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A Rational Aesthetic

Essay by Alan Fowler


First published in Philosophy of Mathematics Education Journal no. 24, December 2009

This article, written from the viewpoint of an art historian rather than that of a mathematician, examines the influence of mathematics in the work of several British  artists who began working in a constructivist mode soon after the Second World. They became known as the Constructionists, a group formed in 1951 around Victor Pasmore,  of whom Mary and Kenneth Martin, John Ernest and Anthony Hill  were those whose work most clearly involved some form of underlying mathematical ‘logic’ or reasoning (John Ernest also joined the later Systems Group). Works by each of these four artists are used in this article to illustrate this approach. While these artists drew on various aspects of mathematics and geometry in the structuring of their paintings and reliefs, it is important to recognise that their aim was not to produce mathematical illustrations. To quote Anthony Hill: “The mathematical thematic or the mathematical process can only be a component: one is calculating or organising something that is clearly not mathematical.”1  And the ‘something’ was an object with a visual aesthetic, or as Hill put it: “an aesthetic of objective invention and sensation, distinctly rational and determinist …”2

The rationalist approach to abstraction developed by these artists, in which the art object is its own essentially non-representational subject, represented a largely new approach in the history of British abstract art. Abstraction in Britain had been dominated by the lyrical landscape-related work of the St Ives artists and by gestural painting influenced by American Abstract Expressionism. In contrast, the work of the Constructionists and later Systems Group artists needs to be seen as related to the long history of European Constructivism. This constructivist tradition originally had a strong socio-political rationale – a new art for a new and revolutionary society. This involved, as an essential element, the ideal of a synthesis of painting, sculpture, design and architecture in the evolution of a new egalitarian and inspirational living environment. By the mid 1960s, however, there had been a retreat from this Utopian concept in the face of disillusionment resulting from the dilution of post-war social idealism, growing fears about the Cold War and the growth of market-driven consumerism. Constructive artists began to focus instead on the internal logic of the art work, although in their reliefs and paintings they continued to display the formal characteristics of constructivist art. In essence, these features, which continue to characterise the work of most surviving members of the Constructionists and Systems Group to the present day, are:

©Copyright Patrick Morrissey and Clive Hancock  All rights reserved.


1. Anthony Hill, article in Structure magazine, February 1961 pp 59 – 63

2. Anthony Hill, statement in Nine Abstract Artists, Lawrence Alloway, London, 1954, p. 28

3. Marcus du Sautoy, review of ‘The Numerati’, the Observer, 23rd November 2008, p. 21

4. Hill, op cit, Nine Abstract Artists

5. Mary Martin, article in Structure magazine, no.1 1961, p. 16

6. John Weeks, letter to Eugene Rosenberg, quoted in Rosenberg, Architect’s Choice,  

        London, 1992, p. 17

7. Mary Martin, exhibition catalogue, Mary Martin, Kenneth Martin, Arts Council, London,

     1970, p. 11

8. Kenneth Martin, exhibition catalogue, Kenneth Martin, Tate Gallery, London, 1975, p.7

9. Kenneth Martin, transcript of radio interview with Andrew Forge, 1973,  Tate Archive

10. Kenneth Martin, interview with Lawrence Alloway in Statements, ICA Gallery, London,

      1956, unpaginated

11. John Ernest, article in Structure magazine, no.2 1961, p. 49

12. Ibid

13. See information on mathematical biography and mathematical papers in exhibition

       catalogue, Anthony Hill, Arts Council, London, 1983, pp 86 &88

14. Anthony Hill, comment in Statements, ICA, London, 1957, unpaginated

15. Anthony Hill, article in Leonardo magazine, vol. 10, 1977, p. 7

16. Alastair Grieve, exhibition catalogue, Anthony Hill, Arts Council,London, 1983, p. 44

Regardless of their mathematical content, Hill’s art works should in no way be thought of as mathematical illustrations. As he put it, a rational, non-figurative work of art such as those he constructed was no more than “one in which some aspects of its organisation 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”.15   And as Alastair Grieve has pointed out: “Judgement by eye was always of paramount importance to him and mathematical systems were only starting points or tools used in the creation of an harmonious object”.16   

The same point applies to the work of the Systems Group, founded at the end of 1969 by Jeffrey Steele and Malcolm Hughes. Members of this group specifically eschewed the Utopian objectives of original Constructivism and emphasised the rationality of the  constructional processes involved in their reliefs and paintings. Although Hill declined to join this group, several of its members retained close contact with him – a relationship of mutual understanding and respect  which continues to the present day. In 1974, Jeffrey Steele described the objectives of the Systems Group as “the production and presentation of art in which a rationally determined system is of central importance”.15  It is the contribution of mathematics to rationality which is the key to the work of artists such as Hill, rather than any attempt on their part to produce visualisations of specific mathematical concepts.

Prime Rhythms, Fig 7, is based on a manipulation of the sequence of numbers from 1 to 100 from which all even numbers are removed, leaving 1  3  5  7  9 …. 99. In these remaining 50 numbers there are 25 primes and 25 composites (numbers.divisible by more than 1).  Of  these 25 primes, 15 are consecutive – e.g. 29/31, and there are 15 black elements in the relief. (ignoring the bottom black bar which is on a lower level and serves to anchor the relief visually). The white areas in the relief equate to the composites. The sequence, width and spacing of the black and white elements in the construction derive from this arithmetic manipulation, while the whole work achieves asymmetric balance by the ‘weight’ of the black and white elements being the same in each vertical half of the relief.

The relief shown in Fig 8 is one of a number in which Hill used pieces of an industrial material – 1200 extruded aluminium sections.. The work consists of two juxtaposed tetrakti, each made up of ten units in four rows, and achieves visual vitality partly by being mounted in a diagonal format, partly by each tetraktus being the reverse of the other. It is important to recognise that although Hill (together with other constructive abstractionists) often used classical mathematical concepts such as the tetraktus and the golden section, this in no way implied any association with the mystic qualities which the Greeks, for example, attached to certain numbers. As Hill wrote in 1957: “Constructionist art is purely inventive and concerned with manipulating real entities. It is neither Academic, nor Phantasist, Classical or Romantic: it is Realist”.14  The emphasis is on structure, with various mathematical phenomena being used to evolve rational structuring processes to produce visually satisfying and self-contained objects, not as symbols of any thing external to the work itself.

In later years, Hill’s work drew on somewhat wider mathematical concepts than those in the two works shown above, with a particular reference to graph theory,  though still less complex than in his purely mathematical papers. His use of industrial materials – a feature of original Constructivism – continued, with the addition of engravers laminated plastic into which lines could be incised to produce precise linear imagery. Fig 9  is an example. The dedication to Khlebnikov  in the title of this work refers to the Soviet poet, aesthetician and figure in the Formalist Movement, Velimir Khlebnikov (1885 – 1922) whose writings Hill much admired. There are no figurative or representational implications in this title.

Fig 8. Anthony Hill, Relief Construction, 1969, aluminium, pvc & plastic, 68 x 68 x 6.4 cms

Groups in mathematics can be thought of as consisting of a number of elements (the number is the order of the group) together with a rule for combining  pairs of elements, the result of such combination being itself a member of the group. The work shown in Fig 5 (or more accurately the central black and white square) is an order 8 group table with some refinements. The sequence of elements heading the rows is not the same as that heading the columns, although this is not an essential group feature. The elements are arranged as repeated motifs in two broad bands which intersect, transforming themselves square by square.

The large wood structure – Fig 6 – was triggered by a long-standing interest of Ernest’s in mapping (in the mathematical sense) and the geometry of surfaces. – particular examples being the characteristics of the Klein Bottle and the Moebius Strip. Ernest experimented with a number of Moebius models and it became clear to him that the so-called ‘magic’ property of a twisted strip of paper remaining in one piece despite being cut apart along its centre-line was a property of the physical object and not a unique characteristic of a mathematical surface. So if true for a strip of paper it must also be true for a more substantial solid – the only condition being that the new solid must have the shape of a torus. The final outcome of this thinking was a square torus with a continuous slot cut entirely through its body in a closed curve to form a Moebius strip – the torus remaining in one piece.

These, and indeed all, Ernest’s works illustrate a characteristic duality of most systems-based art – the image as an entity appealing to the eye in a visual aesthetic sense, while close study of its structural logic has intellectual interest.

Ernest worked with Anthony Hill on a number of purely mathematical studies, an occupation which Hill developed to a level which eventually led in 1970 to him being elected Honorary Research Fellow in the Department of Mathematics, University College, London. This had been preceded in 1963 by the publication in the proceedings of the Mathematical Society of a paper to which Ernest had contributed - ‘On the number of crossings in a complete graph’. Other papers followed, several on different aspects of graph theory, while in the catalogue of his retrospective exhibition at the Hayward Gallery in 1983, Hill listed 21 professional ‘mathematical mentors’ who had contributed to his fascination with, and knowledge of, mathematical concepts.13   However, the depth and complexity of his mathematical enquiry and research was not reflected to the same degree in his art. This can be illustrated by the two works below, of which there were a number of variants. Each drew on relatively simple and age-old mathematical phenomena – prime number sequences in  Fig 7, and the ‘divine tetraktus’ combination (1+ 2+ 3+ 4 = 10) in Fig 8.

Asked by Lawrence Alloway in 1957 for a more general explanation of his concept of the relationship between his art and mathematics, Martin replied: “Mathematics in its simplest form is part of the nature of man, which he has developed as a tool for communication, for the comprehension and management of nature, and for pure delight. It is developable, not static. For instance, the concept of space in the Renaissance is not the concept of space today. My work is concerned with space and so it has something of the spatial concepts of today about it. Therefore it has something of the mathematics of today about it, and these mathematics stimulate my imagination and help me to comprehend some of the laws of the creation of form..”10   

Martin’s mention of ‘delight’ in this quote can be related to a statement by John Ernest, who wrote in 1961: “I suppose I am trying to achieve some of the beauty of a formal mathematical system in a visual experience, for it is this kind of beauty in mathematics – where the lovely abstract machinery goes into action – that moves me most deeply”.11  While stressing that it was not the purpose of mathematics to provide an aesthetic experience, Ernest explained that the aesthetic response he obtained from mathematical thinking came from its association with a formal procedure which produced a unified whole from the ordering of a number of elements. In an insightful account of the relationship between art and mathematics, he went on: “If, from a number of elements you create a consistent structure of relations, and then, returning to the elements you reconstruct these entirely in terms of the structure they form, the resulting object is composed of utterly relevant parts and is perfectly economical. Such an object cannot be contemplated without giving rise to an aesthetic emotion.”12   

Ernest, together with his friend Anthony Hill with whom he worked on several mathematical papers, had a more extensive knowledge of mathematical principles and theory than the Martins and consequently drew on a wider range of mathematical concepts. This is illustrated by the two works of his in Figs 5 and 6, below, the first of which is linked to group theory, the other to topology.

Kenneth Martin helpfully distinguished constructivist art from abstraction from nature (the characteristic of the St Ives abstract artists) and from minimalism. He wrote: “It is not the reduction to simple forms of the complex world before us. It is the building by simple elements of an expressive whole.”8  He produced a number of large constructions for permanent display in public spaces (see Fig. 4 below) and commented on the importance of these works having a significant environmental presence. In these constructions, as in his extensive series of mobiles, he emphasised the importance of the work having a logic which mathematics could provide, but explained that: “I couldn’t go into any sort of higher mathematics or topological ideas. Everything had to be reduced to simple progressions.”9  For example, in a mobile constructed of brass bars of differing widths rotated around a central rod, the number of times each width occurred was determined by the Fibonacci series (1 1 2 3 5 8 etc)  and the interval between the bars varied in relation to their widths in a range from  1/8th” to ½” at 1/16th” intervals. In other works, arithmetic progressions to govern the spaces between individual elements or to determine relative sizes of elements  included the series 2 3 5 8 12 17etc, together with numerical permutations of the number of ways two  forms could be joined – such as two brass washers joined in either a flat plane or at 90o. Basic arithmetic criteria of these kinds were used to determine the varying size of individual elements in works of the kind in Figs. 3 and 4 below, and to calculate their rotational sequences.

It would be a mistake, however, to give an impression that all her work had a direct architectural link. Most of her output consisted of reliefs in plaster, wood, stainless steel and aluminium, dominated by the square, the cube and the rectangle.  She described her constructional process by explaining that:  “I start with a drawing, often suggested by a mathematical idea, which I carry forward to a precise concept of shape and form …starting from one unit, subjecting it to a logic and accepting the result without any artistic interference”7 The mathematics principally involved the arithmetic of proportional relationships, permutational systems, the golden section ratio and various number series. The use of the Fibonacci series can be seen in the dimensions of her Climbing Form, Fig.2, below.

Fig.1.  Mary Martin, Wall Screen, 1957, Musgrave Park Hospital, Belfast; stainless steel, brick, painted plaster; 203 x 228 x 34.3 cms

Fig.2. Mary Martin, Climbing Form, 1954 – 62, wood, Perspex, stainless steel, 55.2 x 34.2 x 8.2 cms

Fig 3. Kenneth Martin, Screw Mobile, 1953, 83.8 x 30.5cms  Engineering Laboratory, Cambridge

Fig 4. Kenneth Martin,Construction in Aluminium, 1967, Engineering Laboratory, Cambridge

Fig 5.  John Ernest, Iconic Group Table,painted wood, 213.6 x 213.6 x 58.4 cms

Fig 6. John Ernest, Moebius Strip, 1971/72

1977, gouache on wood structure, 89.5 x 89.5 x 8 cms

Fig 7. Anthony Hill, Prime Rhythms,1960, plastic, 31 x 29 cms

The engraved lines in this shallow relief are the four long vertical and horizontal lines plus nine crosses in three rows within the work which are partly obscured by the nine pieces of plastic on which they are centred. Each piece of plastic consists of a variant of three L-shapes. The nine shapes in this relief are derived from the 65 possible variants of a mathematical tree which has seven points and six lines in an orthogonal format. From these 65 possibilities, Hill selected the nine which shared two common features – each has five right angles and each can be drawn by connecting three L shapes as in the example in Fig 10, below.       

Fig 9. Anthony Hill, The Nine – Hommage A Khlebnikov No. 2

1976, engraved laminated plastic, 91.4 x 91.4 cms

Fig 10. One of the nine possible configurations of the basic tree used by Hill in the work shown in Fig 9. Taken from his article in Leonardo, vol. 10, 1977:  ‘A view of non-figurative art and mathematics and an analysis of a structural relief’, p.11

Both these links with mathematics influenced the thinking and work of Mary and Kenneth Martin. Describing constructive art as architectonic, Mary Martin was interested in the extent to which art and architecture formed a distinct dualism or alternatively might achieve a degree of synthesis. Writing in 1961, she suggested “whereas the work of art in architecture formerly often had a dualism, the fact that the constructionist can use the architect’s materials, dimensions and concepts opens up the possibility of dissolving this dualism and producing an architectonic unity”.5  She had been able to put this view into practice in 1957 when she designed a wall screen in a Belfast hospital, working closely with the architect John Weeks. Weeks later explained that he and the Martins “had been speculating about the possibility of a totally integrated art work and architecture for some years, and this was the first opportunity of trying it”.6  In determining the dimensions of this work (Fig. 1), Mary Martin applied the same arithmetic properties as the modular grids on which the building itself was designed – as well as using only the materials available on site.

The involvement of mathematics in constructivist art can be seen as having two main derivations. One stems from the concept of ‘construction’ itself, which points to the link with architecture which was one characteristic of the original constructivist philosophy.  Architects are concerned with the articulation of space through the design of buildings constructed from constituent elements in accordance with pre-determined proportional relationships. Calculation and measurement – the use of mathematics – is integral to this process. Constructivist artists are similarly concerned with the manipulation of space within and around an art object and the proportional relationships between the elements from which the work is constructed - with the necessary corollary of calculation.

The second link with mathematics is more fundamental and relates to the essence of mathematics itself. Marcus du Sautoy recently suggested  that “mathematics is all about spotting patterns and finding the underlying logic of the seemingly random and chaotic world around us”. 3 For artists searching for an approach which emphasised rationality or logic in opposition to what Anthony Hill once termed “the subjectivist limbo in which the greater part of art today is submerged” the use of mathematics as a structuring element was clearly apposite.4 It provided a basis for a logically systematised process for evolving patterns of imagery, and could also be argued as relating the constructed abstract art object to a deeper reality of the world (its underlying mathematically defined structure) than could be achieved by any form of representation or interpretation of visual appearance.