Friday, February 12, 2010

Geometry of the Rhizome: Models for Art

A new geometry for the rhizomatic, the impossible figure. The reversible perspective cube or the Necker cube and the Penrose cube both of which are impossible figures. Louis Albert Necker, the inventor of this optical illusion was also a crystallographer. Is it more than just coincidence that Gilles Deleuze uses the metaphor of the crystal to discuss the temporality of cinema in The Time Image? There is no way to know with certainty but there does seem to be a strong affinity toward his collective assemblages of enunciation, the body without organs and the “impossible picture” (p. 7)[i]. In the Penrose cube the informative regions are apparently coherent yet are revealed to be invalid in three dimensions. Glanced at one corner at a time the figure maintains somewhat of a semi-stable identity. The artist Maurits Cornelis Escher was intrigued by these types of visual paradoxes and made them the study of many of his own works (see below).

We tolerate the inconsistency of the impossible figure because of the way our perception constructs predicative assumptions about outlying visual information. As Hochberg shows detail falls off as a function of the distance from the center of the fovea where our gaze is focused so that toward the periphery of our gaze things are not very detailed (click on figure below to enlarge) (p. 241)[ii]. Let us take the example of the Penrose cube (second from top). The illusion depends in part on the distance over which integration proceeds between one set of intersecting lines, one dihedral corner, and another. In other words our ideas about the figure require us to move our eyes from one point to the next and adjust accordingly. Thus it has a lot to do with the measurable relation between things: that state of transition, which separates one view from the next. This is what I believe Deleuze is essentially talking about when he states that, “the units of measure are what is essential” even though he is not speaking directly about perception here (p.4)[iii]. The figure is continually changing between one configuration and another depending on the movement of our eyes. There are more than two cubes present at once, their inner dynamic constituted by a centrifugal or de-centering force. Hence the method of the rhizomatic, which de-centers things onto other dimensions and registers.

Even when we are told in advance that the figure is impossible we persist in seeing the Penrose cube and the Necker cube as three-dimensional objects. Why is this? Hochberg conjectures that “our perceptual systems seem more tolerant of inconsistency than thy would if they mirrored faithfully the couplings found in the real world” (p. 244)[iv]. It shows that our understanding of the world is a generative act and not merely the passive state of recognizing forms in nature. This idea of perception works well with the description of the rhizomatic, “a semiotic chain-like tuber agglomerating very diverse acts, not only linguistic, but also perceptive, mimetic, gestural and cognitive.” In short we can comprehend and make sense of varying versions of reality, a testament to the flexibility of our minds and the fabric of space-time. Try to grasp the blade of grass in the middle and it is all but impossible. The impossible cubes are in a state of perpetual flux and this is how they can maintain their own inner logic (n-1) in spite of their apparent and rational impossibility. If we enumerate the traits of the rhizomatic, such as the principles of connection and heterogeneity, multiplicity, asygnifying rupture, cartography and decalcomania we find them analogous with those that constitute the perceptual illusion of the Necker and Penrose cubes, which are made possible in turn by the expanding connection between the picture and our neural apparatus, which co-exist on the plane of immanence. It constructs the unconscious and does not just trace or illustrate a representational figure or concept. What further implications might the Necker/Deleuze models have for practicing artists? Can we observe these structures in any currently existing works of art? The question remains open to investigation.

caldwell l.

[i] Deleuze, Gilles, and Guattari Felix. A Thousand Plateaus: Capitalism and Schizophrenia. University of Minnesota, 1987.

[ii] Hochberg, Julian E. In the mind's eye Julian Hochberg on the perception of pictures, films, and the world. Oxford: Oxford UP, 2006. Print.

[iii] Deleuze

[iv] Hochberg


  1. We will be discussing some related material in a few week's time when we read the chapter from A Thousand Plateaus called "The Smooth and the Striated." As we will see, what Deleuze and Guattari refer to as smooth space is influenced by the concept of the manifold by the 19th century mathematician Bernard Riemann. Riemann is a key figure in the rise of non-Euclidean forms of geometry. These new kinds of geometry (e.g., differential geometry) would stimulate the development of new spatial models, such as found in topology, not to mention the new model of space-time proposed by Einstein. Deleuze, for his part, often draws a parallel between the conception of space found in Riemann and the work of the filmmaker Robert Bresson. (See discussion of "Any Space Whatever" in Cinema I: The Movement-Image.) This link between Riemann and Bresson may strike some people as odd (for me, it works well if one focuses on late Bresson: Lancelot du lac, Le Diable probablement and L'Argent), but Deleuze's larger point should not be obscured: for Deleuze, cinema, among other modern art forms, is to be understood as a participant in the new topological models, the new forms of spatiality, that follow in Riemann's wake.

  2. I find the use of mathematical principals in Deleuze to be especially interesting and an increasingly common thread through the readings we have done so far. I just watched a Manuel DeLanda Lecture on youtube from EGS that gives a very detailed review of the mathematics that Deleuze seems to draw from. (


One larger question that I would like to explore as the semester continues is the relationship between computation, mathematics, and the differences between transcendental vs. immanent philosophy. I wonder to what extent Deleuze discusses computation and information sciences in his book "The Fold." Leibniz is credited with both the development of calculus as well as initial theories about computation. I understand that Deleuze characterized information sciences as "arborescent systems [...] which still cling to the oldest modes of thought in that they grant all power to a memory or central organ." (A Thousand Plateaus, 16) I wonder to what extent computation can be characterized as transcendental or can certain forms and organizational principles in computation be rhizomes? Is the increasingly decentralized and deterritorialized internet essentially like a large hierarchical tree or something more complex?

  3. Ariel, it would definitely be interesting to explore the mathematical basis of Deleuze's work in more detail, especially in The Fold, which I find fascinating but elusive. In many ways it seems to be a summation of Deleuze's vast oeuvre, but its complexity/density has yet to be adequately addressed. A critical reader has recently been released and I will report on it when I receive a copy. I have a few good essays on The Fold and a few good essays which address the mathematical elements of Deleuze's work which I could make available to you. In regards the latter there is also a recent book dedicated to this subject: Virtual Mathematics: The Logic of Difference. This collection includes an excellent introduction to Riemann and manifold by Arkady Plotnitsky, who has also written a number of other essays on Deleuze and science. One of these, on Quantum Field Theory, is available at [you know where].