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The Life and Work of Godfrey Clive Miller, 1893 - 1964

PhD thesis by Ann Wookey

Chapter 5. Miller's application of mathematics in painting, ca 1929 - 1942 (link to notes)

Miller's mature paintings such as Triptych with figures [Plate 75] are generally characterised by underlying geometrical matrices [emphasised in Plate 76]. These grids served various purposes, a major one being as design matrix for the painting's composition. This chapter examines when Miller began to use design matrices or the "skeletons" of his work, the incidence and type of mathematical notions in Western art, Miller's thought about mathematics during the 1930s and his appropriations from mathematics into painting. Evidence of Miller's gradual adoption of structural matrices lies in the early transitional paintings. The theoretical approach he came to rely upon in establishing structure and design can be designated proportional rhythm theory, with dynamic symmetry dominating in the late transitional and mature work.

Hidden agendas: early design matrices in Miller's art

That Miller never dated his work has made even more difficult the historian's task of establishing the origins of the mathematical grids that mark much of his art. The tracing of these origins in turn has accentuated the significance of other disciplines for the art historian for it was through co-operation with conservators that Miller's earliest interest in design matrices became known. Four paintings were involved: Dusk, Warrandyte (Early landscape, Warrandyte) [Plate 5], Warrandyte [Plate 16], Still life, London (Table group, London) [Plate 12] and Jug and egg [Plate 14].

An early version of the artist's mature matrices is seen in Dusk, Warrandyte [plates 5 and 6]. Documentary evidence from 1935 largely supports my belief that the picture is a Melbourne work from the 1920s over which, quite possibly late in the 1930s, the grid has been transcribed in pencil and the whole then sealed with varnish. The canvas is small and thereby consistent with Miller's recalling in October 1935 that his earlier Melbourne paintings tended towards the diminutive. A mid-1930s dating for the matrix becomes possible, although indirectly, from other passages in his letters. The structure partly takes its impetus from the system of dynamic symmetry propagated by Jay Hambidge from 1919. As noted previously, in September 1935 Miller had three paintings underway that coincided almost exactly with, as he termed them, "dynamic rhythmic" principles copied from a book Dynamic Rhythmics, which were 'less than 1/8 inch out, in 14 inches' [35 cm] 1. Measurements exclude Dusk, Warrandyte from being the work to which he referred. Nevertheless, the pattern discernible for the artist's progression towards using matrices of this type means that date remains a possibility for this application. An alternative is that Miller applied the grid in the late 1920s before he left Melbourne for London for the first time, which, as my introduction indicates, is Eagle's view. Alternatively, it could have been applied experimentally within a broader exploration of design matrices made after 1929 and into the very early 1930s - perhaps over his first London days, or on the brief return to Melbourne in 1932, else once back in London after 1933. Finally, as we will see, as late as 1938 or thereabouts is a possibility.

When late in 1988 Dusk, Warrandyte surfaced in Melbourne it initially caused some consternation since the National Gallery of Australia holds the almost identical painting, simply entitled Warrandyte [Plate 16]. Consequently a comparative assessment of the two paintings was made at the National Gallery in Canberra. In so doing, a matrix was recognised as probably having been used for Warrandyte [Plate 16], its double. The diagram of Plate 17 clarifies this structure. The discovery of the hidden matrix in Warrandyte caused a search for confirmation in other of Miller's earlier paintings. This came with the examination professionally for conservation purposes of Jug and egg [Plate 14] and my less professional study of Still life, London (Table group, London) [Plate 12]. In Jug and egg small "pin" holes located around the edge of the canvas can indeed be joined to give the structural matrix of Plate 15. My mapping of Still life, London (Table group, London) generates the matrix of Plate 13 2. The patterns extracted are as consistent in their organisation and directional regularity as that for Warrandyte [Plate 17]. These diagrams hold interest on two grounds. First, the linear patterning of each is firmly established on diagonals that lie at or close to 45o. In other words, the grids are distinctly regular.  Secondly, a number of interlocked Ø-ratio proportions exist within the diagonal patterning present; instances of these are indicated only in Plate 15. An important point to make is that unlike later matrices, these tend not to be compositional grids at all since they do not especially determine Miller's placement of forms. However, the matrix of Still life, London (Table group, London) [Plate 13] provided some compositional guidance since the upper left to lower right diagonals give the direction of the banana and the knife.

In the absence of documentation from before late-1934 for Miller's development, circumstantial evidence becomes significant. There are three main aspects: the Slade and New English Art Club, the contemporary exhibition scene and the influence of proportional rhythm theory. Firstly, the Slade and New English Art Club: as noted previously, Still life, London (Table group, London) holds similarities to Sweet's My supper table [Plate 161] hung with the New English Art Club in 1930. Miller also probably attended Sickert's lecture on drawing given around London over 1929-30, which was, Laughton records, 'very explicit on how to draw, and then how to square up and transfer the drawing to canvas for painting' 3. The matrix of Warrandyte [plates 16 and 17] can be explained as only a variation on the squaring-up grid since the pattern does not readily associate with the image as a compositional matrix. For Miller, it enabled the exact copying of the original, Dusk, Warrandyte (Early landscape, Warrandyte) [Plate 5]. These factors suggest 1929-31 as a possible date for Still life, London (Table group, London) [Plate 12], Jug and egg [Plate 14] and Warrandyte [Plate 16]. However, because of their colour treatments a date around 1932-34 is more likely. In addition to the reasons previously proposed for accepting this later dating, Miller's letters from 1934 onwards indicate his interest in structure and mathematics.

Secondly, early Italian painting and contemporary School of Paris work on view in London galleries in the early 1930s provide the clues to Miller's subsequent development of mathematical structural matrices. Italian wood-panel painting from the 13th to the 16th centuries is characterised by the generation of geometric form using compass and ruler, especially for architectural detail. Pictures in London's National Gallery, for example, Sassetta's Saint Francis and the wolf of Gubbio, 1437-44, gave Miller a ready historical model for adopting a like practice 4. In addition to the use of geometric method to establish this painting's architectural detail, an incised parabola describes the flight grouping of birds in the right-hand panel. Meanwhile, School of Paris painting on show in London was influencing the move towards abstraction by British artists. This thesis has argued that Miller's impetus to using structural matrices partly lay in Synthetic Cubist practice. In essence he took the planar attributes of Braque's early paper collages and Synthetic Cubist pictures and systemised their organisation by submitting the planar patterning to a more rigorous mathematical method. The resultant skeleton structures were further adapted into the structural design system that typifies his subsequent work. Miller's absorption of Cubist practices was matched by his increasing dissatisfaction with representational painting. A general involvement by the artist at the time with scientific and mathematical matters has been noted. His experience of the stringed mathematical models at the Science Museum at South Kensington also may have inspired the incorporation of mathematics into his work, just as these had for Henry Moore. Like Moore and others of the London avant-garde during the 1930s, Miller was turning away from the naturalism encouraged at the Slade towards the Modernist persuasion. In so doing, a variety of contemporary practices began to emerge in his work. And one of these was the use of mathematically-based structures in establishing design.

How and when proportional rhythm theory came to influence Miller has various possibilities. Miller's acquaintance with Tonks after 1933 provides an opportunity for his learning of two-dimensional schematisation. In the previous year Tonks had written to another ex-student:

Now as to Dynamic Symmetry. Do not altogether dispense with it . . . [T]here are certain proportions which are eternal, largely based on the series of numbers . . . 1 . 2 . 3 . 5 . 8 . 13 . 21 . 34 . etc found all over the animal and vegetal kingdom. Man has used them . . . since the Greeks 5.

Sweet knew by the late 1920s of Ghyka's Esthétique des Proportions dans la Nature et dans les Arts of 1927. Although this allows for the two friends having discussed its contents at that time, a mid-1935 date is more likely 6. Ghyka allocates a full chapter to Hambidge's dynamic symmetry method for organising a plane surface. Hambidge first published his work as a monthly magazine, The Diagonal, over 1919-20 and as the book, Dynamic Symmetry: the Greek Vase, in 1920. Proposed as a means of understanding natural design law, the system reflects Hambidge's realisations of parallels between Greek architecture and art and design in nature. Further volumes followed 7, while adaptations appeared through the 1920s and early 1930s from other design theorists such as Ghyka. Consequently the method was a significant influence on design practices through these years. The 1927 Ghyka volume could accordingly have been Miller's source for the pencil grid in Dusk, Warrandyte (Early landscape, Warrandyte) [plates 5 and 6]. However, Dusk, Warrandyte is strictly not a dynamic symmetry application since the outer proportions of the painting reduce to a ratio of 0.79, the √Ø rectangle, which lies outside Hambidge's √2 √3 √4 √5 rectangular system. Miller's method may derive indirectly from Hambidge, but a more direct source may be a still later book by Ghyka. In Esthétique des Proportions dans la Nature et dans les Arts from 1927 Ghyka had provided a construction for the √Ø rectangle, but apart from guide lines his diagram is in outline only 8. The internal patterning of Miller's matrix accords more with the complexity Ghyka introduced for a differently proportioned shape, the Ø rectangle [Plate 170, left hand figure] in Le Nombre d'Or of 1931 9. A similarly complex diagram for the √Ø rectangle [Plate 171] was not published until 1938 10. The claim in The Oxford Companion to Twentieth Century Art that Miller, in 1936 is inferred, read Ghyka and Valéry 'on the hidden geometry of art' points again to this 1931 volume 11. Le Nombre d'Or commences with a letter from Valéry 'de l'Académie Française'. The date of around 1936 in turn accords with the artist's own statement from 1935 on having three paintings underway that coincided almost exactly with "dynamic rhythmic" principles. Moreover, in September 1938 he made a further two important admissions. Firstly, that he knew Hambidge's writings - because of a reference in a "French book". Secondly, that he had been engrossed for the past two years in evolving:

. . . a system made of a blending of a logic basis or framework on to which intuition or personality can be placed. I have studied a particular line of mathematics consisting of a dynamic rhythme [sic] of rectangles . . . [A]mong other qualities these rectangles are related to pentagons, etc. With these impressed on the mind I look at things such as trees [and] mountains and contrive to see . . . a demonstration . . . of what I had previously done with compasses and lead pencil. In a painting the logic basis shld [sic] be just apparent giving . . . the genesis of the picture-idea. This relationship or poise between logic and . . . personal intuitiveness is the great creative thing . . . [Y]ou won't see the practice of the principle in most painting 12.

A fine example of Miller's work demonstrating this fitting of rectangles, the circle and the pentagon to landscape, although from a later date, is Untitled (Tree and mountain series) [plates 73 and 74].

The second half of the 1930s remains the most likely time for Miller coming to work with a particular structural system. However, matrices that lie outside the system such as those of Still life, London (Table group, London) [plates 12 and 13], Jug and egg [plates 14 and 15] and Warrandyte [plates 16 and 17] can be dated some years earlier.  Above all else these hidden grid structures of the more representational Modernist canvases indicate the attraction to Miller in the early 1930s of using some systematic structural method when painting. But he had not as yet alighted on a suitable method. His progress is marked by the types of structures found in these pictures being better resolved in the slightly later work, Growth [plates 35 and 36]. By the time of this canvas Miller had moved his structural matrices into full display as an integral element of the paintings. As he explained in 1938, 'I like the idea of having the genesis of the painting at least discernable [sic] coupled with an expression of individuality. I have in unadvanced stage some paintings in this manner' 13.

The artist's schematised approach to design was unusual for Modernist painters of his day. The golden section had found adherents from the Cubists onwards but few other painters worked as did Miller. Two reasons explain his being so strongly drawn to the technique. First is the design assistance given to composition. The other is the meaning carried into these works by the symbolism intrinsic to the particular system he adapted. While the dating of Miller's pictures remains contentious, recognition that a structural approach informed these naturalistic paintings contributed greatly to clarifying Miller's development of the geometrical design matrices that so characterise his mature paintings. To understand Miller's interest in making matrices inherent to his expression, his practice must be placed in the context of the æsthetic and practice therewith of Modernist art in the mid-1930s. This in turn means tracing the types and incidence of mathematical approaches in painting, especially through the late nineteenth and early twentieth centuries.

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Mathematics and painting: the late nineteenth and early twentieth centuries

The mathematics handled by artists is almost completely geometrical in kind. Given the philosophical interest by early twentieth century avant-garde intellectuals and artists in ideas of archetypes and universals, the first issue that bears examination is the view held at that time of geometrical forms. There are two alternatives - as innate a priori knowledge or as learned phenomena. If the latter, their use in artistic expression could not have come, as they did, to constitute archetypal prototypes or universals for later artists like Miller.

Kant argued geometrical axioms to be synthetic a priori knowledge. During the nineteenth century new geometrical conceptions entered Western thought, namely non-Euclidean geometry and the geometry of n-dimensions. Henderson maintains that non-Euclidean geometries challenged Kant's view through their substitution of a variety of ideas of space. Helmholtz, for one, held that the existence of the new geometries proved that geometrical axioms were empirical in origin and therefore not a priori. Helmholtz held also that space, whether Euclidean or non-Euclidean, could be intuited 14. Helmholtz's work significantly influenced Charles Henry, thus allowing the possibility of these views entering Neo-Impressionist doctrine 15. The important late nineteenth and early twentieth century French mathematician Henri Poincaré equally emphasised the importance of intuition to spatial understanding. In addition, as Henderson discusses, Poincaré believed geometric axioms to be neither synthetic a priori nor empirical knowledge but conventions. His premise derived from the unmeasurable nature of space itself [only bodies in space can be measured]. Henderson notes that 'the question, "What is the true geometry?"' came thus to be proven meaningless. The conception of absolutes was under challenge with the result that ideas of relativities of knowledge gradually gained wide acceptance 16. This enabled the concurrent holding of significantly different views within a single social milieu, such as by the School of Paris artists both before and after World War I. In so doing it engendered the freedom of individuals to derive their methods from wherever was personally found appropriate. Thus, for Analytic and Synthetic Cubism a single commonly held mathematical system cannot be isolated. Most often the position assumed for the geometrical play with planes that is the signature of Synthetic Cubism arose from the artist following his or her intuitions and not from a specific convention-bound geometrical system as such. The attitude found theoretical justification in Wilhelm Worringer's writings as early as 1908, which argued that the urge towards abstraction and the creation of geometrical form in artistic expression was purely instinctual 17. Thus the return to order and structure reflected in art post World War I was for the most part a return achieved by intuitive means. Few of the artists, excepting perhaps Gris, attempted anything of great mathematical complexity in their adaptation of geometrical ideas into their art 18. Usually the application of the golden section [Ø ratio] alone appears as a proportional device. The golden section is a classical mathematical notation believed intuitive because of its capability for describing some life forms in nature. The device thereby came to symbolise growth, eternity and the Godhead. Its use was thus most appropriate for artistic aspirations over the late 1910s and through into the 1930s towards an intuitively structured and ordered universe. Lastly, within the new understanding of mathematics, the forms of plane and solid geometry became intuitively known symbols rather than absolutes of knowledge. This allowed them to be conceived as representations of archetypes or universals by the Modernists of the 1930s and to be internally consistent [in the philosophical sense].

Historically, mathematical methods can be claimed to have played a major role in Western art. As implied above there are two modes by which this happened. One is the artist's use of proportional devices, or the holding to simple dimensional ratios, 2 to 3, 3 to 5, 3 to 6, 3 to 7, etc, in constructing a form or area. The second comes through recourse to elements from planar and solid geometry [the square, the rectangle, the triangle, the circle, the cube, the cylinder, the cone, the sphere, etc] in creating the form expressed. The two modes are not necessarily mutually exclusive. Proportion naturally enters the realistic description of solid forms on the plane surface in their aspects of linear perspective or in application to their dimensions in general. Linear perspective itself is a convention based on the optical evidence that parallel lines moving into distance merge to a single central point, or to two points or more, on the horizon. To these lines are relegated all planar movements into distance, while all other planes lie parallel to the picture surface 19. Proportion enters the system through the requirement of exact representation to scale. As an example, in the realistic representation of a scene in which a hut a hundred metres distant appears somewhat smaller to the eye than the same hut ten metres away, this proportional diminution would be maintained. Equally, if a tree-trunk forks mid-way up, this relationship between the parts and the whole would reflect in the artist's representation of the form.

Traditionally from Renaissance times through into the twentieth century, systems of proportion and perspective were taught in artists' workshops, academies and schools of art. In the nineteenth century textbooks about proportion focused mostly on the human body and architecture. A variety of systems may be identified. Some approaches were based around body proportions then recently analysed in Egyptian art or known from classical Greek art, while others seem quite individual to the author concerned 20. For architecture, systems based around Egyptian, classical Greek and musical ratios emerged. Often one of these was examined in conjunction with either one other or both others 21. Through their application in architecture, systems of proportion became carried over to design in general and so into the decorative arts. Macdonald stresses the mostly arithmetical nature of these systems 22. Their accompanying æsthetic ideal, the achievement of rhythm, harmony and symmetry in artistic expression, was therefore most commonly understood in terms of regular and equal proportional division.

This tradition of interest in proportion was continued into the twentieth century. At the Slade, in 1929-30 lectures given by A T Porter on perspective and solid geometry were compulsory unless the student was exempted by Tonks. A textbook by Porter, The Principles of Perspective and their Application to the Representation of the Circle and the Sphere, is held still in University College Library. Porter there refers to a cone found in every art school that was constructed to show its three types of section, the triangle, the circle and the oval 23. Miller alluded to this object in letters dated to mid-1935 recognising that although the triangle and circle were the same [that is, sections of the cone], addition of the parts would not add up to the whole 24. These letters show his exposure to traditional teaching about perspective and proportion.

In 1930 Bayes succeeded Porter as lecturer on perspective and solid geometry 25.  Bayes' The Art of Decorative Painting of 1927, which was written for the "average artist", introduces isometric drawing as an alternative to linear perspective. Isometric drawing assumes 'an infinite distance point' so that the "eye" [in an idiosyncratic sense indicative of an artist's perception] becomes:

. . . not a point of consciousness but a plane parallel to and of equal dimensions with the plane on which the drawing is assumed to be done, so that the rays conceived as passing from the object represented through the wall to that eye would be parallel . . . rather than as in an] ordinary perspective arrangement . . . [as] converg[ing] towards the eye 26.

The method introduces perspective distortion into images without fully flattening them and a 'lengthening of dimensions away from the point of sight' 27. This lengthening of proportion away from what is more properly the "plane of sight" [see footnote] provides a similar representational outcome to that obtained by applying Miller's conviction that parallel lines diverged into distance. Alongside this advice, Bayes suggests that spatial arrangements be constructed in small cubic blocks. This practice indeed was taught at the Westminster School of Art in London during the 1920s when he was principal. He attributes this 'designing in solids under the guise of painting in area' to Piero della Francesca, Mantegna and Hogarth, and so as not new 28. The approach starts with a grid of equal area squares or rectangles onto which the cubic forms are then placed. A kind of Cubism is the outcome shown by illustrations in The Art of Decorative Painting. The sense of the forms of solid geometry found in Analytic Cubism is there, but now set in a fully three-dimensional space rather than the uncertain artist's space alluded to by those Cubists. There is no direct evidence for a connection between Bayes' teaching and Miller's work but there are similarities of thought. These include the observation about lengthening dimension in distance given above. Another is Bayes' recourse to contemporary building construction processes using steel girders as an example of the "exciting" complexity of three-dimensional transparency [that is, space as volume] 29. As we saw, this metaphor coincides almost exactly with a similar description from Miller in 1936. Lastly, Bayes' bibliography included important texts for the arts regarded elsewhere in this thesis to have held Miller's interest. These are Chevreul, Goethe and Rood on colour, and Hambidge on construction and design [that is, proportion and perspective] 30.

The use of regular grid matrices against which to design a picture, or organise the elements of an artist's vision, is an important difference in application from using the grid for squaring-up so as to enable an exact copy of another work. Organisational matrices are found in the nineteenth century decorative æsthetic concerning architecture and the applied arts 31. A similar use of construction or design matrices entered the Modernist painter's æsthetic only with the revitalisation of the French avant-garde in the aftermath of World War I. Nineteenth century theoretical discourses on systems of geometrical frames, or design matrices as I denote them, most often emphasised a connection between nature and the geometrical approach under discussion. An exception occurred prior to World War I when actual geometrical form [in contrast to "frames"] entered the æsthetic and artistic practice of the Futurists in Italy, artists like Léger in Paris, the Rayonists in Russia and the Vortists in England. Broadly the geometry in this art was a metaphor for modern industrialised society. Regular structural matrices based upon geometry were used by only some of these artists; for instance, the British painter David Bomberg. More generally simplified geometrical forms were depicted. These forms resulted from the connection perceived between modern life, man's industrial achievements and machine design. The last of these is based in geometry, hence these artists' recourse to the forms of plane and solid geometry and the occasional organisational matrix. After the war the machine æsthetic was no longer hallowed by the avant-garde in general since they perceived that man's industrial achievements had ultimately led to appalling human waste and psychological distress 32. Rather, the patterns of nature once again came to be believed to more appropriately hold the key to accomplishing harmony in artistic expression, and to spiritual salvation. Geometry could still assist the artist - when the geometrical forms and systems that were incorporated into creations referred to their occurrence in nature. The changed aspirations of the post-war era towards the ordering and structuring of man's "natural" experiences might then be reflected.

Christopher Green identifies Gris, Jacques Lipchitz and Metzinger as working with "geometric armatures" during 1916 and 1917 to produce the increasingly "synthetic" style of Cubism that he, Green, has denoted "Crystal Cubism". After the war the group came to include María Gutíerez Blanchard, Herbin and Severini, together with Braque for a short while. Of these artists, Gris, Lipchitz and Severini occasionally incorporated the golden section into the structure of their art works 33. Green defines the synthetic process of twentieth century art in which the artist no longer works directly from the motif but rather builds up 'with a range of elements set free from any origins they might have in observation' 34. He writes also that the work of the Crystal Cubists:

. . . made a material statement of its resistance to the transience of everyday life, and by doing so accentuated the gap between the æsthetic and the non-æsthetic with a new and deliberate insistence. Everything about it was to be seen as stable, enduring and pure; everything about it invited that most adhesive and durable of analogies, the classic analogy, with all that it implied concerning art and the ideal 35.

With this move towards synthesizing design and form as artistic expression more artists came to incorporate elements of planar and solid geometry in their pictures. The tendency culminated in the fully abstract art that first found a Parisian audience towards the close of the 1920s and into the early 1930s. Moreover, as Green observes, although Gris in 1916 still built his work from observation,

. . . the possibility is unmistakably announced of the abstract geometric structure itself as the starting point, its main lines generating configurations of flat shapes which need not be given a specific denotational meaning until the pictorial idea is quite far advanced 36.

This assertion can be expressed in another way - that Gris' work then foreshadowed the acceptance in the fine arts of a design practice, the deployment of geometrical matrices as a commencement point for the artist's creations. This was the position that Miller reached in the 1930s. Like others of his generation, Miller's inclinations here were likely to have been spurred by his experience of the Great War. Partly in reaction to the despair about the human condition engendered by the destruction and horror of that war, French artists had afterwards turned towards the ordering and structuring of experience. A similar reaction partly explains the interest in design from the late teens through the 1920s and into the 1930s found among British contemporary artists 37.

The application of regular [arithmetic] ratios in establishing proportional harmony was not the only approach examined by theorists and adopted by artists. The irrational [logarithmic or geometric] ratio of 0.617, the golden section, was recommended in England as early as 1856 in Jones' design propositions 38. His The Grammar of Ornament was disseminated as a key reference in public art schools during the latter half of the nineteenth century throughout Britain and her dominions. Jones was thus critically influential on design practice through into the twentieth century. Although the Slade remained independent of the public art school system [both through its founding bequest and as an institution devoted to the fine arts], his theory can be expected to have been taught there 39. Jones was also available in French translation and consequently was known across the Channel.  In France the golden section was one of several ratios propagated in 1921 by Paul Sérusier in A B C de la Peinture, for instance, although alternative sources are also probable 40. Publications of the 1930s and later attest to the continuing preoccupation of the international Parisian avant-garde with mathematical ideas in art and with the golden section in particular. The Manifeste Dimensioniste of 1936 by the French painter Charles Sirato had as signatories twenty-six Modernists including Nicholson, Miró, Arp, Robert and Sonia Delaunay, Marcel Duchamp and Kandinsky. Sirato argues for a non-Euclidean [N+1] dimension in art equivalent to the recent space-time dimension in European thought 41. Another publication of note for the decade is E Beothy's La Série d'Or. Theorie et Methode Practique in which, similarly to Miller's beliefs, harmony in life and art are given as the union of conflicts [or opposites] 42, becoming an æsthetic that corresponded to the uncertainties of the late 1930s. The golden section was one means to this achievement. Although too late to be a primary source for Miller, Beothy's book helps establish the continuity of European curiosity about the topic. Interest in the golden section through contemporary European culture perhaps culminated in a conference devoted solely to the notion held in Milan in September 1951, although publications on the topic continue 43.

There is no reason to expect that French art school teaching during the late nineteenth and early twentieth centuries placed a lesser emphasis on the appropriateness of these mathematical applications in painting than their English equivalents. However, by the turn of the century, along with the rejection by the French avant-garde of all in the nature of the "academic" or rules in art, came the rejection by some artists of one and two point perspective in visual expression. Since the system was based on conventions that could only approximate reality, perspective amounts to illusion in art, and illusion was the faux pas monumental of the Modernist's creed. A "patterning" of surface replaced the painter's previous recourse to perspective, a practice that lay very close indeed to that in the decorative arts. This in no way meant a lack of awareness by later avant-garde artists of the roles mathematics could assume in expression.

The illusory nature of mathematically-derived perspective in visual expression was reason, then, to turn some contemporary artists towards alternative practices with artistic space. Others held mathematics to be an artist's tool just because of the success with which it contributed to explanations of reality [as mathematics could for science]. There were also the nineteenth century geometrical descriptive schemas not based solely in Euclidean mathematics [as was traditional perspective] to tantalise the avant-garde imagination. These were non-Euclidean geometry and the geometry of n-dimensions. Non-Euclidean geometries developed from the belief that Euclid's fifth postulate involving parallel lines was self-evident for only one type of surface, the flat surface, and thus could not be regarded as a geometric axiom 44. This premise allowed the existence of other geometrical systems besides that of the ancient Greek geometer. Nineteenth century mathematicians invented two alternatives. Lobachevsky-Bolyai geometry held for surfaces of negative curvature [a concavely-curved cone, for instance] and Riemannian geometry, for surfaces of positive curvature [the outer surface of a sphere, for example] 45. Naturally Euclidean geometry continues to provide appropriate geometrical descriptions of phenomena on a flat surface. The excitement these new conceptions raised for the visual arts concerned the possibilities of variously shaped space that they entail [Riemannian geometry especially]. Henderson notes Metzinger's Cubist landscape of 1911 where the landscape curves in on itself as an attempt at incorporating Riemannian geometry into artistic expression 46. A further implication of her writings is that the deformation found elsewhere in Analytic Cubist form equally represented the idea 47. Few of these points about non-Euclidean geometry can be recognised in Miller's pictures or writings. Certainly by the mid-1930s deformation had entered his artistic vocabulary. However, rather than representing non-Euclidean space in the manner of the Analytic Cubists, deformation was an adjunct to the abstracting of archetypes or universals from groups of objects in nature, and therefore from the three dimensional world known to the human senses and not these more imaginary spaces 48. A similar claim cannot be so readily made for Miller and n-dimensional conceptions, particularly the fourth dimension. As we have seen, the means of visually realising these ideas extend beyond mathematical description: if indeed Miller did represent a fourth dimension, he most likely used colour methods. In other words, Miller's matrix configurations convey neither the fourth dimension nor non-Euclidean geometrical precepts.

In summary, Miller's introduction of mathematical structures into his art during the 1930s was consistent with contemporary artistic practices. His base remained traditional Euclidean geometry rather than the more contemporary methodologies drawn from non-Euclidean geometry and n-dimensional theory. As we will see, his recourse to a system where the golden section is generic served to instil symbols of order, purity and idealism into his expression. He thus came to represent, but at a later date, very similar conceptions to those which had inculcated the viewpoints and practice during the late-1910s of those European Modernists who had returned from World War I to civilian life as painters working in an increasingly abstract mode.

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Miller on mathematics and his sources

'[T]he Universe Maker was a master of Mathematics' 49. This phrase written towards the end of 1934 is Miller's first reference to "mathematics". The metaphysical allusion is crucial to understanding the artist's intention for the sub-structures that made their appearance in his pictures - those linear matrices over which he orchestrated line and colour. Miller's appreciation of the role that mathematics and geometry occupied in man's intellectual life and of their relationship with nature changed subtly around the mid-1930s. Many ideas mentioned in his letters can be traced to the literature of the period and the more technical aspects of this literature appear directly in his paintings, underlying their surface action. As outlined previously, by the close of 1934 Miller held existence to comprise organic and inorganic form. The instinctual attributes of growth and motion characterised organic form. By contrast, the intellect understood inorganic form, a "fixed" thing, which could therefore be described by geometry. A necklace where the string signified life's instinctual flow or motion and the beads, the step implicit to rational thought, provided his analogy for the unity inferred 50. The probable source of this analogy was Bragdon's The Frozen Fountain from 1924:

Only by schematization can multiplicity be resolved into unity, and unity split up into multiplicity without itself being lost in the process: like the string in a pearl necklace, or the movement of the baton of an orchestra-leader, schematization is an agent of unity, the preserver of union 51.

Miller's recourse to the analogy in late 1934 implies his awareness of Bragdon's work by that time. Importantly, Bragdon went on to prescribe a two-dimensional schematisation system, dynamic symmetry, precisely that to which Miller came. Doubtless Bragdon's preference for the philosophically "organic" structures of dynamic symmetry resulted partly from his architectural background and partly from his Theosophical beliefs. The same could be argued for Miller. Furthermore, over the following eighteen months Miller altered his view of geometry as a "fixed" construct of "inorganic" value to encompass the organic.

It seems certain that by 1935 Miller was in the necessary impressionable frame of mind to absorb a schematised approach to design. In April 1935 when already searching for an alternative to naturalism, his experience of Moorish buildings and gardens led to a belief that geometrical form had enhanced the Moor's expressive ability. He appreciated their use of geometrical form both as symbolic invention and as an organising principle in design capable of infinite variety. He also admired their lack of a 'slavish adherence to accuracy' in the formalised geometry he encountered in Spain and the 'distinct . . . unity of thought' present 52. Miller's observation of Moorish courtyards as 'the neuclus [sic] whence all radiates and to which all runs in parallel terms' 53 is reflected in Cityscape [plates 32 and 33]. The surprising variety among the internal structural matrices that he went on to create also occurs in traditional Islamic art.

Into mid-1935 Miller continued to associate geometry with the inorganic. But its sense had seemingly expanded to encompass architecture [the Moorish gardens, for instance] and man-made objects, in other words items structured by man's intellect. As he wrote in May of that year when characterising line, 'The fish line is not going to have anything in common with geometry - that puts out architectural lines, streets and such as pots, bottles, etc' 54. The study of Chinese painting led him to remark in July that:

[I]n painting there is an abstract construct element . . . [I]n a Chinese painting it is possible to imagine all local things . . . as removed with the result that a substructure is left. This substructure cld [sic] be used for an entirely different picture . . . [A] mathematician had left his blackboard with ellipses, parabolas and lines drawn on it - and the pupils . . . make a bridge of the ellipse, a river and hills of the lines and curves and a sky rocket of the parabola . . . The same lines . . . cld [sic] have been used as a structure for quite different things. Constants and variants or - Plato's "actuality flowing into ideas". This is the basis of the painting - not English. The ordering gives comprehension and unity of vision 55.

Henceforth in Miller's letters the rationalised "stages" of intellectual process became increasingly disparaged as "arithmetic". Now "mathematics" was reserved for processes to do with achieving the metaphysical ideal of unity amongst separateness 56. This is much the same as Bragdon's argument about schematisation as necessary for the resolution of multiplicity and unity. Two months later in September 1935 when Miller first referred to dynamic symmetry, he sought in mathematics 'the basis of the evolution of design' as a firm base from which to move to free-form expression 57. It had rapidly become a necessity for Miller that mathematics should have an organic association. He had come to desire that geometrical method be able to describe movement and rhythm, those instinctual and intuitive attributes of life and existence. The elements the artist commanded, structure, form and colour, would then be merged in the work of art into a representational unity that in itself symbolised life or creation.

This conception of geometry as symbolic expression is encountered often in British writings on art of the 1930s. Wilenski in The Meaning of Sculpture published in 1932 justifies a significant mathematical bias by drawing on Plato's definition of beauty from Philebus with its recourse to geometrical form as the highest form of beauty. Wilenski then identifies two types of "geometricisation": one based on "visual experience", by which he presumably means the forms of planar and solid geometry, and one on "natural structure", which symbolise natural form. The photographs by Blossfeldt of spiral forms in nature provide his example. Wilenski also recognises relations of geometrical form as symbols of 'a universal analogy of form' 58. This emphasis on relationships would later make an appearance in Miller's thought. It is quite probable, then, that the artist was aware by mid-1935 of Wilenski's work. Brooks, whom I have discussed previously, argues more broadly than Wilenski to carry these symbolic connotations of geometry over to the picture itself 59. Writings of the early 1930s such as these were obviously pertinent to Miller's own development. His mathematical schema from the late 1930s onwards drew mostly on dynamic symmetry where such symbolic undertones are integral. The associated theory may be designated proportional rhythm theory.

Miller's introduction to proportional rhythm theory perhaps came through Tonks and Sweet. Alternatively, Bayes may have lectured on the method at the Slade and Miller learnt of it there. Given Miller's reference to the art school cone in mid-1935 it seems probable that he had attended Bayes' lecture course. The most likely account combines Tonks as briefly introducing the approach, Miller's attendance at Bayes' lectures during 1934 or early 1935, and Sweet bringing Ghyka to his attention in the second half of 1935. Support for a mid-1930s dating comes from Miller's later accreditation of Sweet as influential in having talked about geometry and the golden proportion over coffee, from the claim that Miller read Ghyka in 1936, and from his own admission in 1938 to having worked with the system over the previous two years. Ghyka's Le Nombre d'Or is almost certainly the source of the claim noted previously that Miller read Ghyka and Valéry 'on the hidden geometry of art' in 1936. Miller's awareness by 1938 of Hambidge's work would equally have been known from Ghyka's writings.  Dynamarhythmic Design, a book published in 1932 by Edward B Edwards [republished as Pattern and Design with Dynamic Symmetry] is almost certainly the book "Dynamic Rhythmics" of which he wrote in September 1935. Edwards appears to be the only writer in the field to include the 1.5388 rectangle, which was a dimensional structure Miller came to deploy. Moreover, Edwards details the construction of logarithmic spirals and gives a diagram for proportionately increasing the size of a rectangle without changing its shape [Figure 2a following] 60. Miller admitted to drawing logarithmic spirals in mid-1938 while a diagram of similar type to Edwards' version appeared in a letter from 1940 [Figure 2b] 61. But then again, Hambidge has a similar diagram [Figure 2c]. However, all these writers freely acknowledge Hambidge and most provide Hambidge's methods of construction for root rectangles [see figures 10 and 11] 62.

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a                                           b                                         c

Figure 2. Duplication of the rectangle; a from Edwards, 1932, b from Miller, 1940, and c from Hambidge, The Elements of Dynamic Symmetry, 1948.

The evidence given here suggests that Miller read a number of sources on proportional rhythm theory and dynamic symmetry, despite his reticence in admitting so. Many of the books have very similar contents and diagrams. In my analysis of the system that follows, therefore, various of these sources have been drawn upon.

Dynamic symmetry was in vogue among architects, designers and artists in the 1920s and 1930s. The Australian painters Grace Crowley, Rah Fizelle, Weaver Hawkins, Ralph Balson, Mora Dyring, Sam Atyeo and Frank Hinder all knew of it 63. De Maistre who, like Miller, worked in London during the thirties also used linear structural grids at times, as is seen in The drawing class [Plate 163] 64. Power may have been similarly inspired - because of the Pythagorean star with which he inscribed most paintings after the late 1920s [in the lower right corner of the painting reproduced as Plate 162]. Miller's involvement by 1936 with proportional rhythm theory, and with dynamic symmetry in particular, was therefore not uncommon among artists in Britain and Europe. Through this approach their work gained a design framework that at the same time symbolised the growth and dynamism innate to all creation.

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Proportional rhythm theory, the design system Miller adapted to his artistic expression

In the various arts, whether music, poetry, painting, sculpture or architecture, rhythm is synonymous with proportion between and within the parts and the whole. As Irma Richter points out in Rhythmic Form in Art published in 1932, while rhythm in music and poetry is an organisation through time, in the other arts it is organisation through space and in painting through the surface of the canvas:

Artistic Space should be conceived as containing a work of art just as a block of marble potentially contains a piece of sculpture, for it pervades the empty space as well as the plastic figure and shares with either the quality of invisibility, intangibility and pervasion. It constitutes the unifying element in a work of art, for in it all things find their relative shapes and places 65.

The book is the study of composition in past art and so of artistic space as just defined. It emphasises principles of harmonic proportion and is thereby mathematical in approach.  Richter derives her arguments back through the Renaissance treatises on proportion by Piero della Francesca and Fra Luca Pacioli to the Greeks, Euclid, Plato and Pythagoras. The dominant philosophy presented thus reflects the Pythagorean principle that, as Ghyka would quote from Iamblichus, '"Everything is arranged according to Number" and the vision of the Universe as a harmoniously ordered Whole' 66.

Pacioli's De Divina Proportione from 1509 was a mathematical treatise on the proportional ratio widely known as the golden section. Richter states the golden section proposition, which 'relates to the proportions of three magnitudes', simply as: 'The first part is to the second part as the second part is to the whole or sum of the two parts'. Pacioli had attributed the phenomenon mystic significance, its a priori interrelation of terms perceived to be analogous with the Trinity 67. This religious symbolic aspect may well have attracted Miller who was intensely religious, if in an unorthodox manner. To the Pythagoreans the golden section was the 'perfect proportion . . . no invention of man, but the masterpiece of the Great Arithmetician of the Universe Himself', as a later theorist Eric Bell writes 68. The similarity of this last description to Miller's 'the Universe Maker was a master of Mathematics' provides the evidence of his indebtedness to Pythagorean mysticism from as early as 1934.

Most writing on harmonic proportion and dynamic symmetry before the mid-1930s included discussions of the golden section together with instructions and diagrams for its geometrical determination. These appeared in a letter by Miller from December 1936. He believed:

[T]he factor which gives the greatest sense of Eternal rest to the Human inner sense . . . [is] Proportion. That great harmony of Distances. Once Proportion is secured, almost nothing short of obliterating the perception of the proportion can conceal its Influence 69.

He went on to discuss the "Golden Division" as known in ancient Greece and determined more contemporarily by psychological experimentation, and to outline its construction [Plate 120 reproduced as Figure 3 following]. Miller found the method to be quick and easy and was amused by his discovery that the results of his instinctively dividing lines coincided with those obtained using geometry. Evidently by this time he could judge the golden division fairly accurately by eye alone. Significantly in relation to the variety of structural matrices that were to enter his work, his reflections closed with his noting there were numerous 'more . . . similar geometric devices' 70.

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Figure 3. Derivation of the golden section ["Golden Division"], after Miller, 1936 [see Plate 120].

Miller couched his algebraic and diagrammatical explanations of the method in terms closest to those given by Richter for the configuration of Figure 4 following:

In order to find the divine proportion of a given line OW.

From centre O with radius OW describe a circle.

Divide the circumference into four equal parts at W, N, E and S.

Bisect OE at A.

From centre A, with radius AN, describe a circle cutting OW at B.

So that AN=AB.

Then OW is divided at B into the divine proportion.

Moreover, OB is the side of a regular decagon inscribed in a circle, and NB is the side of a regular pentagon inscribed in the circle 71.

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Figure 4. Derivation of the golden section ["divine proportion"], from Richter, 1932.

Traditionally the ratio is represented by the Greek letter phi, Ø, and expressed mathematically as a/b = a/a+b where FB = a and AF = b [figures 3 and 5]. Arithmetically the ratio's value is given by √5+1/2, an incommensurable or irrational number having as its positive value, 1.618034 . . . . . . and its negative, 0.618034 . . . . . . [the reciprocal of the positive - the ratio is the only one with this property]. a/b = a/a+b becomes 0.382 : 0.618 :: 0.618 : 1.000, each term being 1.618 times larger or 0.618 times smaller than its adjacent term. The ratio emerges between adjacent terms in any summation series in which each term is the sum of the two preceding terms, becoming more closely approximated as the series ascends into higher values. The Fibonacci series 0  1  1  2  3  5  8  13  21  34  55 etc is that referred to most frequently. Artists can readily approximate the golden section in the proportions of their work without resort to complex mathematics or geometry - a canvas dimensioned as 3 units by 5 units or 5 units by 8 is so proportioned. Many of the books under discussion give the series 72.

A characteristic of the Ø ratio that underlies its importance in antiquity and in organic structuring is indicated by Figure 5. Here, if FB is divided by the ratio at G, then the whole length becomes divided in the golden section in the opposite direction to when F established the proportion.

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Figure 5. The division of a line into its golden sections, from Howat, 1983 73.

Only the Ø ratio has this property, one that allows infinite extension in either direction to produce an interlocking harmoniously balanced proportional system. Figure 6 demonstrates this.  AB is the unit line. The points lying beyond B are obtained by applying the multiplier 0.618 to each integer AB, BC, CD, DE, etc, and the points within AB similarly [the integers there are AB, AC', AD', etc].

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Figure 6. Derivation of the golden progression in one dimension.

Naturally the system has application beyond the linear, for instance, to plane surfaces. Figure 7, which is constructed upon the unit square ABQP, demonstrates its application as rectangles 74. This is how it enters painting - used either to establish the dimensions of the painting's outer limits or as an organisational tool for the internal spatial structure.

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Figure 7. Derivation of the golden progression within a plane surface.

ACRP is an Ø rectangle as the proportion base : height is ~1.618. D'BQS' is another but in reciprocal;  that is, the ratio base : height is ~0.618, proportional in area to ACRP but turned through 90o. If a painter were to establish this system of harmoniously proportioned areas based on the Ø ratio, the application of any segment therein to his work could be seen as securing, as Miller put it, 'that great harmony of Distances' in a way inherently tied to secular and mystical ideas concerning nature and man's existence. Ratios from Figure 7 that appear through Miller's paintings [in the outer dimensions of their work areas and to establish proportional balance within] are:

RC/AC = D’B/QB = ~0.62    the Ø rectangle

E’B/QB = ~0.76

F’B/QB = ~0.85

G’B/QB = ~0.91

H’B/QB = ~0.94

SM/SO =   1.00    the square

The ratio series is known as the golden progression 75.

Another ratio connected closely with the Ø ratio approximates 0.79, or √Ø. This incidentally occurs most frequently among Miller's structures. Only Ghyka seems to discuss it. √Ø is the height of the "Great Pyramid" right-angled triangle, also known as the triangle of Price, with base = 1.00 and hypotenuse = Ø.  One construction derives from the Ø rectangle as shown in Figure 8. PQMO is the Ø rectangle where QM = Ø and OM = 1.00. The inscribing of the arc from Q, radius QM, gives SM = Ø where PO is cut at S. SO = √Ø.  OMNS is thus the √Ø rectangle 76.

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Figure 8. Derivation of the √Ø rectangle, from Ghyka, The Geometry of Art and Life, 1946.

SM/SO = SO/OM = √Ø = 1.273 where Ø=1.617, and

0.786 where Ø=0.617

= 51o50' 77

Other ratios used by Miller in establishing the outer proportions of his work and for inner proportional balance approximate to 0.58, 0.65 and 0.71. Once again all three are accorded significance in the contemporaneous literature. These ratios expand to:

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Figure 9. Dynamic symmetry ratios and rectangles.

The first is termed the root-three rectangle, the last, the root-two rectangle. As noted previously, the 1.5388 format seems to be provided only by Edwards, who gives it to be the one rectangle into which a logarithmic spiral can be fitted together with its inverse form without a break in the curve 78.

The derivation and signification of the root rectangles and dynamic symmetry principles have a long tradition that became "updated" in the early twentieth century. Some traditional applications were noted by Sérusier in A B C de la Peintre, namely that the root-two rectangle belonged with landscape painting, the Ø-ratio format to seascapes and the doubled Ø-rectangle to portraits 79. More "modern" literature of the times handles the root-rectangles within the concept of dynamic symmetry, and so makes them part of a design system without any particular thematic significance attached to any one of them. Hambidge identifies "plan schemes", or structural design, as consciously used in much art of the past. He defines the method as:

. . . symmetry, using the word in the Greek sense of analogy; literally it signifies the relationship which the composing elements of form in design, or in an organism in nature, bear to the whole. In design, it is the thing that governs the just balance of variety in unity.

He recognises investigations since the turn of the century into design forms as having identified two types of symmetry in nature and art, the static and the dynamic. As the name implies, static symmetry is regular in nature and, as in the crystal, presents fixed and orderly arrangements of form. Such static arrangements are based, Hambidge observes, 'upon the pattern properties of the regular two-dimensional figures such as the square and the equilateral triangle', with areas 'divided into even multiple parts, such as a square and a half, three-quarters, one-quarter, one-third, two-thirds, etc'. The ratios established by rational and regular numbers constitute an arithmetic series. Hambidge recognises such static symmetry in post Classical art, noting in particular 'the Copts, Byzantines, Saracens, Mohammedans and the Gothic and Renaissance designers' 80.

Hambidge identifies the alternative approach of dynamic symmetry as having been present in ancient Hindu, Egyptian and Greek cultures in connection with altar ritual, with which he associates 'the Delian or Duplication of the Cube problem' 81. He holds that the Greeks discovered that 'this symmetry was . . . of the growth in man' and of nature.  There it is the irregular orderly type of arrangement of form as found 'in a shell or the adjustment of leaves on a plant . . . The dynamic is a symmetry suggestive of life and movement', he writes. Its value to design lay 'in its power of transition or movement from one form to another in the system . . . [T]he only perfect modulating process in any of the arts'. He gives three sources that enable the study of dynamic design principles. These are firstly, Egyptian and Greek art and secondly, plants and man with the skeleton 'the source par excellence for the artist . . . the chief source of the most vital principle of design'. Lastly are the five regular geometrical or Platonic solids, the cube, tetrahedron, octahedron, icosahedron and dodecahedron. In art, Hambidge continues, 'the control of reason means the rule of design'. In turn, design allows instinct and feeling to be directed out of chaos and away from slavishly following nature 82. The importance of ordering instinct and feeling for the post World War I generation and in Miller's writing from around 1935 have been noted.

Scientific studies from around the turn of the twentieth century provided Hambidge with one impetus to the system proposed. Examination of spiral phyllotaxis in plants [that is, the arrangement of leaves on a stem or the seeds in a sunflower head, etc] had established that the most descriptive mathematical model was the Fibonacci series. This convinced Hambidge that the curves of shells could be described similarly, that is, '[by] the equiangular or logarithmic spiral curve of mathematics' 83. After reducing the curve to straight lines set at right angles whose lengths follow a geometrical progression [see Figure 12a], he argues for the use of this system by artists. A relationship with nature would then be maintained in their work.  Application of the principle to rectangles led him to the concept of the 'reciprocal of the rectangular shape' or forms generated in conjunction with their diagonals. Measurements there then caused his awareness that where reciprocals establish some even multiple of the parent form in area, such as 1/2 1/3 1/4 1/5, the parent forms constitute the root-two rectangle, root-three rectangle and so on 84.

Hambidge provides a series of lessons on these root rectangles derived from 'the square and its diagonal'. Figure 10 reproduces their construction 85.

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Figure 10.  Construction of the root rectangles, from Hambidge, The Elements of Dynamic Symmetry, 1948.

The diagonal AB inscribed as arc BC creates the √2 rectangle ACDK, the diagonal AD as arc DE, the √3 rectangle AEFK, the diagonal AF as arc FG, the √4 rectangle AGHK and the diagonal AH as arc HI, the √5 rectangle AIJK.

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Figure 11.  Construction of four root rectangles within a square, from Hambidge, The Elements of Dynamic Symmetry, 1948.

The four root rectangles are also constructed within a square [Figure 11] 86. Here, on the quadrant arc CD, the diagonal AB where it cuts the arc at E establishes the √2 rectangle ADGF, the diagonal AG at H, the √3 rectangle, AI at J, the √4 rectangle and AK at L, the √5 rectangle. Root-five was selected as the furthermost extension in the system since Hambidge analysed the larger root rectangles as rarely found in Greek art. The root-five and 1.618 [Ø, or "whirling squares"] rectangles derive from the square and the diagonal to its half. Figure 12a demonstrates for the 1.618 format. The area of the root-five rectangle equals the 1.618 [Ø] rectangle plus its reciprocal, the 0.618 rectangle, and thus relates intimately to leaf distribution on plants.

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Figure 12.  Derivation of the Ø rectangle, from Hambidge, The Elements of Dynamic Symmetry, 1948.

In the Ø rectangle the 'continued reciprocals cut off squares and these squares arrange themselves to form a spiral whirling to infinity around a pole or eye' [Figure 12b] 87. The similarity between this construction and the Greek fret has been noted previously, as has Miller's postcard inscribed "Hachette" on which such a squared-up spiral is lightly sketched.

Some of Hambidge's thinking is obscure and his conclusions not always obvious. Edwards is clearer about the benefits of applying the system to composition and design. The rectangles give rise to what he terms "structural skeleton forms". He writes, in disagreement with Hambidge over which rectangles formed the dynamic system, that the 'substance of dynamic symmetry rests on the principles of continued proportion and divisibility into similar shapes that are inherent in any rectangle'. He appreciated Hambidge's analysis mostly for the placement of two diagonals in the rectangle. One is corner-to-corner and the other commences in a corner and cuts the first at right angles, so generating the reciprocal of the parent rectangle. These define 'the principle of continued proportion of end to side within the limits of the space . . . [T]he beauty of nature does not depend on any one combination of ratios, but on the principle of continued proportional growth and on similarity of form', he observes 88

Proportional rhythm theory was thus a means of ordering spatial design according to harmonic and dynamic principles. Its application in painting lay in planar design or the ordering of the picture surface. Principles of harmonic proportion are not readily engendered in depth. Depth moves to infinity, a phenomenon visually indescribable on the plane except through systems of perspective proportion. Hambidge for one argues that too much stress on perspective led to 'excess of realism and consequently to loss of design'. Design was the more important 89. This emphasis in turn moves proportional rhythm theory into conjunction with the contemporary idea as to the integrity of the picture plane, or the Modernist recognition of its flatness. The significance of that for Miller's work has been discussed.

Finally, a closer look at the pentagonal qualities of the system is warranted. Intimately related to Ø-ratio or golden section geometry alone and not the other root rectangle derivations of dynamic symmetry, these are of particular significance in Miller's organisation matrices. The circle, the pentagon and its "Sublime Triangle" [or triangle of the pentagon], and the decagon are the geometric configurations involved. Incidences of the Ø-ratio permeate these regular polygons, the pentagon and decagon. Through them, therefore, proportion enters the circle.

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Figure 13. The pentagon located within a circle, from Richter, 1932.

To illustrate, in the pentagon ABCDE of Figure 13 the diagonals cut each other in the Ø-ratio and each side is equal in length to the major part of the diagonal as cut. Thus, for the diagonal AC,

AC : AG :: AG : AF :: AF : FG    [ratio 1.618]

AC : FC :: FC : GC :: GC : FG    [ratio 1.618]

AG and FC both equal AB.

These diagonals form the pentagram or Pythagorean star, so called because the figure was favoured by the Pythagorean school 90. Bell argues that two properties account for the significance of the pentagram to the Pythagoreans: its unicursality or ability to form the star from a continuous uninterrupted point and its numerological connection with the Greek five-lettered word for health, which enabled the attachment of these letters to the pentagram's five points. This allowed the figuration to signify health 91. Doubtless the incidence of the Ø-ratio throughout the figure mattered also. Richter writes also of the pentagram symbolising Christ and his "salutiferous cross" [that is, conducive to health], its use as a stonemason's sign in the Middle Ages and its later adoption by the freemasons 92. These last usages suggest the Pythagorean star may have passed through history by way of master masons and so architects. Whether this path provides a clue to Miller's introduction to the structural practices he adopted is an intriguing possibility. Constructions of the pentagon using compass and line given in the literature of the day were relatively simple and well within Miller's capabilities 93. The presence of the pentagon in his painting is easily seen in The four seasons [plates 43 and 44].

The "Sublime Triangle" is the form given by DCA in Figure 13 above, an isosceles triangle measuring 36o at its vertex with each base angle equal to 72o. Ghyka has a diagram [Figure 14] in which the sublime triangle's two-fold repetition side-by-side generates a rectangle of ratio ~0.76, the first in the golden progression series. The configuration occurs in Miller's work. The matrix of the late transitional Madonna from the JGL Collection is an instance 94.

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Figure 14. The generation of the 0.76 rectangle from a pair of "Sublime" triangles, from Ghyka, The Geometry of Art and Life, 1946.

For the inscribed decagon [Figure 15], the radius of the circle to the side is in the divine proportion, that is, OB : AB = Ø, etc, and the diagonal BE to the radius BO equally, that is, BE : BO = Ø. Therefore, 'the side is to the radius as the radius is to the diagonal'. Sublime angles join the radii to each side. The figure can be formed by inverting two pentagons onto each other, as illustrated in Figure 16. In Figure 17 the side of the decagon inscribed in the outer circle equals the radius of the next circle in, a proportional phenomenon repeated through each diminishing concentric form. Therefore, as Richter points out, 'the radii of the four circles are related in divine proportion' 95. A harmonic circle series is to be recognised in Miller's Cityscape [plates 32 and 33].

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Figure 15. The inscribed decagon, from Richter, 1932.

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Figure 16. The inscribed decagon generated from two pentagons.    Figure 17. Harmonic circles series, from Richter, 1932

The proportional rhythm system is very flexible. Plates from Ghyka's writings reproduced here illustrate well the variety of harmonic structures possible for just two of the system's rectangles, the Ø rectangle [Plate 170] and the √Ø rectangle [Plate 171] 96. The others are found throughout dynamic symmetry literature. Application of these ideas to composition results in what can be termed "modal" composition. Across the picture plane emerges a series of foci where constructional lines intersect. These are readily apparent in Miller's work.

In summary, Miller's perception of mathematics underwent a fundamental alteration over the years from 1934 to 1936. He had initially viewed mathematics as a logical and rational thought process valuable in "inorganic" applications like architecture [and by inference, mechanics]. In this his conception approximated the geometrical views that artists held immediately before World War I. Inside two years Miller grew to perceive mathematics, and geometry in particular, as a means for symbolising the intuitive experiences of growth and movement inherent to organic life. This later position was in keeping with other artists and designers of the 1930s. By applying proportional rhythm theory, the correspondence maintained with "life" at the most basic level in creative endeavour, design, meant that their art gained the potential to become more fully Symbolist [in the nineteenth century sense] than that of their Modernist forebears. At the same time, their expression remained rooted in the classical tradition - because the golden section or Ø-ratio lies at the core of the geometrical systems used to convey the æsthetic [equally the presence of the golden section allows metaphysical symbolism to be an essential feature]. Given Miller's architectural training, his interest in sculpture and the drift in his thinking away from naturalistic expression towards a more symbolic representational mode, the appeal of the approach to him seems most natural. Miller drew upon various of the many art and design theorists writing of these geometrical applications, but eventually Ghyka became his principal mentor. Consequently Miller's methods of schematisation went beyond Hambidge's original dynamic symmetry system. This broader methodology is what has been nominated proportional rhythm theory. Having now examined these principles as espoused during the 1920s and 1930s, the nature of their appearance in Miller's work, especially the late-transitional painting, can be addressed.

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Dimensional proportion in Miller's work

In establishing the extent of his picture areas Miller applied the full range of proportional ratios that have been discussed.  Intriguingly, this observation holds for the work of his earliest Australasian period before his contact with the proportional rhythm theory as such 97. Information about his artistic development can be gleaned from the incidence and variation in the spread of these ratios between his different career phases. The pattern present signifies Miller's specific design interests at any one time together with his developing skills. Briefly stated, over the earliest years in Melbourne organisational matrices were of little or no interest. Next, a developing interest and knowledge of structural design systems begins to show, accompanied by a tendency to concentrate on structural possibilities as these became known to him. At maturity, by then fully versed in the theory, a great deal more structural variability entered his work.

The spread of the external dimensional ratios to Miller's pictures encountered for both phases of his transitional period between 1929 and 1942 has no ready explanation 98. Three points can be made. First, Miller had a pronounced preference for square canvases then in comparison with the early and mature periods. Second, another pronounced preference marks the early transitional years  - for surfaces of the order of the √2 rectangle [0.71]. Third, in the late transitional period this changed to the √Ø rectangle [0.79]. The pattern provides clues to the specific books Miller read as the decade progressed. His themes broadened in the early transitional years to focus mostly on landscape, cityscape and still life, with just a few figure studies and abstract pictures. There is no real pattern to the incidence of themes across the dimensional ratios present 99. Miller thus remained oblivious through the early 1930s of traditional French practices matching proportion to theme. Never, in fact, did he use working areas of certain proportions for any single theme in the manner Sérusier advocates. For the late transitional phase, landscape and cityscape predominates, with still life, figure studies and abstract paintings found in almost equal numbers and with religious painting making an appearance 100. In maturity after 1942 this pattern would change again.

Five characteristics typify the skeletal design matrices Miller came to deploy in the late 1930s. First is their very obvious inclusion as an integral element of the work. Secondly, a high incidence of golden section [Ø] proportion is generated within them. Thirdly, Miller turned in the late 1930s exclusively to the matrices of dynamic symmetry and the peripheral configurations there from. Next, the structures assist the picture design. Lastly, the paintings had symbolic notions induced by Miller's recourse to these matrices. To demonstrate these points we will examine a few of his late transitional pictures. These late transitional works have been isolated into four sub-groups, each of which will be looked at in turn. The first three are grouped together as applications derived from proportional rhythm theory in general; the fourth comprises pictures whose matrices come from dynamic symmetry.

The first late transitional category comprises paintings having no obvious structural matrices such as Figure composition [Plate 29]. A compositional semblance exists between this work and early transitional paintings such as Still life [Plate 22] by way of an emphasis on vertical sectioning 101. Figure composition bisects into vertical parts of practically equal width through the figure placements and the slightly angled tree stem running down the canvas almost at the centre.

The second late transitional category consists of paintings with refined structures that are in progression from those of early transitional pictures like Jug and egg [plates 14 and 15] and Warrandyte [plates 16 and 17]. Examples are Growth [plates 35 and 36], Cityscape [plates 32 and 33], Palette design [plates 49 and 50] and The four seasons [plates 43 and 44] 102. The systems of parallel horizontals and diagonals present connects these to the earlier matrices. Growth, Cityscape and Palette design take the square as format while The four seasons is a √Ø rectangle.

The grid of Growth [plates 35 and 36] clearly generated a guide to the picture's composition. The mountain peaks and tree-trunks lie within the two equal width horizontal bands across the centre of the canvas, while these are equalled in width themselves by the bands top and bottom. The elliptical egg-form placed firmly on the upper right diagonal takes its direction in space accordingly. The addition [by myself] of the circular bounding motif encompasses the picture's movement [generated through the curves in the lower section] and its dominant focus. The structure has little other significance. However, its regular diamond-like aspect is similar to matrices deployed by Miró in the mid-1920s. Tête du paysan Catalan, 1925 [Plate 159] shown at the 1936 International Surrealist Exhibition in London has one 103. Miller's Growth corresponds again with Tête du paysan Catalan, even if invertedly, by way of his thin triangular mountain peaks supporting the procreative egg and Miró's seven thin and curvilinear tentacle-like forms of various lengths bounded by an equilateral triangle pointed downwards.

Unlike Growth the structural matrix of Miller's Cityscape [plates 32 and 33] has identifiable symbolic significance. It resembles a Ghyka diagram, the 'Diagramme traversa « modulaire » de la cathédrale de Cologne' after Lund [Plate 169] published in 1927 and a similar one from Richter [Plate 173] 104. As seen previously, the circles each are in the golden section proportion to the next. Four pentagons generate two decagons and twenty equidistant points on the outer circle's circumference. A small number of sublime triangles are formed. For Ghyka, pentagonal and decagonal configurations symbolise the microcosm and macrocosm 105. Golden section breaks on the vertical and horizontal [indicated by the Ø-marks on border lines] have compositional significance for the placement of form. The introduction of centered axes on the vertical, the horizontal and diagonally, corner-to-corner, is also noteworthy. Ghyka claims the morphological and rhythmic correspondences between man and the universe to have been reflected in ancient Greek culture through the intermediary of the temple. They derived from the Greeks learning of Egyptian practices with pyramid construction. In turn, he writes, these practices would inform the architectural treatises of Vitruvius, Gothic cathedral building and Renaissance architecture 106. Miller's forms indeed suggest the pillared arches of mosque interiors and cathedrals. There is a reflection even of the architectural detail of Fra Angelico's Annunciation [Plate 127], which he admired 107. The Cityscape matrix was earlier linked to Miller's description of Moorish courtyards where water is an inherent component. Richter found the concentric configuration suggestive of waves extending to infinity in every direction. She also regarded the circles as spherical cross-sections, allowing the figure to take on a signification related to early notions about the heavens and the movements of the spheres 108. The theme of Miller's painting may therefore be interpreted as microcosm [man] and macrocosm [the universe] in unity.

That the matrix of Palette design [plates 49 and 50] is equally symbolic is even more easily established. This matrix of parallels has a direct source in a diagram [Plate 172, left] published by Ghyka in 1938. This is a schematic geometric representation based on the golden section proportions of the young male athlete [Plate 172, right]. Ghyka comments of these diagrams:

Now, in the human body, in effect a symphony of proportions both very complex yet . . . fully unitary, the golden section and related proportions (especially √5) are revealed as constant leitmotif.

The human body, or microcosm, can be very well symbolised by the pentagram (star pentagon), which gives a recurrent indefinite series and "pulsations" of the powers of the golden section (Ø = √5+1/2 = 1.618 . . . . ) and which came to be the rallying sign of the pythagorians as symbol of Life and of harmonious health . . .

The body scheme (Viennese athlete) represented in plate I is framed within a square (plate II only reproduces the truly remarkable abstract armature for plate I) . . . [T]he navel divides the body vertically in the proportion of the golden section 109.

Miller has used the matrix to structure a fully abstract work 110. Heavily saturated with the Ø-ratio throughout and with two sublime triangles inverted into each other in the vertical panel, the painting comes to symbolise man as microcosm, an integral element in the greater macrocosm of being. Equally growth, balance and rhythm, and ideas of the sublime were introduced.

The compositional significance of the structural matrix in The four seasons [plates 43 and 44], a √Ø rectangle, is clear. The upper half of the structural circle present carries the rainbow motif, and alterations in the colour's density and treatments take place on the central horizontal and various of the structural diagonals. A system of parallel diagonals similar to that of Palette design [plates 49 and 50] is evident. The pentagon and decagon are as visually evident as in Cityscape [plates 32 and 33]. However, because the circle that contains them is placed centrally on the golden division vertical located along the base line, these forms are more philosophically important here than in Cityscape. The intersection of the Ø-section vertical and the central horizontal axis generated two 1.5388 rectangles on the left and two squares on the right. As we have seen, Ghyka derives the √Ø proportion from the Great Pyramid 111. Other connections that the painting holds with Egyptian mythology allow the picture to be interpreted as a representation of creation. While the structure remains indebted to Miller's earliest structures that comprised series of parallel diagonals, dynamic symmetry is emergent here.

Paintings of the third late transitional sub-group are distinguished by their being made up of rectangular equal-area planes. Untitled (Tree / forest series) [plates 51 and 52] illustrates this, its dimensional format of 0.91 coming from the golden progression series. These very regular structural matrices, here of four-by-eight equal area rectangles, reflect Hambidge's "static symmetry" idea. Certainly the picture does not have the dynamism, the sense of movement and rhythm, that Miller's application of dynamic symmetry principles instilled into many paintings. Compositionally the vertical struts were used to determine the symmetrical placement of the trees each three columns inwards from the sides. Horizontals within this armature mark major changes in their growth, a splitting into several trunks or branches, a move from vertical growth to a diagonal direction. Lastly, the diagonal present sets the slope of the small branches and the direction of much of Miller's brushwork for the foliage. Occasionally a semblance of dynamic symmetry structure exists alongside a "static" configuration. When this has happened, as in Cityscape [plates 57 and 58], the claim as to a lack of dynamism is no longer sustainable. Regular equal-area breakdowns are part of dynamic symmetry structuring, the square into four equal parts, the √2 rectangle into two, the √3 rectangle into three and so on, but a proportional relationship reflecting dynamic symmetry principles is maintained always. No such relationship was carried into Untitled (Tree / forest series) [plates 51 and 52] where the internal rectangles each approximate 0.42 and thus come closest to being √5 rectangles 112.

It is on turning to the final late transitional sub-group that direct evidence of Miller's involvement with the proportions and rhythms of dynamic symmetry is found 113. Firstly, the existence of a diagram on tracing paper either constructed or copied from a book by the artist [Plate 121] requires acknowledgement. Miller may have used Hambidge's construction of the four root rectangles within a square [Figure 11] as the basis for his diagram since some resemblance exists between the two. Doubtless such diagrams guided the artist with his structural matrix construction for whatever dimensional proportion and internal rhythmic system was wanted. A mechanical device like the pantograph, an expanding parallelogram comprising as few as four levers, would readily have enabled Miller's transference of any one of the proportional rectangular matrices present to a larger surface 114. The diagram contains a number of different constructive possibilities within itself. The full format is not a square as is Hambidge's, but rather a rectangle of proportional ratio 0.89, practically the 0.91 ratio of the harmonic progression series. Horizontals present allow the breakdown of the vertical into two, three, four or five equal parts, and vertical lines of the horizontal similarly. The notations I have added in Plate 122 assist in demonstrating the other constructive possibilities present. Combinations of vertical and the full-width base line generate a variety of forms - the √Ø rectangle [OABC], the √2 rectangle [OADE], the 1.5388 rectangle [OAFG] and the √3 rectangle [OAHI]. Combinations using OJ, the base line diminished to equal length with the vertical, produce the square [OJKL] and the √Ø rectangle [OJMG], the 1.5388 rectangle [OJNI] and the √3 rectangle [OJPQ]. The golden section is present in abundance 115.

That a matrix structure appears on Dusk, Warrandyte (Early landscape, Warrandyte) [plates 5 and 6] has been indicated. As noted then, the painting's theme, the brushwork and the nature of its colour all combine to suggest the work was originally painted during the 1920s when Miller was in Melbourne. The matrix is believed to have been added later. Its detailed analysis was not included earlier since this is more appropriately discussed here. The picture size of 22.8 x 28.2 cm [ratio 0.81] approximates the 0.79 ratio or √Ø rectangle. The matrix itself is most notable for its "uncertainty" by which is meant the number of duplications of line present and their poor resolution around any point of significance on its boundaries. This poor resolution in turn generates a plethora of indeterminate modal points through the composition. Despite this indeterminacy the structure approximates a diagram first published by Ghyka in 1938 [Plate 171]. As we saw, Ghyka first gives a simple construction of the √Ø rectangle, along with those of the Ø-ratio and other root rectangles, in 1927. No complex harmonic "decomposition" of a rectangle was seen until 1931 when the figure on the left in Plate 170 appeared. This 1931 "decomposition" could, therefore, have been Miller's model for the matrix of Dusk, Warrandyte. However, two factors suggest otherwise, namely the difference in proportion of the two rectangles and the lack in Ghyka's 1931 diagram of diagonals between border Ø-points and corners. These diagonals are present both in Dusk, Warrandyte and Ghyka's diagram of 1938. Moreover, other of Miller's struts in Dusk, Warrandyte coincide with Ghyka's "decomposition" of 1938. The lines Ghyka generated then from the Ø-breaks along the vertical and horizontal boundaries find an equivalent on Miller's vertical border and an uncertain replication on his horizontal. From the Ø-point verticals Miller generated two 0.76 ratio rectangles each overlapped to the other's inner vertical border [the Ø-lines]. In this regard the structure differs from Ghyka's configuration of 1938 since there the overlapped rectangles are of dimension 0.79, the √Ø rectangle again. Miller introduced two verticals and three horizontals that also are not found exactly in Ghyka's 1938 figure. One extraneous horizontal ranges along the lower limit of the leafy masses of the trees closest to the foreground [the lower horizontal Ø-line delineates their tops in a general sort of way and the upper horizontal Ø-line, the full extent of the complete central band of leaf greenery];  the pictorial space above is a √3 rectangle. Another extra horizontal 1.1 cm above the base line does not hold any design significance, but it reduces the full canvas to more closely approximate the Ø-ratio of 0.79. No proportional interplays exist between any of these lines other than between the Ø-breaks. Although differently positioned, Ghyka's harmonic scheme of 1938 also contains lines extraneous to the generations from Ø-points, namely those shown more boldly which include the horizontal marked Q. In this case, the rectangle created above the lower of these two horizontals is a √2 rectangle. These similarities and differences between the matrix for Dusk, Warrandyte and the various diagrams Ghyka published between 1927 and 1938 suggest that it was Ghyka's diagram from 1938 that Miller used there, and not those of 1927 or 1931. They also point most strongly to Miller's continuing in experimental mode with dynamic symmetry applications after 1938. This conclusion is consistent with the evidence provided by other of his matrices of the late transitional period. Other similarly unresolved dynamic symmetry structural grids include those of the 1.5388 rectangled Paddington terrace (Street scene study) [plates 45 and 46] and Still life with oranges (Objects on a table) [plates 55 and 56], a √3 rectangle.

The structural matrix of Cityscape [plates 57 and 58] is among the most complex encountered in Miller's œuvre. The picture is the largest known square canvas by the artist. Its square format analyses down to a collection of equal-area rectangles [the dotted lines present in Plate 58 were induced by myself] reflective of static symmetry. However, these rectangles are all of the √2 format, thus introducing the dynamic. What is more, the square generates from the two Ø rectangles present overlapped to the other's Ø-point, these also being the golden section breaks on the base line. The two overlapped [side-on] equilateral triangles, their apexes meeting the vertical boundaries of the canvas at their mid-points, create the six-pointed star known as the seal of Solomon. Vertically positioned through the centre is a sublime triangle. Both golden section breaks on the base line and the upper one on the vertical edge are present. The lines hold design significance. As to the picture's potential symbolism, Cirlot identifies Oswald Wirth to have in Le Tarot de Imagiers du Moyen Age [1927] designated Solomon's seal the "star of the microcosm", 'or a sign of the spiritual potential of the individual who can endlessly deny himself. In reality', he continues, 'it is a symbol of the human soul as a 'conjunction' of consciousness and the unconscious'. The last association comes when the triangles in the star are placed vertically - the upward pointing triangle represents fire and the inverted triangle, water, both of which are, 'according to alchemic theory, subject to the principle of the immaterial' [an imaginary aspect is implied here] 116. Since Miller's star is positioned on the horizontal axis Cirlot's alchemic meaning cannot be imputed into the picture. However, that of the spiritual value of denial in life can be.  In addition, the upright star appears in the Theosophical Society's emblem. Its appropriateness to Miller's personality and religious leanings requires little elucidation 117.

By the time Design, Young Street [plates 59 and 60] was painted Miller had resolved his dynamic symmetry matrices. The subsuming of the major compositional rectangle, here of √Ø dimension, within an outer rectangle, a √2 format, is encountered elsewhere, particularly in the Madonna paintings. Although almost fully abstract, Design, Young Street is difficult to see as anything other than a cityscape. The inner rectangle is the more important since it bounds the apexes of the structural triangles and diagonals present. The horizontal bar of the cross on the central axes of the painting forms through the overlapping of two Ø-rectangles to each one's vertical Ø-point. The vertical boundaries of this cross come from the two Ø-points of the base line, these in turn being the Ø-breaks of the two segments [corner to Ø-point] generated. In other words, a play of harmonics has been introduced through the painting. The structure is noteworthy for the preponderance of pentagon angles situated along the internal horizontal boundary lines. This high incidence of sublime angles is again common to the Madonna paintings. Beyond noting these characteristics and attaching the meanings of "creation" and "life" that come from golden section ontology and those of growth and movement as particular to dynamic symmetry theory, the imagery of the painting carries little symbolic significance.

Miller introduced quite specific symbolic meaning into his paintings built on proportional rhythm theory. These geometrical applications were believed to correspond with life principles of growth and movement as they reflect in nature. In particular, their golden section proportions induced a spiritual aspect, one that associated with life as well. In a very full sense, therefore, these pictures all convey the interactive process of "creation". That is to say, life's beginnings, changes and progress, and finality, all encompassed together. The artist's comments from 1938 about his use of dynamic symmetry principles and their significance for him are worth recalling at this point:

[F]or the past two years I have been . . . evolving a system made of a blending of a logic basis or framework on to which intuition or personality can be placed. I have studied a particular line of mathematic consisting of a dynamic rhythme [sic] of rectangles . . . [A]mong other qualities these rectangles are related to pentagons, etc. With these impressed on the mind I look at things such as trees mountains and contrive to see . . . a demonstration . . . of what I had previously done with compasses and lead pencil. In a painting the logic base shld [sic] be just apparent giving . . . the genesis of the picture-idea. This relationship or poise between logic and . . . personal intuition is the great creative thing 118.

To sum up, the results achieved by Miller's application of dynamic symmetry principles are as follows. First, ratios there from for the most part describe the external dimensions of the late transitional and mature paintings. In many works the crossed corner-to-corner diagonals and the centrally-orientated vertical-horizontal axes appear the starting points for the internal matrix determinations. In part if not whole, Cityscape [plates 32 and 33] and Still life with oranges (Objects on a table) [plates 55 and 56] all demonstrate this. Within the structural matrices, the Ø-ratio often breaks the space between any two neighbouring horizontal or vertical lines and may contribute further in the occasional play of interlocked harmonic proportions in and between these. These phenomena are best seen in a later canvas like Triptych with figures [plates 75 and 76]. Internal rectangles frequently take harmonic progression and dynamic symmetry ratios as well.  The point was made with Cityscape [plates 32 and 33]. Further golden sections are repeatedly established by modal points generated along diagonals. To Miller, as noted, modal points represented places where forces changed or contrary forces met. Even there, then, a symbolic meaning was instilled. Miller used these structural armatures not only to position form across the painting, but also to establish the directional movement of the fine linear latticing which frequently orchestrates the surface and for the slope of his brush strokes. Finally, it must be strongly emphasised that application of proportional rhythm theory enables a very great variety of structural formats.

In conclusion, Miller's progress with design structures is judged to have proceeded from his adoption, probably in London around 1933-34, of the squaring-up grid of the unseen matrices of representational canvases like Still life, London (Table group, London) [Plate 12], Jug and egg [Plate 14] and Warrandyte [Plate 16] to the irregular, non-systematic matrices of his earliest Modernist pictures such as Still life [Plate 22]. Within a year or two he allowed these to become obvious as in Design [Plate 28]. From there, in about 1936, his more regular schematisations made their appearance, as in Abstract [Plate 31], Cityscape [Plate 32] and Growth [Plate 35]. Ghyka's writings came to his attention at about the same time and Miller's first efforts using dynamic symmetry and other proportional rhythm matrices emerged. Likely canvases here are Untitled (Trees / Forest series) [Plate 38], the probably inaccurately named Paddington terrace (Street scene study) [Plate 45], Untitled (Trees / Forest series) [Plate 51] and Still life with oranges (Objects on a table) [Plate 53]. From after 1938 almost certainly comes the surface matrix applied in pencil to Dusk, Warrandyte (Early landscape, Warrandyte) [Plate 5] and the grid of Palette design [Plate 49]. Once in Sydney after 1939 and approaching his artistic maturity, Miller can be expected to have refined his technique to the fully resolved matrices typical of The four seasons [Plate 43], Cityscape [Plate 57] and Design, Young Street [Plate 59]. Finally, with maturity those occasional armatures that lay outside dynamic symmetry methodology such as in Palette design [Plate 49] and The four seasons [Plate 43] disappeared from his practice. Perhaps Miller's lessened exposure to contemporary international design theorists because of Australia's cultural isolation with the outbreak of World War II was responsible for his changed approach. By the time cultural links were re-established in the mid-1940s Miller's years of experimentation were over and his art practices mostly settled. Paintings like Triptych with figures [Plate 75] were commenced.

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