Even the linkages between music and geometry in Islamic culture are suggestive of this.

Byzantine ‘surface-aesthetic’ was inevitably derived from arithmetic since all harmonies in form or colour were the echoes of an incorporeal music, the harmonies of Pure Number.[6]

[The interior of Hagia Sophia]

Again, this echoes the Arabic idea of unifying music and geometry, and perhaps derives from Pythagorean notions of a scale which could be expressed numerically or musically. Medieval and Renaissance attempts to link number and music, often with magical purpose (such as that found in Michael Maier’s Atalanta Fugens) also spring from this root.[7] As described in detail below, the work of the Abstract Animators aimed to produce both visual and musical material from the same numerical data, thus bringing the mathematical basis of these arts in line with ancient thinking.

[Michael Maier produced his Atlanta Fugiens in 1617 as an emblem book with 50 engravings with epigrams accompanied by 50 fugues.]

Quite apart from its utility in the field of perspective drawing, which led to the development of perpsective geometry, mathematics found a role in early twentieth century abstract art. The Suprematists and other Russian avant-garde artists were influenced by discoveries in multi-dimensional mathematics, attempting to link them with Theosophist notions of higher spiritual planes; and Malevitch in particular tried to depict four-dimensional figures.[8] Later Constructivists and allied artists also made use of mathematical discoveries: Max Bill is much quoted by Jean-Pierre Hébert on the subject of connections between his art and mathematics. Yet Bill, who deployed mathematics to produce much of his work, and is regarded by the Algorists as a forerunner, also spoke of the unwillingness of artists to countenance purely mathematical art:

Artists who employ mathematical relationships in the organization of their work have spoken of the beauty and interest of mathematical models but at the same time have disavowed them, asserting that they are not influenced by mathematics […][9]

Later, Bill speaks of mathematical art as being not pure mathematics as such, but the assembly of forms according to systems, which resembles Hébert’s approach to his algorithmic work. The resultant art “can, perhaps, best be defined as the building up of significant patterns from the everchanging relations, rhythms and proportions of abstract forms, each one of which, having its own causalty, is tantamount to a law in itself.”[10] Such rhythms and proportions were the foundations for the system propounded by the theorist Joseph Schillinger, which connects mathematics with musical composition and visual form.

The geometric abstract art typical of the pioneering computer artists in the 1960s has links to mathematical and rule-based art systems, because their visual qualities were readily translated into a computer-based environment. This early use of the computer may  itself derive from a very ancient urge to formalise images and subject them to rules. Once art is predicated on a series of rules is much more amenable to production by a calculating machine. Perhaps this outcome is inevitable at some point. Of course, this would probably never have occurred to the earliest creators of such formalised images, but the urge to contain the image within a set of defined possibilities was there from the beginning.

In early Computer Art, the images were also heavily constrained by the available graphics technology: simple shapes and blocky lines proliferated. It was not a self-imposed artistic limitation but an overall boundary to image structure.

Today, the absolute technical constraints are far fewer and it is down to artists to impose their limitations. This may occur simply by concentrating on a particular area, or by laying down rigid guidelines for their art. Certainly, this artistic process could be suggested by the technology, but it remains a choice on the artist’s part. As computer graphics pioneer John Lansdown said: “[One] should also remember Orson Welles’ insight, ‘The enemy of art is the absence of limitations’”.[11] For this reason, various groups have formed around particular approaches to using the computer. In the case of the Algorists, their self-imposed limitation is the use of algoristic, rule-based art.

A paradox with Computer Art is that the mathematical aspect can be overt, expressed as abstract and geometric forms readily identified as “mathematical”; or it can be covert, subsumed into the structure of highly realistic, even organic-looking, 3D imagery. The degree to which the mathematics makes itself obvious informs the viewer’s reaction to a piece of art, determining whether they see it as “computerised” and “mechanical” or “naturalistic”.

Perhaps it is the underlying rationalisation of form that underpins both the geometric forms of Computer Art and its ancient precursors. Rudolf Arnheim sees similarites between the rational shapes imposed on the external world by craftsmen and engineers, and the rationalising impulse that led to rules for sonnets and haiku, or Dante’s structured cosmology at work in the Divine Comedy:

Everywhere the mind craves the rationalization of shapes and, if necessary, produces tools to achieve it.[12]

Although many artists since the Romantics have striven against the imposition of such systems, Lansdown notes that these rules may serve to further art through the very restrictions that define them. He mentions the orders of Greek architecture, the use of mathematical and randomising techniques in music, and the theories of Joseph Schillinger. As Lansdown says:

Systems of architectural proportion and other formal systems (such as the Orders) might be thought of as restrictions on rather than aids to creativity. This would be a mistaken view. Systems of proportion provided an accepted framework within which architects could develop their ideas […][13]

The need, or desire, for controlled and constructed art long preceded the invention of a machine capable of implementing it, just as the desire for realistic images preceded the camera.[14] In neither case did this desire presuppose the existence or the desirability of a machine to produce art; rather the machine emerged as an unforeseen consequence of other developments. In the camera’s case, it proceeded directly from mechanical attempts to capture reality; the computer, by contrast, was adopted indirectly because it seemed to fulfil earlier expectations of automation in art.

Immediately prior to the inception of Computer Art, in the 1930s and 40s, artists and mathematicians experimented with Lissajous figures using weights and pendulums. [15] The images were inscribed on paper or drawn in sand, and some artists attempted to automate the process by making mechanical computers aimed at producing Lissajous figures.[16] By the 1960s, there were numerous examples of harmonographs and related mechanisms, such as Ivan Moscovich’s Lissajous machine, John Ravilious’s pendulum drawing machine and D.P. Henry’s mechanical analogue computer.[17]

[Plate XII: DP Henry’s drawing computer c.1955]

[Plate XIII: Ivan Moskovich’s Pendulum Harmonograph]

[A typical harmonography in action]

As these pictures demonstrate early drawing machines could produce highly detailed abstract images resembling those from drawing programs on early digital computers. However, as Ravilious noted, all mechanical devices were limited by their physical construction to a certain type of drawing, no matter how much its parameters could be altered.[18] Thus their output was, in a sense, inherent in their construction and though they were “computers” in that they could be “programmed” with certain functions, they were also single-purpose devices. Nonetheless, they were important evolutionary stages in the development of fully computerised art, a fact recognised by Richard Land when he explained how the term “Computer Art” was commonly restricted to electronic computers only:

Although machine-produced art is in some elements indistinguishable from Computer Art, present-day opinion would likely restrict the term to works produced by […] computers designed for general logical operations […] The trend of modern thought on the subject is inclined also to eliminate from the category of Computer Art those instruments which seem to have artistic expression as their sole purpose.[19]

Thus mechanical devices certainly informed the development of Computer Artforms, but more importantly they also prepared the ground conceptually and visually.

[1] El-Said, Issam & Ayse Parman, Geometric Concepts in Islamic Art, 1976

[2] Interview with Larry Cuba, Los Angeles, August 2001.

[3] See Gervase Mathew, Byzantine Aesthetics (London, 1963) “The Mathematical Setting” p23

[4] Mathew, ibid, p28

[5] Mathew, ibid, p24

[6] Mathew, p26

[7] For instance, see Joscelyn Goodwin, The Harmony of the Spheres (1992)

[8] Manuel Corrada, “On some vistas disclosed by mathematics to the Russian Avant Garde: Geometry, El Lissitzky and Gabo”, LEONARDO Vol.25, No.3/4, pp.377 384, 1992

[9] “The Mathematical Approach in Contemporary Art” Max Bill, quoted by Hébert at, web ref, p148.

[10] “The Mathematical Approach in Contemporary Art” Max Bill, ibid.

[11] “Artificial creativity: An algorithmic approach to art” John Lansdown, Centre for Electronic Arts, Middlesex University

[12] Rudolf Arnheim, “The Tools of Art – Old and New” New Essays on the Psychology of Art (1986, Los Angeles), p128

[13] John Lansdown “Artificial creativity: An algorithmic approach to art”, in Beardon C (ed) Digital Creativity: Proceedings of CADE 95, University of Brighton, pages 31-35

[14] Rudolf Arnheim, New Essays on the Psychology of Art (Los Angeles, 1986)

[15] Herbert Franke, Computer Graphics, Computer Art (1971), p60

[16] S. Tolansky, “Complex Curvilinear Designs from Pendulums” Leonardo Vol.2, pp267 -- 274, 1969

[17] Jasia Reichardt, ed. Cybernetic Serendipity (1969), pp48-50

[18] John Ravilious, “Limitations and general design features of a pendulum drawing machine”, Reichardt, ibid, p49

[19] Richard I Land, “Computer Art: Color-Stereo Displays”, Leonardo maths & comp issue