a view of non-figurative art and mathematics and an analysis of a structural relief

Leonardo

A View of Non-Figurative Art and Mathematics and an Analysis of a Structural ReliefAuthor(s): Anthony HillSource: Leonardo, Vol. 10, No. 1 (Winter, 1977), pp. 7-12Published by: The MIT PressStable URL: http://www.jstor.org/stable/1573619 .

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Leonardo, Vol. 10, pp. 7-12. Pergamon Press 1977. Printed in Great Britain

A VIEW OF NON-FIGURATIVE

ART AND MATHEMATICS AND

AN ANALYSIS OF A STRUCTURAL RELIEF

Anthony Hill*

Abstract-The author discusses some interpretations of the term mathematics and what can be meant by mathematical abstract or non-figurative visual art. He then presents an analysis of the mathematical content of his structural relief entitled 'The Nine'-Hommage d Khlebnikov, which he made in 1976. The mathematical notions employed are from the domain of combinatorics.

I.

S. Giedion coined the slogan 'mechnization takes command' [1] for application to architecture in 1948 and, for some time now, it has become equally applicable to mathematics as a result of the development of digital computers and of computer science. The trend of mathematization has long been visible in philosophy (Leibnitz's Mathesis) and, since the times of Copernicus, Galileo and Newton, in the physical sciences, and in recent decades it has played a steadily larger role in the life sciences and technology, as well as in psychology and the social sciences.

In France, mathematics is called an 'exact science', as is presumably the domain of logics, but the term science in French has not meant the same as science in English, for in France the term refers to any kind of systematic scholarly study of questions, not only those the answers to which require either experimental or observational verification. Whether anything of value is achieved by calling mathematics a 'science' in the English sense of the term seems doubtful to me.

Consider for example the following passage: 'The only example known to me of a scientific doctrine which is unlikely to be shown to contain mistaken assumptions is the mathematical theory of the 230 crystal groups. This rests only on the three-dimensionality of physical space and the spatial repetition of equivalent and equivalently placed units' [2]. This passage raises a number of questions, for example, am I to accept that 'scientific doctrine' (I assume this to mean 'scientific theory') can be synonymous with a 'mathematical

* Artist living at 24 Charlotte St., London, Wl, England. (Received 5 Aug. 1976.)

theory'? Clearly, in the English sense of the term, it cannot.

It seems equally clear to me that, while defini- tions of mathematics are useful if they point to important features of the activity, few are con- fronted with the problem of preparing an all- embracing definition. I shall briefly consider some attempts that have been made. The French group of mathematicians called Bourbaki [3] has agreed that 'it is a study of abstract structures or formal patterns of connectiveness'.

The Swiss mathematician Paul Bernays says that 'it can be regarded as the theoretical phenomeno- logy of structures' [4] and the philosopher C. S. Peirce (U.S.A.) maintained that it is concerned with the 'observation of artificial objects of a highly recondite character' and that 'as the great mathematician Gauss has declared-algebra is a science of the eye' [5]. Peirce also took a lively interest in topology, which he described as 'the science of spatial connections', adding that this branch of mathematics would be better called 'synectics'.

In this article I shall discuss the meaning of a kind of non-figurative or 'abstract' visual art that can be described as being, in part, 'mathematical'. This property is not to be taken as a physical characteristic or parameter of this art, such as the dimensions of length, time and mass of different kinds of art media. However, a work of art classi- fied as 'mathematical' is one in which some aspects of its organization can be shown to involve the manipulation of concepts or ideas that, at least in part, stem from those found in some branch of mathematics.

I shall not present my views of the historical precedents for 'mathematical' visual art nor attempt to show whether much that is claimed to be an art of this kind really is, such as art objects

7

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Anthony Hill

that incidently include common geometrical shapes or mystical numerological concoctions.

In some non-figurative pictures artists employ a kind of 'calculation' to organize the composition of shapes [6] and/or of colours, others take into account what psychologists of visual perception have learned about visual illusions, drawings of impossible objects and colour after-image pheno- mena. 'Calculation' may, incidently, be necessary when artists apply scientific and technological developments, such as holography and digitial computers.

There may also be a kind of 'calculation' in- volved when artists make use of the visual material provided by scientific and technological research, especially photographs obtained with the aid of microscopes and telescopes of various kinds. Such material is, of course, meaningless without a framework for making sense of it, although some people may obtain aesthetic satisfaction from it by misconstruing the sense of it.

The popular words one finds in texts on 'scienti- fically oriented' art tend to be drawn from the latest 'sensational' press reports on new developments in science and technology. The new disciplines of cybernetics, semiotics and computer science have especially fascinated some artists and writers on art. What is called General Systems Theory has had a particular attraction for a number of artists. In my opinion, the visual fine arts are more likely to breed pretentious claimants of effective innovations than music. Musical works of this century have also undergone radical changes in form and content, although I doubt that some of them will withstand the test of time. The organization of musical material has in the past offered a broad scope for 'systematization'; even some Oriental music takes advantage of this possibility. But I shall not pursue this aspect of the arts further here.

In 1961 I exhibited some of my constructional or structural reliefs in the British Section of the Paris Biennale. The then Minister of Culture, Andre Malraux, on looking at them expressed the opinion that they appeared to exemplify the so-called French 'System D' (The'D' stands for debrouillard, which means resourceful in getting what one wants, including resorting to tricks to get out of a diffi- culty). The remark was, of course, meant in a denigrative sense and I do not think that it was at all germane. Two of the works shown were the subject of an analysis published in 1966 [7]. One of these, 'Prime Rythms', which I made in 1958, is shown in Fig. 1.

II.

One of the artists who shaped an outlook on the visual arts of this century was Marcel Duchamp. In 1919 he referred to himself as a'chess maniac' [8]. Although it has not been said explicitly either by him or by his admirers that he took up chess very seriously because he was bored by his art or because he wanted to show that the brain of an

b

Fig. 1. 'Prime Rhythms', relief structure, laminated plastic, 91.5 x 91.5 x 1.9 cm, 1958 (Photo: Cooper, London.)

imaginative artist need not be addled by alcohol and egomania, I think that these two reasons featured largely in his decision to spend so much time at the chessboard.

I have not been attracted to chess, nor to any other board or card games; life and my art work seem 'games' enough for me. On the other hand, I would not like to be thought an artist who indulges in 'art gamesmanship' or in the mathematical Theories of Games and of Metagames.

This is not true of John Ernest, whom I met in London in 1954, the year in which we began making relief structures instead of non-figurative paintings. He did not, however, convert me into a chess enthusiast, although he was then a fair club chess player.

I did learn from him some techniques for making art works of better craftsmanship. We also shared an interest in mathematics and in the ways certain of its notions might be applied in relief structures. Before taking a closer look at qualitative mathematics, we explored mathematical formulations that have been used in art, such as the Golden Ratio. (I had come to this in 1950 after reading Le Courbusier's book entitled Le Modulor [9].

At that time, a number of English painters (Victor Passmore, Kenneth and Mary Martin, Adrian Heath and I) formed a loosely knit group to investigate notions behind constructivist works ranging from those of Max Bill to those of Charles Biederman. We were interested in geometrical non-figurative works and abhorred Tachisme and Action Painting. While my interest centered on a kind of synthesis of Neo-Plasticism and Con- structivism, there was then, as there is now, an aspect of my personality that draws me to the study of the works by Marcel Duchamp, whom I consider

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A View of Non-figurative Art and Mathematics and an Analysis of a Structural Relief

to be this century's most daring innovator in the visual arts [10].

Duchamp's artistic innovations were numerous. I consider the most important one to be his introduction of the idea of a 'ready made'; a utilitarian object, for example a urinal, upon which an artist confers the status of an art object. This does not apply to natural objects, such as a pretty stone. While this idea is usually taken to be an anti-art Dadaist trick, the ramifications of the idea can be interpreted to go much further. For example, if an artist finds that a 3-dimensional or pictorial mathematical model is for him charged with visual aesthetic properties, he may appropriate it as it exists, call it an art object and ask money for it. Of course, mathematicians could enter this form of business also.

In my article Constructivism-the European Phenomenon [11] a sculpture by Naum Gabo is shown together with a line drawing taken from Encyclopeadia Britannica that clearly was the source of inspiration for his work. He has not denied my finding. Another interesting example is the illustration (Fig. 2) in an essay by the celebrated English mathematician J. E. Littlewood that represents what he calls '. . . "the hippopotamus", a well-known character in the theory of "prime- ends", but only now baptized in imitation of the crocodile. . .' [12].

In 1958, Ernest and I became engrossed in a problem that had not been formulated before in the mathematical literature and to which we supplied a tentative solution that still has been neither proved nor disproved [13]. The problem is stated as follows: If more than four arbitrarily placed points on a plane are joined by straight lines so that each point is joined to the others, it is possible to join the points so that a certain minimum number of the lines are crossed. In the case of five points, one crossing is unavoidable. What is the minimum number of crossings when one has any specified number of points larger than five?

The unexpected pay-off of our venture lead me into what I might describe as a one-man gambling syndicate in mathematics. I say 'gambling' because, on looking back, this seems to express an important aspect of the venture. The motivation certainly was not that of monetary reward. In fact, the time and

Fig. 2.

effort I have spent working in the field of mathematics that might best have been left to the pro- fessionals has since then jeopardized the income I earn from my vocation as an artist.

Have I done this to escape the boredom of art ? Was I trying to prove that an 'illiterate' in mathematics, an innumerate, could show interesting notions to mathematicians of which they were not previously aware? My reply to both questions is, in part, yes.

III.

I have presented the above discussion as a pre- amble to the following analysis of a recent work of mine in which I shall point out the kind of mathematical notions I have 'put to work' within the context of present-day 'mathematical' non-figurative or abstract art.

The work is a low relief structure that I call 'The Nine-Hommage a Khlebnikov' (Fig. 3), which was made between 1975 and 1976. It consists of a white plastic laminated octagonal panel on which are graved the lines shown in Fig. 4. Within this graved configuration are attached 27 L-shaped pieces of the same white plastic material. Each L shape measures 2 x 1 in (5 x 2.5 cm) with a thick- ness of i in (3 mm). The sides of the L-shaped pieces are black. A photograph taken from directly in front of the work does not provide sufficient information on its characteristics. They can be observed only when a viewer looks at it from different angles and notes the black sides of the L- shaped three-part clusters.

I have dedicated the work to the Soviet poet and aesthetician Velimir Khlebnikov (1885-1922). He was an important figure in the Formalist Movement in the arts. His poetry is well known in the U.S.S.R., but his theoretic aesthetics are not available at

Fig. 3. 'The Nine-Hommage A Khlebnikov', relief structure, laminated plastic, 91.5 x 91.5 x 1.9 cm, 1976.

9



Anthony Hill

tion differ, but each of them is mathematically identical to the tree variant A.

Fig. 5. 'Co-structure, No. 2, Hommage i Roberto Frucht' (Version 3), welded stainless steel, 1975.

(Photo: R. Scherman, London.)

Fig. 4. Location of graved lines on the panel of the relief structure (cf. Fig. 3).

present. What I know of his work and of the Formalist Movement in general interests me greatly. It is noteworthy that a line of development from this Movement runs from Roman Jacobsen, one of its founders, who left the U.S.S.R. to settle in Prague, where he headed the 'Prague School'. He then emigrated to the U.S.A. where his linguistic work influenced, among others, Noam Chomsky. I have dedicated my structural relief 'The Nine . . .' to Khlebnikov in admiration of his work and so I hope it will not be taken as a pretentious act.

Earlier, I dedicated a piece of sculpture (Fig. 5) to Roberto Frucht, a German mathematician living in Chile, who greatly helped me to develop ideas in which he was much interested.

'The Nine . . .' (Fig. 3) is the last of three relief structures bearing the same title. The other two have a square format and the same L-shaped elements are employed. With the help of the diagrams in Fig. 6, I shall explain what are the characteristics of the nine configurations of the L- shaped elements in Fig. 3 and why there are only nine distinct configurations. This explanation involves the mathematical domain-Graph Theory. A good introduction to the subject can be found in Ref. 14. Readers of Leonardo will find Refs. 15 to 18 helpful as regards the meaning of the special terms that are used below.

In Fig. 6, the trees labeled A, B and C are, by definition, the same mathematical type of tree, but they differ in appearance because each is embedded (or laid out) in a different type of grid or lattice- that of A is hexagonal, of B, square, and of C, tri- angular. The hexagonal lattice permits the tree A to be embedded in it in eight different ways. If each of these is rotated in all possible ways around the nodal points (a nodal point is a point from which at least three lines emanate), one finds that there are a total of 8 x 12 or 96 trees whose configura-

A A , B

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Fig. 6. Tree configurations.

C

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IL.

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A View of Non-figurative Art and Mathematics and an Analysis of a Structural Relief

In the case of tree variant B, one obtains 8 x 65 or 520 trees of different configuration and for tree variant C, the product of 12 x 733 or 8796 trees of different configuration.

The numbers 96, 520 and 8796 are a 'function' of the symmetry group of the chosen tree type, which in this case is asymmetric, that is, its automorphism group is restricted to that of identity (it has no other automorphism). In Graph Theory such a tree is called an identity tree and the type I have chosen (A, B and C) is the smallest of group order 1, having seven points and six lines, Fig. 8.

Fig. 7. A k

1 4 2

5

Fig. 9.

Fig. 10.

The seemingly simple observation that the smallest tree having group order 1 has seven points and six lines was communicated ina well-known paper by the Hungarian mathematician George Polya in 1937 and independently by Frucht in 1938. Polya also showed that the smallest asymmetric graph, is unicyclic (as can be seen in Fig. 7, it has one

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Anthony Hill

cycle or circuit and the same number of points and lines). The asymmetric tree was the basis of a recent relief I made and the eight configurations associated with tree A in Fig. 8 served as the basis of a series of reliefs that immediately preceeded the relief I have analyzed (Fig. 3).

I shall now explain what is common to the nine trees shown in Fig. 6 D, which were selected from the 65 associated with the tree B. Each of the nine has two common features, each has five right angles (Fig. 9) and each can be drawn by connecting three graphs of L shape, thus allowing sets of two lines to be within an area that can be colored (Fig. 10).

In my serigraph 'Parity Study' (1971), which is based on the above ideas, the areas enclosing the trees are colored red and blue. The relief 'The Nine . ..' is white throughout, except for the black sides of the L's. The decision to use in 'The Nine .. graved fragments of a lattice and the four long lines emanating from alternative corners of the octogon, a cyclical symmetry (Fig. 4), thus contrasting with the 3 x 3 array of configurations made of elements of L-shape, were made on the basis of my aesthetic trails.

I shall not here present other aspects of 'The Nine.. .', such as the way the nine configurations of L-shaped elements are individually disposed in the array; they will be seen to follow discernable rules.

I hope that readers will be helped by my analysis to understand the process I used to 'put to work' in an art work an abstract idea from the domain of mathematics. As I pointed out, decisions of an artistic character were made and they were arrived at on the basis of my personal experience. I took a structural theme, which is essentially qualitative, and realized it in a presentation that involves measured modulations. The theme derives from the domain of combinatorics (which embraces

Graph Theory), but I have also made use of some of my own small mathematical discoveries.

REFERENCES 1. S. Giedion, Mechanism Takes Command: A Contri-

bution to Anonymous History (New York: Oxford Univ. Press, 1948).

2. L. L. Whyte, The Unity of Visual Experience, Bull. of the Atomic Scientist, p. 20 (Feb. 1959).

3. N. Bourbaki, General Topology (Paris: Hermann, 1966). 4. P. Bernays, Comments on Wittgenstein's Philosophy of

Mathematics, Ratio I (No. 1, 1959). 5. C. S. Peirce. Collected Works, P. Weiss, ed. (Cambridge,

Mass.: Harvard Univ. Press, 1931). 6. C. Johnson, On the Mathematics of Geometry in My

Abstract Paintings, Leonardo 5, 97 (1972). 7. A. Hill, The Structural Syndrome in Constructive Art,

in Module, Proportion, Symmetry, Rhythm, G. Kepes, ed. (New York: Braziller, 1966) p. 168.

8. F. Le Lionnais, Marcel Duchamp as a Chess Player, Studio Int., p. 23 (JanlFeb 1975).

9. Le Corbusier, Le Modulor (Paris: Collection Ascoval. Editions d'Architecture d'Aujourd'hui, 1950).

10. A. Hill, The Spectacle of Duchamp, Studio Int., p. 20 (Jan./Feb. 1975).

11. A. Hill, Constructivism-the European Phenomenon, Studio Int., p. 144 (April 1966).

12. J. E. Littlewood, The Zoo, in A Mathematician's Mis- cellany (London: Methuen, 1953) p. 214.

13. F. Harary and A. Hill, On the Number of Crossings in the Complete Graph, Proc. Edinburgh Math. Soc. 13, p. 333 (1963).

14. R. J. Wilson, Introduction to Graph Theory (Edinburgh: Oliver and Boyd, 1972).

15. F. Harary, Aesthetic Tree Patterns in Graph Theory, Leonardo 4, 227 (1971).

16. F. Harary, A Mathematical Approach to Nonfigurative Modular Pictures, Leonardo 9, 215 (1976).

17. A. Hill, Art and Mathesis: Mondrian's Structures, Leonardo 1, 233 (1968). See also Program: Paragram: Structure, in DATA: Directions in Art, Theory and Aesthetics, A. Hill, ed. (London: Faber & Faber, 1968) p. 260.

18. F. Nake, Asthetik als Informationverarbeitung (Berlin: Springer, 1974), see Mondrian Strukturel-Topologische, p. 311.

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a view of non-figurative art and mathematics and an analysis of a structural relief

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