Blueish yellow

The mirror (c) 2017 Samantha Groenestyn (oil on linen)

The connection between colour and geometry demands some attention. Richard Heinrich (2014: 41) argues that ‘there is always a tension between … colour space and the geometry of colour,’ that conceptualising colour in terms of space is not as simple as unearthing the underlying geometric principles that will take care of everything. He is, of course, correct in this. There are many rich and nuanced ways of conceiving of colour spatially, from Aristotle’s delightfully plain string of colours, which Newton (1672) eventually closed into a circle, which has been expanded both theoretically and experimentally into various three-dimensional schema that are as idealised or roughly-hewn as their methods dictate (Briggs, 2017). A geometric conception of colour space, like that of Philipp Otto Runge (1810), approaches colour from a purely theoretical side, permitting us the sharp analytical divisions of conceptual midpoints and the elegant polish of a sphere as the theoretical limit. The reality of colour, both for the physicist and the painter, is much rougher at the edges, much more irregular, much grittier. But this does not mean that some abstracted principles, deliberately divorced from the messy realities of light and pigment, cannot be united with the practice in an instructive way. Indeed, such conceptual clarity can help the practicing colourist organise her approach to colour, while still allowing the flexibility to adapt those principles to experience.

But this is not really the disjunction that Heinrich is getting at. Rather, he is concerned that a geometric model for colour tries to explain both our perceptual experience and our concept of colour, and that this uneasy compromise tends to destroy our concept of colour (Heinrich, 2014: 41-42). We establish a working web of relations, but relations between possibly infinite coordinates of hue-value-chroma, none of which bear any greater significance over any other such that they attract the familiar and seemingly meaningful titles of ‘red’ or ‘yellow.’ This is true, but it points to the greater underlying problem that our concept of colour is desperately flawed. That we conceive of colour so misguidedly despite our firmer scientific grasp on it has only negative implications for painters. Most pressingly, there is a pervasive and false belief that colour cannot really be taught, which lends it a certain mysticism both in philosophy and in art schools. This mysticism is only compounded by the fact that colour is persistently mistaught on the basis of our flawed conception of it. We need to reconfigure our concept of colour or, if that is too extreme, to at least separate out a working theory of colour that practitioners–painters–can rely on from a more experiential understanding of it. This, I think, is not so outlandish: physicists operate with a different set of primary colours without threatening our habitual perceptual ideas about colour. What needs to be teased out is the psychological conception of colour, dearly-held but quite unrelated to the models most useful to artists and physicists.

From Runge, 1810: Farbenkugel

The primary colours are a good place to start, especially given Heinrich’s justified criticism of Runge’s development of the colour sphere (Farbenkugel). Runge moves deftly from a triangle (picking out red, yellow and blue) to a star which incorporates orange, green and purple, smooths them into a familiar colour-wheel and fleshes the whole thing out into a ball. The dubious move (which Heinrich (2014: 38) does not let him get away with) is that he begins with certain geometric parameters but quietly dissolves them along the way. The triangle is made of points, marking out the primary colours, which are connected by lines, which signify the gradations between them. The triangle says that conceptually, we grasp the idea of a ‘pure’ red–it tends neither towards yellow nor blue, it is not in the least orange or purple, it holds a privileged status as a colour (hue) that every orangish red and purplish red does not. It says that while there are many oranges, there is only one pure red.

We can, however, conceive of a middle-orange, one that appears equally red and yellow, and a green that is no more yellow than it is blue, and likewise a perfectly balanced purple. Runge (1810) thus bisects each line and places each of these so-called secondary colours at the midpoints, forming a small inverted triangle. Perhaps what starts to go awry here is that the lines from green to orange, from orange to purple, from purple to green, do not really signify anything–just a gradation of muddy browns. Runge expands this second triangle without explanation, presenting us with two triangles which we could not, on geometric terms, distinguish, though they represent vastly different ideas: the hierarchy is dissolved. To gloss over this fact, Runge removes the points altogether, and it is this that Heinrich (2014: 40) particularly objects to. The model abandons its initial claims about the significance of some colours above others and drops into a fluid mass of relations.

Runge’s Farbenkugel development

Runge’s move is questionable, but the result is perhaps not so catastrophic. This is not only because in practice, one can navigate colour more nimbly and efficiently when one thinks only in terms of relations rather than absolutes (for example, recognising that this mix should be bluer than that mix, rather than trying to match a particular fixed shade on a colour chip). But also because our attachment to the primary colours might be unjustified. Runge’s initial choice of red, yellow and blue–even as conceptual ideals–could be as arbitrary as his model ultimately suggests.

As David Briggs (2017) describes, the concept of a primary colour is itself somewhat muddy. We generally bring to it the idea of an ‘unmixed,’ ‘pure,’ or ‘primitive’ colour. But these intuitions bring various assumptions, mostly derived from paint, which are simply nonsensical when we describe colour in terms of light. In light, common colours compound the reflectance: green does not ‘defile’ red, but their shared components yield yellow and their differing components cleanly cancel out. Another enduring sense of ‘primary colour’ is a colour from which all others can be derived. This would already force us to branch colour into two separate realms, one of paint and one of light, which revolve around different base colours: subtractive and additive primaries, respectively. Briggs (2017) assiduously notes that this formulation brings conceptual dangers of its own, particularly that ‘it is a small and slippery step from the observation that all hues can be made from three primary colours, to the assumption that all hues are made of those three colours,’ which would be another paint-oriented bias.

To further complicate the idea of a primary colour, Briggs (2017) rightly points out that in fact we cannot derive all colours from just three. For the painter, purple is notoriously elusive because red pigment is still too yellow, thus the mixture of red and blue tends to result in an unsavoury brown. Painters resort to other pigments such as a rose (suspiciously magenta-like) or to outright purple pigments. Perhaps even more shatteringly, the additive primaries are no more certain, they do not correspond to any specific red or green or blue wavelengths; rather, Briggs describes them as optimal ranges of wavelengths. Defining primary colours at all turns out to be a hazardous and imprecise enterprise; at the very least this should cause us to question what reason we have to insist on points in our geometric model of colour.

Copy after Mestrovic

That reason might have something to do with our perception. Ewald Hering (1878) describes another set of primaries: the four psychological primary colours of red, yellow, green and blue. These four colours are privileged for having a ‘mentally unmixed’ status, while all other colours seem, to our minds, to be gradations between adjacent colours. This is why an orange can satisfactorily be described as a yellowish red, but we feel uncomfortable to describe a green as a yellowish blue. This seems to be the unrelinquishable ‘grammar of colour’ that Heinrich (2014: 41) particularly wants to hold onto: the sense, based in our experience of colour, that these colours are distinct and in this way primitive. This stance seems as arbitrary and as defensible as any: green is rigid and present in our experience in a way that orange is not. Or as Heinrich (2014: 41) puts it, ‘we will have to admit that green lies between blue and yellow in a fundamentally different sense as orange between yellow and red.’ But for the painter, green remains a mixture of yellow and blue, just as red may be a mixture of rose and yellow, depending on her pigments. And for the physicist, green is the absence of blue and red, while orange is a more complex array of light. Our mental divisions–what we project onto the world and how we break it down–do not correspond to the ‘input into our visual system’ and the stimulation of our rods and cones (Briggs, 2017); nor do they correspond to the pigments that happen to be available to painters. And that might be just fine.

What I propose is to keep these three types of colour systems distinct, while acknowledging their intersections. Runge’s colour sphere perfectly captures the fluid conceptual relations between hues and their values and chroma for the painter. Since it is advantageous to think relationally rather than in absolutes when trying to establish a harmonious colour context in a painting, an idealised, geometric model of three-dimensional colour space proves a useful tool for the painter. Such a tool, being relatively simple, yet rich and adaptable to any situation, empowers the painter both to organise her observations and translate them into paint, and to teach a coherent and systematic approach to colour to her students.

Copy after Belvedere Apollo cast

Physicists, meanwhile, may continue to measure wavelengths, discuss energy, and optimise their additive primaries of red, green and blue. Since the physicist is concerned with describing what light information enters the eye, his measurements do not undermine or contradict the relational model of the painter’s pigments. Rather, the two conceptions intersect unexpectedly beautifully: the complementaries of the additive primaries (red, green and blue) are cyan, magenta and yellow. These last three are used in printing to achieve the maximum range of mixed colours, and can be shown to yield a broader gamut of colours in paint than red, yellow and blue. This elegant inversion, identified by Helmholtz (1852a), perhaps gives us a firmer reason to fix cyan, magenta and yellow as the optimal subtractive primaries, if indeed we would rather retain points in our geometric model of colour space. At the very least, we might revise our pedagogical practices and stop teaching painters colour theory based on the psychological primaries rather than on the actual properties of light and pigments.

A painter does not need to understand the physics of light in order to manipulate paint. The systems remain conceptually distinct. But I think it would be correct to say that not only is the painter’s system inversely related to the physicist’s; it is also subordinate to it in the sense that after the pigments are applied, a painting, too, is simply an object reflecting wavelengths of various frequencies into the rods and cones in our eyes. In this sense, as Briggs (2017) argues, the painter works with light. He offers a particularly nice example that bridges the two systems in the practice of painting. A painter can drag paint roughly over dry paint of another colour such that the colour underneath sparkles through the gaps, or lay small strokes of different colours next to each other as the Impressionists did. The eye mixes these physically unmixed colours in an additive manner. Scientifically, it would be called ‘additive averaging mixing;’ painters call it ‘optical mixing’ and use it knowledgeably to great effect. Briggs (2017) further argues that the painter works with perception, and that what the spectator perceives remains largely geared around the four psychological colours, by which he makes sense of the painting.

And so we return to the ‘concept of colour’ that Heinrich is reluctant to dissolve into the more sophisticated systems. Drawing on Ludwig Wittgenstein, he relates it to a ‘grammar of colour,’ which modestly and openly captures something but not all of our experience of colour (Heinrich, 2014: 41). This is the key: none of the systems of colour we have discussed capture everything of our experience of colour; each operates in its realm without excluding or invalidating the others. An artist might comfortably talk of a ‘blueish yellow’: her vivid cadmium yellow paint is redder than the mental ideal of yellow; she can physically add blue to it to make it more yellow. But for the spectator, who now sees an ideal yellow in the painting, no feat of mental dexterity seems to allow him to imagine a blueish yellow. The slightest introduction of blue slides the colour irrevocably into the lush spectrum of greens. That is simply the mental category of green. And since, mentally, green is opposed to red, our brains cannot grasp a red that leans towards green, or a green that leans towards red. The curious thing is that yellow and blue, though they complement as strikingly as red and green, merge effortlessly into a pleasing colour. This says very little about how light or pigments operate, but it says a great deal about what we project onto what we see. Perhaps a phenomenology of colour would treat of questions like these.

Copy after Mihanovic

In any case, as spectators with firm mental categories for colour, the are things we can say about colour, and things that we cannot. Wittgenstein (LWL, 8) is not so facetious to suggest that certain models of colour–such as his favoured colour octahedron–are ‘really a part of grammar… It tells us what we can do: we can speak of a greenish blue but not of a greenish red etc. … Grammar is not entirely a matter of arbitrary choice.’ Grammar has its role, and need not be threatened by geometrical schema designed to help the painter navigate colour space, any more than it should be threatened by physics. A grammar of colour seems to attempt to describe our intuitions about colour based on how we perceive it, just as the grammar of a natural language attempts to explain how we structure our expressions, even though it may consist more in explaining exceptions than syntactic regularities (Chomsky, 1965: 5). Perhaps the intersection between a geometric colour space and a grammar grounded in a phenomenology of colour would reveal yet more rewarding insights, perhaps as beautifully connected as light and paint have proved to be.

Briggs, David. 2017. The Dimensions of Colour: Modern Colour Theory for Traditional and Digital Painting Media. Accessed November 2017, <www.huevaluechroma.com>.

Chomsky, Noam. 1965. Aspects of the Theory of Syntax. Cambridge, Mass.: MIT.

Heinrich, Richard. 2014. ‘Green and Orange – Colour and Space in Wittgenstein.’ In: Frederik Gierlinger, Stefan Riegelnik (Eds), Wittgenstein on Colour. Berlin, Boston: De Gruyter.

Helmholtz, H. 1852a. ‘On the Theory of Compound Colours’. Philosophical Magazine, Fourth Series, 4(4): 519-34.

Hering, Ewald. 1878. Zur Lehre Vom Lichtsinne. Wien: Gerolds Sohn.

Newton, Isaac. 1672. A Letter of Mr Isaac Newton, Professor of the Mathematicks in the University of Cambridge; Containing His New Theory about Light and Colours: Sent by the Author to the Publisher from Cambridge, Febr. 6. 1671/72; In Order to Be Communicated to the R. Society. Philosophical Transactions of the Royal Society, 6, 3075,8.

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

Wittgenstein, Ludwig. 1980. (LWL) Wittgenstein’s Lectures, Cambridge, 1930-32, from the Notes of John King and Desmond Lee. Lee, Desmond (Ed.). Oxford: Blackwell.

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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.

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