Cubist Geometries

In Patterns of Connectedness, Allen Fisher writes, In the second half of the nineteenth century Riemann, Bolyai, Lobachevsky, Gauss, and others had demonstrated alternatives to the linear geometry of Euclid’s Elements.’  These names also appear in Olson’s ‘Equal, That Is, To The Real Itself’:

Within five years [of Keats’ negative capability letter], Bolyai and Lobachewsky, weren’t any longer satisfied with Euclid’s picture of the world, and they each made a new one, independently of each other, and remarkably alike.  It took thirty-one years … for the German mathematician Riemann to define the real as men since have exploited it: he distinguished two kinds of manifold, the discrete … and, what he took to be more true, the continuous. (Olson, p. 46)

Reading Marvin Jay Greenberg’s Euclidean and Non-Euclidean Geometries, I found out that János Bolyai’s description of a non-Euclidean geometry was published as an appendix to his father Wolfgang’s Tentamen in 1831.  When Wolfgang Bolyai sent a copy of the book to Carl Friedrich Gauss, Gauss replied that he also had been working on a similar geometry.  In 1817, the year of Keats’ negative capability letter, Gauss had written to W. Olbers:

I am becoming more and more convinced that the necessity of our [Euclidean] geometry cannot be proved, at least not by human reason nor for human reason.  Perhaps in another life we will be able to obtain insight into the nature of space, which is now inattainable.  (Greenberg, p. 145)

Meanwhile, Nicolai Lobatschevsky had already independently published an account of non-Euclidean geometry in 1829.  In 1854, Bernhard Riemann, a student of Gauss, went on to develop the geometry of n-dimensional space known as the Riemannian manifold, which would form the basis for the spacetime of Minkowski and Einstein.

I came across references to non-Euclidean geometry in the theories of Cubism.  Albert Gliezes and Jean Metzinger wrote in Cubism (1912): ‘If we wished to refer the space of the [Cubist] painters to geometry, we should have to refer it to the non-Euclidean scientists; we should have to study, at some length, certain theories of Rieman’s [sic].’ (Chipp, p. 212)  While in The Cubist Painters (1913), Guillaume Apollinaire wrote:

Today, scientists no longer limit themselves to the three dimensions of Euclid.  The painters have been led quite naturally, one might say by intuition, to preoccupy themselves with the new possibilities of spatial measurement which, in the language of the modern studios, are designated by the term: the fourth dimension.  Regarded from the plastic point of view, the fourth dimension appears to spring from the three known dimensions: it represents the immensity of space eternalizing itself in all directions at any given moment.  It is space itself, the dimension of the infinite… (Chipp, pp. 223-4)

The concept of a fourth dimension had been elaborated from Riemann’s non-Euclidean geometry of higher dimensions.  In 1880 Charles Howard Hinton had popularised the idea in an article entitled ‘What is the Fourth Dimension?’.  Edwin A. Abbott fictionalized the contemporary discussion of dimensions in his 1884 parable, Flatland.  One frequent visitor to Cubist gatherings was the mathematician Maurice Princet who contributed to the conversations by his speculations on the possible relations of Non-Euclidian geometry, and even of the Fourth Dimension, with the new concept of space in Cubist painting.’ (Chipp, p. 223n)  It’s often claimed that Princet showed Picasso Esprit Jouffret’s Elementary Treatise on the Geometry of Four Dimensions (1903), which described and depicted hypercubes and other four dimensional shapes, and that this led to the discovery of Cubism.  Metzinger, however, reported it the other way around, writing in 1910 that Picasso ‘defines a free, mobile perspective, from which that ingenious mathematician Maurice Princet has deduced a whole geometry.’ (Chipp, p. 223n)

Whichever came first, clearly such ideas were in the air.  What interested me was why the Cubists were attracted to ideas of non-Euclidean geometry and the fourth dimension.  In The Fourth Dimension and Non-Euclidean Geometry in Modern Art, Linda Dalrymple Henderson says that ‘non-Euclidean geometry signified a new freedom from the tyranny of established laws’ and that ‘this recognition of the relativity of knowledge was a powerful influence on early twentieth-century thought’ (Leonardo, Vol 17, No 3, 1984, p. 205).  Henderson goes on to suggest that ‘belief in a fourth dimension encouraged artists to depart from visual reality and to reject completely the one-point perspective system that for centuries had portrayed the world as three-dimensional’ (p. 205).

It was a realisation that the established laws of Euclidean geometry are not the only way of understanding the world.  Non-Euclidean geometry gave artists license to depart from the three-dimensional illusion of perspective, which had been a convention in Western art since the Renaissance.  Three-dimensionality is also characterised as repressive, with non-Euclidean geometry as offering liberation from this ‘tyranny’.

Georges Braque, ‘Still-life with Metronome’ (1909-10)

But the Cubist painters don’t apply non-Euclidean geometry in the way that, for example, M.C. Escher does – systematically.  It’s not the purpose of Cubist painting to illustrate or approximate four-dimensional space.  It was against the tradition of representing depth that the painters got rid of perspective in the first place.  Instead, Cubist painting seems to remove depth, emphasising the fact of the two-dimensional canvas.  Pictorial space is reduced by a dimension rather than increased.  The world we are used to seeing in three dimensions is flattened.  Everything crowds the surface.  There are no gaps.  Space becomes solid.

I wanted to see how these ideas impacted on poetry so I looked at some books about Pierre Reverdy.  Reverdy lived in Paris from 1910-1926, where he knew many of the Cubist painters and wrote about their work.  In Six French Poets of Our Time, Robert W. Greene suggests some ways in which Reverdy’s poetry parallels the non-Euclidean effects of Cubist painting:

In former times one-point perspective had provided the painting with a mode of transition between foreground and background, between subject and setting, while discursive logic, traditional syntax and neatly modulated thematic development had provided comparable links or transitions in poetry.  Now we are to see instead fragments on a canvas or on a page. (Greene, p. 32-3)

Here Greene is connecting the removal of perspective in art to the break-up of syntax in poetry.  Reverdy’s poems are composed of fragments of images and thoughts and these isolated line-units with the abrupt transitions between them might well be compared to the angular planes of Cubist paintings.

For example, Reverdy’s ‘The Same Number’.

There is a flatness, a two-dimensionality about this poem which is typical of Reverdy.  The lack of transition between phrases keeps the attention at the surface.  There are phrases that are not full sentences – ‘the hardly open eyes / the hand on the other shore’ – a sentence begins and breaks off.  At other times the syntax remains intact but the sentences are extremely short and direct: ‘It gets warmer,’ ‘We were happy’.  The language is plain and literal.  Reverdy is not interested in metaphor or simile, but works by a process of paratactic juxtaposition which he calls the image: ‘…the image for Reverdy is the bringing together of two equally important objects of attention not normally associated with each other…’ (Greene, p. 33)  Phrases might be placed together on the page, but one does not take precedence over the other as in a metaphor or simile where one phrase is the main subject and the other only functions as a comparison.  The image works on the tension between the lines, as Reverdy explained:

The image is a pure creation of the spirit.  It cannot be born of a comparison but of the bringing together of two realities which are more or less remote.  The more distant and just the relationship of these two realities, the stronger the image – the more emotive power and poetic reality it will have.’ – Reverdy, Nord-Sud, March 1918 (quoted in Waldberg, Surrealism)

Reverdy’s poems are thus made up of fragmentary notations of observations and thoughts.  As he stated in the epigraph to Le Gant de crin: I do not think, I note. (Greene, p. 33)  Emphasising the materiality of writing over thought, Reverdy’s notations lack any organizing narrative voice or consciousness.  In ‘The Same Number’ there’s a ‘you’ and an ‘us/we’, as well as the unnamed ‘somebody’ and the ‘passersby’ who look up.  There are eyes, a hand and a head without any specified owners.  Many objects are named though it is difficult to get a secure sense of where we are, whether we are inside or outside, whether it’s night or day.  Greene delineates the similarity of this effect to Cubist painting:

The viewpoint of the Cubist painter breaks up into many viewpoints, while the subject of his painting sinks back or melts, as in a mosaic, into its setting.  Similarly, the speaker in a Reverdy poem seems fractured, both multiple by reasons of the ever-changing personal pronouns and splintered into shards of incomplete, enigmatic utterances. (Greene, p. 34).

It does seem to me very similar to the way in which a figure or an object can merge with their surroundings in a Cubist painting.  This has clear implications for the philosophy of consciousness and that was also an important issue for the discoverers of non-Euclidean geometry.  Gauss never published his work on non-Euclidean geometry for fear of the controversy it would draw him into.  In 1824, he wrote to F.A. Taurinus:

… it seems to me that we know, despite the say-nothing word-wisdom of the metaphysicians, too little, or too nearly nothing at all, about the true nature of space, to consider as absolutely impossible that which appears to us unnatural. (Greenberg, pp. 145-6)

Greenberg says that the metaphysicians Gauss was worried about were the followers of Immanuel Kant because Gauss’ discovery of non-Euclidean geometry refuted Kant’s position that Euclidean space is inherent in the structure of the mind.’ (Greenberg, p. 146)

Greene, meanwhile, goes on to explain that the Cubist painter and poet ‘creates by externalizing the content of his world-filled consciousness’, and asks:

How far are we here from Heidegger’s idea of man as Dasein, as Being-in-the-world, or from Husserl’s notion of intentionality, in which consciousness never exists as an independent cogito but always as consciousness of something, forever “out there”? (Greene, p. 34)

Non-Euclidean geometry showed that Euclidean space is not innate.  This had previously been used as evidence for the a priori existence of consciousness.  In Cubism consciousness is only ever of external things and doesn’t exist on its own.  Whenever we try to pin consciousness down, it slips away because we can only experience consciousness of something else.  The subject only exists through fragmentary observations of things.  Consciousness is these jottings.

Cubist painting or poetry is not simply two-dimensional.  One perspective, one reality, does not dominate over another.  The space is not static.  Why present a scene or an event from a single perspective when there are many spacetimes active in any situation?  In the Cubist poem or painting, the observing consciousness is allowed to move and change.  In constructing his poems from fleeting instants of consciousness of things, Reverdy points the way to the four- and n-dimensional poetic spaces that poets like Olson and Fisher would go on to explore.


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  1. * allen fisher says:

    There’s a complexity regarding the concept of ‘decoherence’ that might benefit from further thinking. At one place in the literarypocketblog postings it notes that ‘Decoherence is about being ‘in the condition of things’, in uncertainties, mysteries, doubts’. Yet, ‘confidence in lack’ is not a matter of being in this condition, it is recognizing that there has been a phase shift from the positions of Keats and Heisenberg and Olson, a shift that acknowledges elements of their efficacy, but also notes a different ‘condition’. It is notable that there is variety in how ‘decoherence’ is defined, so it may be that my use and reuse of the vocabulary, which is figurative, may be encouraging a red herring (or even a kipper). I think of ‘decoherence’ as referring to a condition where I am confident in the information I have been provided with, through the use of interlocutors and expertise, which contributes to my experience of truth or a multiplicity of truths, but contemporary with this, I am unable to empirically confirm the truths or the information; I am unable to perceive, even with a very powerful telescope, a vast area of the cosmos, but can get information about it through the use of radio signals, the data from which are transformed into a model of what is there. A similar condition applies when I am involved in subatomic particles; aspects of what is being recorded at CERN are not perceptible, even with the most powerful microscope, except through data that is transformed into a model of what is there or, as is as often the case, proposed to be there. In these conditions I am in a state of ‘decoherence’, that empirically, proprioceptively, is a confidence in lack.

    Posted 6 years, 8 months ago
  2. Thanks for your comment, Allen. It’s a useful distinction and something I’ll try to develop in further postings. It’s worth thinking about the difference between your position and that of Olson, between decoherence and the uncertainty principle, and between confidence in lack and negative capability. Is this phase shift you mention an attempt to move beyond the postmodern do you think? I’m thinking of Bernard Williams’ Truth and Truthfulness which you refer to in ‘Confidence in Lack’ (2006). The uncertainty principle might be seen as a relative of postmodern relativism. Decoherence seems to be an appropriate term because it’s an alternative to the uncertainty principle. Do you think that there’s been a shift in your own thinking since you wrote in Necessary Business, ‘They are unstable arrays also, as physicists like Born and Heisenberg made clear, because the reader is not simply an observer but a participant and thus affects what is read’? The observer-participant view seems to say that the reader produces the text and that the text will be different for all readers, in a similar way to how the uncertainty principle suggests that the observer only finds a particle because a particle is measured for. The decoherence view seems to suggest that the ‘classical’, macroscopic world would exist without the observer; the wavefunction would break down anyway through its entanglement with the environment. Could your decision to use the term be related to a dissatisfaction with the relativism of observer-centered uncertainty?

    Posted 6 years, 8 months ago
  3. * allen fisher says:

    I am not sure about the term ‘postmodern’, and need to unpack that. Olson’s use of the term seemed to me to herald a position that carried modernism forward. Subsequent ideas of the term from commentators like Lyotard might help here. His idea of the ‘nascent state’ of postmodernism (‘Postmodernism … is not modernism at its end but in a nascent state, and this state is constant.’ [Jean-François Lyotard, The Post-modern Condition: A Report on Knowledge, trans. Bennington and Massumi, 1984: 79]) which he follows with ‘Yet I would like not to remain with this slightly mechanistic meaning of the word’. This usefully becomes, ‘The postmodern would be that which, in the modern, puts forward the unpresentable in presentation itself …’ (1984: 81) So in answer to the first question, I think that my sense of ‘decoherence’ provides for a position shifted from Olson’s ‘postmodern’ whilst continuing to leave it in place. I do think there has been a shift in my own thinking since ‘Necessary Business’ and, thank you, that is a useful observation. I am weary of the relativistic position, and still want to reconsider what is re-articulated in Bourriaud (Nicolas Bourriaud, Relational Aesthetics, trans. Pleasance and Woods, 2002), where the difficulty, not entirely Bourriaud’s doing, results from an over-simplified reading of him and leads to a weak æsthetic condition.

    (I should say here, briefly, that I am critical of Bourriaud’s reliance on the idea of form as a ‘coherent unit’ (2002: 19), but haven’t fully thought through aspects of his work, much of which I am on board with. The conventional critique of a practice that still factures objects (such as paintings and poems), forgets to comprehend concepts and materials provided by them and appears ignorant of the position that aesthetic production requires æsthetic reception, and thus, in a rather simplistic way, rejects objects. I think Relational Aesthetics falls short because it appears to propose compomise rather than resistance, but, to reiterate, ‘haven’t fully thought through aspects of his work’.)

    Yet another matter comes to the fore, which I am in the process of trying to understand. Aspects of this are implied by Lyotard when be writes at the end of his ‘What is Postmodernism’ essay, ‘We have paid a high enough price for … the reconciliation of the concept and the sensible …’ (1984: 81-82) I notice this after the fact, that is after my own facture and after articulation of my æsthetic positions, but apparently I want to develop this with the state of decoherence, in which there is an inherent critique of empirical possibility. I will need to think this through and I’m working very slowly on this. My initial note, which has only been developed in my artwork and not in retrospection upon it, says: ‘further to the matter of decoherence is the matter of knowing, which for instance need not derive from empirical experience, but can derive from a mathematical understanding; is this in fact true? doesn’t mathematical understanding require an empirical base? This apparent contradiction has been evident for some time in my work, for instance in the distinctions in my work, usually in the same piece of work, between concepts and materials, or better understood as between conceptual practice and the materiality of the work in production. This will become more pressing over the next six months as I engage in working with the formulation of a Portable Allen Fisher (which will need to be different from the typical ‘selected poems’.)

    Posted 6 years, 7 months ago

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