Geometry & painting

Adèle (c) Samantha Groenestyn (oil on linen)

Importing mathematics into painting has some potentially grand implications. The idea makes me flush with uncontainable excitement; it smacks of Descartes (2006 [1637]: 9) and his methodical approach to knowledge, and I would echo his rationalist sentiment: ‘I was most keen on mathematics, because of its certainty and the incontrovertibility of its proofs.’ This unlikely marriage between mathematics and painting is especially dear to me because it offers something steady and dependable in terms of colour and not merely in terms of drawing; it promises to embrace the entirety of painting with its sober orderliness. This systematisation hardly destroys the poetry of painting. Rather, it allows us to sharpen our technical methods, which equips the genius (of the Kantian flavour) to paint something deeply insightful and moving. And it promises a double elegance: the sight of the painting itself, just like the sounds in music, may please us, and at the same time be grounded in delightfully crisp mathematical relationships, just like the improbable mathematical elegance of harmony in music.

These longings for order and systematisation sound rather like seventeenth-century aspirations to elevate painting to a science, or at least to a liberal art, which has much to do with shedding its humble craft status, as a trade practiced by illiterates. Painting has certainly made many efforts in this direction; it may boast of its academic status now that it is so commonly taught in universities rather than in ateliers, now that it defends itself verbally and indeed often consists more in its verbal conception and explanation than in its visual execution. But perhaps these victories are no victories at all: they strip painting of the very things that distinguish it as painting. Painting might have done better to have sought an intellectual ally in mathematics rather than in language, for there it would have found ways to describe its visual concepts succinctly and precisely.

Copy after Rodin, Burgher of Calais

This camaraderie is most apparent when it comes to colour. Colour is the rogue that has been seized by painters who want to defy philosophical discourse, and it is the uncontainable element that philosophy has used to subordinate painting. It seems to defy principles, thus it eludes philosophers, and it seems to operate largely by inspiration, superstition and magic, which seems to be attractive to painters. Across both disciplines, there is general agreement that colour is definitively not rule-amenable, while drawing is. Jacqueline Lichtenstein (1993 [1989]: 4; 62-3), in The Eloquence of Colour, traces this long-standing tension back to Plato and Aristotle, observing that ‘being material, colour has always been seen as belonging to the ontologically deficient categories of the ephemeral and the random.’ Philosophy has, she writes, thus favoured the more conceptually manageable element of painting: drawing (Lichtenstein, 1989 [1993]: 4).

If colour does not lend itself to principles, this has another, more practical, result. Philosophy aside, it means that colour cannot be taught. This lends itself to all varieties of unwelcome mysticism, that I personally would like to see chased out of the discipline of painting. It suggests that painters are ‘gifted,’ that they are conduits for ‘inspiration,’ or that they must operate by chance–all of which deny that painting is a disciplined skill that can be developed and improved and harnessed for aesthetic purposes. This is an unhappy state for painting to be in, for it grants artists license to all sorts of nonsense and self-indulgence, and abuses the viewer with all manner of ineptly executed work. In short, it encourages carelessness and invites decadence. Painting is visibly decaying before our eyes.

Copy after Rodin, Burgher of Calais

In the face of these two apparent deficiencies, I want to argue that the emphasis on drawing–both as philosophically acceptable and as practically teachable–is misplaced. Drawing certainly does lend itself to principles which can indeed be taught, and perhaps this fact is even overplayed. There are elements to drawing that cannot be taught, because each draughtswoman will adapt the learned principles to her own sensibility; she will interpret them, introducing a quality of line that no one else has. And, more broadly, the principles that are discussed and taught are not incontestable facts of existence. This is very clearly described by Panofsky’s (1991 [1927]: 37) contrast of spherical and linear perspective. Lastly, I want to raise a surprisingly little-grasped fact, one that is also popularly rejected by painters: colour is indeed amenable to principles, and there are painters who work with these principles and succeed in teaching them. Colour is very acutely described by geometry. In our infatuation with language, this straightforward ordering of colour has persisted largely unnoticed for at least two hundred years.

Lichtenstein (1993 [1989]: 142) notes that ‘ever since society has set a hierarchy among human activities, their relation to language has been the ultimate criterion for the establishment of a division, both social and philosophical, between the noble arts and the servile trades.’ Because of this, she explains, painting has sought to prove itself by ‘literary credentials;’ in order to do this, it has been expected to ‘satisfy both theoretical and pedagogical objectives,’ as we have already considered (Lichtenstein 1993 [1989]: 142; 151). Since she accepts that colour defies principles, she looks to rhetoric to redeem the intellectual status of painting, a fascinating move that demands more attention elsewhere, but we may here respond with our geometry of colour.

Copy after Rodin, Burgher of Calais

A fascinating little tract by Philipp Otto Runge appeared in the early 1800s. His Farbenkugel, or ‘colour sphere,’ is a mathematically pure way of conceptualising colour. It conceives of the relations between all colours three-dimensionally. He begins with a flat triangle that represents the three unmixed colours of red, yellow and blue. Each line is bisected to indicate that, mathematically, the secondary colours are the halfway points between each of these: orange, green and purple. These six points are extended out to the edges of a circle, which is then pierced by a perpendicular axis at whose poles stand white and black. The mid-point of this pole is, mathematically, a mid-tone grey. As colours move directly across the horizontal axis, they are neutralised by their mathematical opposite, entirely cancelling each other out as grey at the mid-point–yellow becomes, not more purplish, but more grey, as it moves towards purple, its opposite. Green and red exist in the same relation, and orange and blue. The knowledge of these relationships means a painter in fact need not use a black paint to recreate these relationships in paint: grey is not the absence of colour, but the annihilation of one colour in its mathematical opposite–‘alle einander auf derselben Gerade gegenüberliegenden Farben [sind] als Kräfte anzunehmen, welche einander entgegenstehen und sich durch ihre Vermischung zerstören in Grau’ (‘all colours that lay across from each other on the same line are to be assumed opposing forces that, upon mixing, annihilate each other in grey’) (Runge, 1810: 28). The rest of the sphere is filled out by every conceivable mixed colour and in every level of lightness and darkness, vividness and neutrality. The whole thing is most easily grasped visually, and this is the advantage of geometry.

(After Philipp Otto Runge)

It is a very beautiful model, one developed concomitantly with discussions with Goethe, and a living idea still used and taught by artists who appreciate the more rugged borders of three-dimensional colour-space. But more than this, the emphasis on relationships allows a shift in thinking: rather than considering colours as absolutes, bound to precise recipes of two-parts cadmium yellow to one-part prussian blue, they may instead be managed and manipulated as a complex but entirely rational web of relationships. This means, in fact, an emancipation from the types of dogmas that more mystically-inclined painters tend to bark at other painters: it means a shift from objectively defining colours to subjectively experiencing them. It allows a painter to recreate her perceptual experience of seeing colours; it allows for the fact that a certain mixture can appear pink or green, depending on the context it is set in. It marks a dramatic difference between painters who ask ‘what colour this really is,’ and those who ask how they perceive it. The second mindset affords far greater flexibility and dexterity with colour. And it can be taught.

(From Philipp Otto Runge, Farbenkugel)

This kind of dexterity is important because ultimately, while we might define our concept of colour in a pure mathematical way, paint itself does not respond to such precise geometrical divisions, and does not correspond so precisely to light. The painter must cope with two additional overlays to her mathematical concept of colour: the chemistry of paint and how the mixtures are achieved by actual pigments of vastly different physical properties, and the physics of light and the fact that her eyes take in a much broader gamut of colours than her paint is capable of mixing. A swift and nimble understanding of the relationships as geometric proportions is a solid conceptual ground that can be modified empirically as the painter’s experience with using paint and approximating it to what she sees grows. Runge (1810: 62) notes this as an aside to Goethe in one of his letters: ‘Ich kann mich hier nicht über die Praktik ausbreiten, weil es erstlich zu weitläufig wäre,’ (‘I cannot expand upon the practice here, firstly because it would ramble on too long,’) but he mentions that the artist requires ‘den nötigen chemischen wie mathematischen Kenntnissen’ (‘the necessary chemical alongside the mathematical knowledge.’)

Such systems equip us with knowledge, and thus confidence, and in the case of colour, adequately describe and organise the material reality of paint and at the same time accommodate our subjective, perceptual experience of it. Runge (1810: 42; 61) hopes that these pure insights will permit more definite expression; he thinks that being secure in the mental connections of the elements is the only means of setting a painter’s mind at ease, in the face of such superstition and chance. It would be well at this point to remind ourselves not to take the implications of these principles too far, and thus to return to Panofsky.

Copy after Claudel, Vertumne et Pomone

For the principles of vanishing-point perspective, the mainstay of principled drawing, are, indeed, a construction devised during the Renaissance, as Panofsky (1991 [1927]: 27) notes early on. It provides us with a mathematical space that is actually at odds with our perceptual experience of space, but that does not undermine its usefulness to us. Panofsky (1991 [1927]: 29-30) contrasts the visibly rigid ‘structure of an infinite, unchanging and homogenous space–in short, a purely mathematical space’ with ‘the structure of psychophysiological space.’ Our working concept of perspective demands that space conforms entirely to reason, that it is ‘infinite, unchanging and homogeneous’ (Panofsky (1991 [1927]: 28-9); but that demands certain assumptions that deny our experience of it: firstly, ‘that we see with a single and immobile eye,’ and secondly, that a flat plane adequately reproduces our curved optical image–two ‘rather bold abstractions’ from our perceptual experience.

‘In a sense,’ write Panofsky (1991 [1927]: 31), ‘perspective transforms psychophysiological space into mathematical space.’ And there is indeed nothing wrong with that if we recognise it as such, and do not take our theoretical underpinnings too far, thus over-emphasising the theoretical validity of drawing over colour.

Copy after Claudel, Vertumne et Pomone

Beginning with (helpfully visual) geometric principles, we can thus devise rigorous and teachable theoretical systems for both of the equally important parts of painting, for drawing and for colour, describing them in pure, abstracted, mathematical terms, whose constancy is beautiful in and of itself. We can reclaim the liberal art of painting, award it some intellectual prestige, and even ground it in scientific principles that draw on chemistry and physics as well. Descartes’ project might not prove so alien in the murky and superstitious realm of painting.

Copy after Rodin, The sculptor and his muse

Lichtenstein, Jacqueline. 1993 [1989]. The Eloquence of Colour: Rhetoric and Painting in the French Classical Age. Translated by Emily McVarish. Berkeley: University of California.

Panofsky, Erwin. 1991 [1927]. Perspective as Symbolic Form. Translated by Christopher S. Wood. New York: Zone.

Runge, Philipp Otto. 1810. Farbenkugel: Konstruktion Des Verhältnisses Aller Mischungen Der Farben Zueinander Und Ihrer Vollständigen Affinität. Köln: Tropen.


Technical things

It is remarkable how Deleuze has forged the impassioned ramblings of Francis Bacon into a deep and cohesive philosophy. I find there is nothing particularly incoherent about Bacon’s convictions about painting, only that in themselves they are almost banal, and Deleuze has elevated them to a surprisingly intellectual status. Nevertheless, Bacon’s mundane observations, perhaps cryptic to a non-painter, are at least refreshingly down-to-earth and as such offer fertile soil for the creator of concepts—the philosopher. The meta-reading of these interviews, then, is that a philosopher may not need to dig so deep, but to simply meditate on the relations between things, and his own philosophy will emerge organically, firmly rooted in ordinary experience (Deleuze and Guattari, 1994: 90).

For any painter will laugh upon reading Bacon’s solemn answer to David Sylvester’s (1975: 18) inquiry about his decision to stop a painting: ‘the canvas becomes completely clogged, and there’s too much paint on it—just a technical thing, too much paint, and one just can’t go on.’ Is this a technical thing? Indeed, sometimes one piles on so much paint that the thing gets out of control, edges mash that should not, would-be layers collapse into each other; it is better to let the thing dry than to go on today—a thoroughly non-philosophical answer, disappointing to thinkers, entirely obvious to painters, and thus despite its lack of claim to being an active ‘technique,’ possibly something that can indeed be cast into the fearful ‘technical’ category.

In this sense, ‘technical things’ are all those unspeakable, messy processes that happen in secret behind the closed studio door, generally barred from aesthetic discussions that would rather poke at the dry, finished result (preferably from behind glass, and with a very long stick). They could even include preliminary decisions, perhaps in the art shop, about the size and shape of brushes to buy, whether to select natural or synthetic fibres, preferences for the ‘springiness’ of the bristles (tested by an expert hand, but verbally inexplicable). They might encompass having to cope with stiff, old brushes out of sheer poverty. They could include unforeseeable and uncontrollable lighting conditions afforded by uncooperative weather, shifting the hue of the work without the conscious knowledge of the painter.

In any case, before we even come to talk about intentional use of colour or tone, there are many inputs and decisions that steer the course of a painting and directly influence what we like to think of as the aesthetic qualities of the work. Deleuze (2003: 86; 93) would include them in his ‘givens,’ the ‘clichés’ that pollute the canvas before a painting is started. But in Bacon’s (1975: 82) rugged simplicity, he speculates that he is ‘probably much more concerned with the aesthetic qualities of a work’—the technical, as opposed to the psychological, aspects of his painted screams. This is a particularly nice observation. Many artists simply care more for the visual qualities of their work than for conveying something to some audience. These visual qualities, coaxed into existence by a perceptive painter, may finally move some unsuspecting viewer and stir all sorts of lofty thoughts in his contemplative mind. But as the genesis of these thoughts, possibly inextricable from these thoughts, might not these mere ‘technical things’ themselves comprise a very important part of aesthetics? Are they not precisely what we want to talk about?

For a humble painter is one of the few willing to simply confront a work on its own physical terms. She surveys the array of technical choices and chances, the resulting relationships between elements, and considers their degree of success. She is willing to consider what the paint itself may say, not merely what it may allude to or denote or represent. Bacon (in Sylvester, 1975: 61) shrewdly notes ‘that most people enter a painting by the theory that has been formed around it and not by what it is.’ They prefer to approach a painting through an indirect, non-visual route. They could shut their eyes and listen to information instead.

It is in abstract painting that people might find the courage to let mere colours and shapes touch them, rather than to search for ideas outside of the painting. What is clear to non-abstract painters is that the sensory force of colours and shapes is available and able to be manipulated even in very naturalistic painting. Abstract painting ‘can convey very watered-down lyrical feelings,’ scoffs Bacon (in Sylvester, 1975: 60), ‘because I think any shapes can.’ And timid observers can project themselves and most anything they like onto the distilled forms of abstract painting. The point is so plain it is hardly raised among painters. It is simply part of our job to actively design an image, to exert control over it, even if we hide our tracks and make it feel inevitable.

And so Bacon (in Sylvester, 1975: 58) repeatedly explains that he is seeking just such direct contact between the painting and the nervous system—the immediate impact of colours and shapes (and every other technical thing), without the mediation of the brain. His phrasing seems oblique and troubling to Sylvester and deep and insightful to Deleuze. One senses that Bacon has finally thought of some words that best approximate this very ordinary painterly experience that might finally get across to these wordy people, that they swirl around in his head until they take the form of some mystical mantra. His words are exceptionally nice, and give the painter a little jolt: because she, too, knows that the best painting works without intellectualisation, that the body itself responds to an exquisite harmony of colours or a pulsing, rhythmic line. Good painting feels immediate, it does not require deciphering, though it may entice one to look longer, to dwell upon the picture and soak up its sensations.

Bacon (in Sylvester, 1975: 120) is firmly convinced that this immediacy, this freshness, must come about through chance. That the coveted deftness of touch, effortless finish, virtuoso resolution, can only be captured unawares, never intentionally. Though he seeks order in painting, he fears that it will look laboured, and prefers that the work look as though ‘it hasn’t been interfered with’ (Bacon, in Sylvester, 1975: 120). This exposes the naïveté of a painter who does not know how to set himself technical problems and to set about solving them. For while he is right that freshness is compelling, such fluency can most certainly come about through knowledge and disciplined application. More adept painters than Bacon have used their knowledge to produce lucid and nervous-system-gripping works, still driven by their own personal sensibilities.

And this is another nice observation of Bacon’s, painfully unnoticed by too many painters. The inventiveness of an artist lies not in the originality of her techniques, but in the pursuit and cultivation of her own sensibility. ‘I’ve never felt it at all necessary to try and create an absolutely specialised technique,’ Bacon (in Sylvester, 1975: 107) declares, and one must not reflect long to call to mind the futile manner in which artists—now more than ever—try to distinguish themselves, dreaming up novelties external to themselves: watercolours fabricated out of dissected felt-tip pens, drawing in crayon onto torn pieces of cloth, tearing old posters from the street, growing seeds inside a pyramid of fluorescent lights. The novelty of our technique may win us some attention, but it will never remedy a weak sensibility.

Sensibility, of course, being a well-chosen word: it draws our attention to an artist’s sensory intersection with the world. The point warrants attention, because I think a non-painter is content to let most of the visual world wash over him, hardly taking it in. A non-painter uses his senses for gathering relevant information; a painter stops to drink in the pink and blue ferment of the sky and shouts, ‘Look at the clouds!’ while the helpful and oblivious non-painter replies, ‘Don’t worry, the storm is moving away from us.’ A painter, one worthy of the name, is genuinely attentive to visual stimuli, is acutely perceptive, is besotted with sight. She hardly has to invent visually interesting things—she is overcome by the sensory cornucopia of existence and is struggling to survive such abundance by her feeble attempt to instate order through her brush. A painter’s sensibility will most certainly emerge if she works with technique rather than against it, as she comes nearer to her sensory reality as her facility with her techniques grows. Perhaps a furious linear energy drives through the human form; perhaps muscles swell according to certain rhythms. Though she ‘may use what’s called the techniques that have been handed down,’ like Bacon (in Sylvester, 1975: 107), she may use them to create powerful work that has never yet been made, declaring with Bacon: ‘my sensibility is radically different.’

Copy after Steinl, Kunsthistorisches Museum, Vienna

I am not defending the type of fluency that produces indistinguishably polished works. Rather, the painter should use her hard-won ability to investigate, to explore, to forge connections that others might not see. One way to stay alert is to make things harder for oneself. Bacon (in Sylvester, 1975: 91) explains: ‘Half my painting activity is disrupting what I can do with ease.’ His insight has always been made by the kind of painters who prefer to test their abilities and extend them more than they care to show off. David Paulson is the kind of painter who works with stubs of pencils, works with his left hand, intercepting his habits with stumbling blocks that force him to work hard to regain control, or, more accurately, to gain a different kind of control.

Under this kind of self-sabotaging lies a desire to find and apprehend new problems. Bacon (in Sylvester, 1975: 37) discusses the difference between working from photographs of paintings, such as the Velazquez popes he had about his studio, and working from photographs of people, explaining that the paintings present problems that are already solved. An artist makes copies of old masters because there is something to be learned by tracing the solutions of someone more advanced. She recognises the problem she would like to confront, and follows the thought processes of another by mimicking their actions. ‘The problem that you’re setting up, of course,’ says Bacon (in Sylvester, 1975: 37) ‘is another problem.’ It is when we apprehend the physical world through our own senses that we discover problems demanding fresh solutions. And when we find ourselves turning again and again to the same reliable solutions, we must interrupt the process manually, thwarting our usual responses such that we not only respond in a new way, but set up the problem in entirely different terms.

And Bacon rightly recognises that few are sympathetic to this personal struggle. Each new painting, each portrait sitting, offers the opportunity to probe some quietly festering problem. It demands untested approaches, not guaranteed to succeed. The sitter expects an exquisite rendering of their face; the painter relishes the opportunity to wrestle with bold new ideas. The sitter grows apprehensive, gradually becomes alarmed. ‘In what sense do you conceive it,’ what you are doing to their face, ‘as an injury?’ asks the moderate Sylvester (1975: 41). The painter can hear Bacon scowl. ‘Because people believe—simple people at least—that the distortions are an injury to them’ (in Sylvester, 1975: 41). And distortions they must be, in the tussle with the problem, in the trial of new responses. Because of this, it can become unpleasant to work with a model. We must pretend that we are immortalising their appearance, to placate their doubts; we would rather shut them out entirely, except for the bundle of gripping visual problems they represent, and ‘practice the injury in private by which [we] think [we] can record the fact of them more clearly’ (Bacon, in Sylvester, 1975: 41). Sylvester (1975: 43) tries to extract something psychological out of the discomfort: Perhaps ‘what you are making may be both a caress and an assault?’ Bacon assures him he need not make so much of the matter. It is hardly a deep psychological tension, but simply that ‘they inhibit me’ (Bacon, in Sylvester, 1975: 41).

Copies after Titian, Kunsthistorisches Museum, Vienna

The thrust of Bacon’s discussion of painting, however, might be reduced to the omnipresent frustration that paint does what it wants. When he trusts everything to chance, he is giving himself over to the fact that paint is disobedient, that the most controlled stroke defies control. He almost boasts that ‘in my case all painting … is accidental,’ because ‘it transforms itself by the actual paint’ (Bacon, in Sylvester, 1975: 16). All of Bacon’s (in Sylvester, 1975: 97) language about paint—‘such a fluid and curious medium’—suggests the near superstitious reverence of paint familiar to the painter. Paint seems to have agency—perhaps painters secretly believe it. ‘I don’t in fact know very often what the paint will do,’ admits Bacon (in Sylvester, 1975: 16; 54), or ‘how [these marks] will behave,’ as though the paint is another active participant, responding to his choice of a large brush with an unexpected manoeuver. Paint is so deliciously malleable but it does not bend to our every intention; paint is ever the volatile element in a painting (Bacon, in Sylvester, 1975: 93). He fears to invite a story into the painting, in case it should ‘talk louder than the paint’—which, we might presume, is talking too, if softly (Bacon, in Sylvester, 1975: 22). Perhaps it sounds mystical to speak in hushed tones about this silent back-and-forth between painter and paint, to attribute the uncontrollable features of paint to its own will. But Bacon describes something very real to the most experienced of painters, something which lies at the heart of the attractiveness of painting. Painting will always be a challenging and thus deeply demanding and rewarding medium, because of paint.

‘I don’t think that generally people really understand how mysterious, in a way, the actual manipulation of oil paint is,’ Bacon (in Sylvester, 1975: 121) comments, and perhaps here he gives the most profound insight of all. To an outsider, a painter must simply master those tricky technical things, master paint, and put this mastery to good use. But the pleasure and the satisfaction of painting derive from paint’s continual defiance of the painter’s every attempt to constrain it, to impose order, to systematise, to achieve fluency. It is paint itself that is profoundly and infinitely interesting—those mere technical things that scamper at the edges of aesthetics. The non-painter need not dig so deep for profound insights, for they are not so intellectual as might be supposed.


Deleuze, Gilles. 2003. Francis Bacon: The Logic of Sensation. Trans. Daniel W. Smith. 1 edition. Continuum: London.

Deleuze, Gilles, and Guattari, Félix. 1994 [1991]. What is Philosophy? Trans. Hugh Tomlinson and Graham Burchell. Columbia: New York.

Sylvester, David. 1975. Francis Bacon, Interviewed by David Sylvester. Pantheon: New York.