Planes, Projections and Transformations
Painting provides us with opportunities to see the historical development of human perspectives regarding the concept of space symbolically and scientifically. Throughout time, the use of devices to render complex figural arrangements have contributed to the communication of ideologies concerning our relationships to the world. Armbrogio Lorenzetti’s use of the vanishing point in his Annunciation of the Pinacotheque of Sienna from 1344 illustrates an infinite distant point at which all orthogonal lines of the floor meet. Later on, Jan van Eyck used a systematic arrangement of coordinates to represent space reaching forward to the surface plane of the painted canvas. As for projective geometry, the first correct use of one-point perspective is attributed to a painting by Mansolino de Panicale’s, St. Peter Healing a Cripple and the Raising of Tabitha, 1426-27. This technique was introduced in the Renaissance by architect Fillipo Bruneschelli. The projective diagram demonstrates a geometrical method of perspective, presenting a linear mathematics in planar projections that artists could use to depict distance and proportions in painting, and solid forms as they would appear in perspective. With one point-perspective, an artist could organize a universe around a vanishing point located arbitrarily in a distance. Before projective space enters the picture, artists of the late Middle Ages, such as Giotto di Bondonne, implied spatial relations by overlapping figures upon a two-dimensional plane. According to art historian Erwin Panofsky, the aggregate space of Giotto and other artists would, by the Renaissance, come to be superseded by a true central perspective extending infinitely into space. This, he says, demonstrates an abandonment of the idea of a cosmos “with the middle of the earth as its absolute center and with the outermost celestial sphere as its absolute limit.” As a result, the idea that actual infinity exists beyond and outside the physical world became conceivable.
The first shadow tracing was probably not intended to be a study in geometry. Nevertheless, this gesture did represent a mathematical projection. A casted shadow is an example of a type of parallel projection (orthography). It is a transformation geometry, concerned not with elastic transformations, but with straight lines and length. With it, an artist may use converging lines to make enlargements and reductions in the scale of an image by changing the relationships between the distance of an object, a source of light, and the surface onto which the object’s shadow will be traced. In representational work, shadows help a viewer enter an environment. They can direct the eye towards a light source, such as the sun in a landscape, or even light felt to exist somewhere outside the depicted scene. These mappings of light sources are useful in one-point perspective paintings. In Caravaggio, for example, Cartesian space, composed of linear elements that intersect only at right angles, merges the painting’s space with the viewer’s. This illusion of reality made possible the exploration of theatricality by intensifying the contrasts of light and dark (called chiaroscuro) and carefully exaggerating the perspective of the viewer. Depicting the light source in the painting as if it were coming from the space of the viewer, Caravaggio was able to bring viewers into the space, inviting them to become more intimately engaged with the depicted events to which they become an immediate witness.
I paint in a space somewhere between shadow projection and Cartesian perspective, or at least I think that’s a good way to characterize it. This mapping was first introduced in my practice in 2009, after I had come across the word affine in Felix Guattari and Gilles Deleuze’s text Anti-Oedipus: Capitalism and Schizophrenia. They wrote that, in barbaric societies, a woman, mother, could become an affine to her offspring. As “affine,” she was a place wherein a child could position him or herself in the matrix of the cosmos.Locating this geometric space within the body was, as I saw it, a way to place the universe, infinity, back into the world. It is positioned as the meeting of the earth with that which is beyond. Mathematically speaking, I was imagining this to be a site—a structure set in a topological space which makes those categories in it act as open sets of that space. In this site, conditions are loose and flexible, defined by properties of connectedness or continuity. Through intensive visual studies of this geometry I came to see how this matrix would serve as a network structure through which I could explore abstracted figures interacting with elements of their environment—in painting, the environment is mostly represented as color, geometric objects, and light. Abstracting a body into these movements was, I thought, a way for me to immerse a figure in its environment and free it from a singular projective perspective through subversive action.
Affine space is composed by an ordered geometry that maps an intermediacy of an object or form, with no preference of chronological ordering.The etymology of the word affine can be traced to Old French afin, or Latin affinis. The word dates back to the early 16th century and means “closely related” or “close to.” The mathematical sense of the word affine dates to the early 20th century, and it is through the mathematical sense that I have been trying to map it in a visual field. Affine equations transform an object into another translation of the object, sufficiently alike to preserve its likeness. Shadows also map similarities, but the forms a shadow takes may not be unique to that which it imitates. A shadow of one object may be similar to a shape projected by another dissimilar object (for example, a cube and a cylinder may cast a similar two-dimensional shape). Unlike shadow geometry, affine geometry does not lose the likeness of its object and in this way may be said to better maintain the identity of its subject. This geometry’s functions translate points of a given object through a logic that maintains the object’s proportions. It allows for more points of translations that can be mapped in the rigidity of a Cartesian scene. The single vanishing point to which all lines converge in Cartesian mapping does not allow for all the possible projective transformations of affine translations. Thus affine space is more boundless and weightless—or so was the sense I was getting early on from my paintings.
An example of the difference of affine space and Cartesian space can been seen in Counter Bells (2013). In this work, the centered object—a display shelf with nine shelves—has been placed in a space with a vanishing point. The abstracted representations of “bells,” placed on the shelves, and the rays of lights bouncing amongst the architecture of the objects in a room have all been imagined through affine linear mappings. The display shelf is more or less architecturally solid and weighted, while the rest of the objects seem potentially solid, shifting without the constraints of gravity, somewhat in intermediacy.
I was told by a friend who had been living with the above painting for several months that “crazy things happen” when shadows from his window streamed across the surface. “It moves” he said. After witnessing similar events of light and shadows casting across surfaces in my own studio, I have come to the decision that the light and sets of parallel lines cutting across the surface over time create illusions of movements in two ways—the colors change in the varying lights, and the new sets of parallel lines (window shadows) shape the movements of sets already painted on the canvas. In some ways, affine paintings cross the territory of time-lapse imagery and still life.
In an attempt to visualize a pure abstraction of affine movement in relation to a figure, I decided, in 2010, to scale an affine mapping to fit the proportions of my body. In the painting Without the Red of the Woodpecker, my “initial” axis was determined by a vertical line (located off center to the left). I positioned myself in front of the vertical axis and extended my arms and legs towards the corners of the canvas frame. An affine linear map was drawn in relation to the points of my body. In mapping this geometry, I wanted to observe how my “center”—my starting position—would become displaced in the mapping. With each set of parallel lines, the beginning axis was shifted from its initial vertical position. This, visually, created a sense of continual motion and movement in an applied color system whose only condition was that I not paint with the color red.
My understanding of these geometric movements and spaces have recently expanded with the aid of technology. In a project we titled Sticks, programmer Eddie Elliott and I have put together an artwork that allows viewers to interact with a geometric mapping that follows the rules of parallelism applicable to affine geometry. This piece began as a means for us to see how affine transformations may be calculated in computer models. These models, I hoped, could be used to visualize affine translations past the constraints I was experiencing in my two-dimensional renderings. Though we were not able to take it past the two-dimensional plane, our experiment did equip me with new information about the work I was doing. I will briefly mention two noteworthy observations.
The first has to do with the difference between the rendering of the transformation spaces in my paintings and in the computer model. The computer is programmed and works through set parameters, following rules based on Elliott’s coding. This allows for variation, but not the kind generated from “error” or a new situation that occurs by breaking away from the rules of the system. In comparison, the transformation spaces in the paintings are all delineated by my hand and through human calculation. Comparing the geometries drawn by hand versus those generated through computer programming, a keen eye may see how, sometimes, my desire to make the application of the geometric spaces “work” in a painting was overriding what could have been a more pragmatic approach. This was not intentional (I was sincerely trying to correctly map these geometries), but in retrospect I am comfortable saying that I made mistakes. These errors are what some of us painters call “aha” moments—meaning they result in unintentional and unpredictable surprises that add to the overall piece. These are the kinds of gestural errors the computer cannot make.
The second noteworthy observation was made by Giuseppe Longo. We were discussing a mid-20th century Italian painting of a woman, a painting I observed hanging in an exhibit at the Museum D’Orsay, Paris, France. The composition, Longo said, seemed similar to a Renaissance painting he had seen some time ago depicting Mary, the Mother of Christ. As we were talking, I opened the browser on my laptop to show Longo the collaboration Elliot and I made. After demonstrating the interactivity of Sticks to him, he casually says to me “It’s got many vanishing points. You’ve made many Madonnas.” This remark made me realize something I had not really thought of before though now, looking back, seems quite obvious. Affine geometry structures the spatial mappings of many vanishing points for one object. In other words, it presents a visual representation of potential displacements of an object in space—not just one orientation of an object space, but the possibility of multiple orientations; not just one Madonna, but infinitely many. The representation of sequential rotations and translations of objects in affine spaces allows us to create object concepts in an infinity unbounded by a single point, something which can help us construct an idea about coordinate systems in which all the universe is organized. Such cognitive orientations help us navigate in the three-dimensional world. Through art we can see how such ideas have been visualized through the centuries. Just like our representations of it, our intuition of coordinated space, according to 20th-century French psychologist and philosopher Jean Piaget, is developed through a series of thought processes constructed by our actions and interactions with objects in the world.
