art and science similarities, differences and interactions: special issue || the fractal cube and...

Leonardo

The Fractal Cube and the Paradigm Shift in Art and ScienceAuthor(s): Serge Salat and Françoise LabbéSource: Leonardo, Vol. 27, No. 3, Art and Science Similarities, Differences and Interactions:Special Issue (1994), pp. 241-248Published by: The MIT PressStable URL: http://www.jstor.org/stable/1576061 .

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INTERACTION

The Fractal Cube and the Paradigm

Shift in Art and Science

Serge Salat and Franfoise Labbe

All that we see or seem Is but a dream within a dream.

-Edgar Allan Poe

THE PERSPECTIVE PARADIGM The cube is the archetype of spatial representation as elaborated in the West since the Renaissance. From the Trecento to the Quattrocento, from Giotto and Cimabue to Massaccio and Piero della Francesca, cubic boxes and checkerboard pavings infiltrated painting. Checkerboard patterns run un- derneath figures, appear in grids across walls and reach up even to the ceilings, thus becoming the common scale of all

spatial values-that of bodies and the intervals of empty space between them.

It was the architect Brunelleschi who, in an experiment dating from the beginning of the Quattrocento, first elaborated the theoretical model of cubic perspective space. In this

experiment, the viewer peeped through a hole in the back of a painting, which was pierced at the composition's vanishing point, and observed the reflection of the painted image in a mirror. Brunelleschi placed a second mirror in the painting itself, in the area corresponding to the sky, to reflect the real sky and "even the clouds pushed along by the wind, when it was blowing" [ 1 ].

Fig. 1. Brunelleschi's first experiment with the Baptistery of Florence seen from the door of the cathedral. The spectator can compare (a) the direct view of the building through a hole pierced in the panel with (b) the reflection in the mirror of the painted image on the panel.

In this painting, Brunelleschi

represents the baptistery of Flor- ence as it was seen from the inner

porch of Santa Maria dei Fiore cathedral. The eye of the viewer is thus placed in a doorway reflected in the mirror. This found-

ing principle of encasement re- curs throughout this article and in our descriptions of the Fractal Cube.

Brunelleschi's device enables the painting to coincide with

ABSTRACT

The Fractal Cube, developed by the authors, is an installation that projects the grid plan of Renais- sance perspective into the space of higher dimensions and complexity generated by contemporary mathematics. Covering such topics as contemporary mathematical theory and the works of such artists as Brunelleschi and Velazquez, the authors discuss the context that in- spired the various Fractal Cubes they have built.

what it represents. He merges one into the other: the picture and the model. Placed at the point assigned by Brunelleschi, the viewer sees the baptistery of Florence not with both eyes, but with one eye pressed to a hole (Fig. 1). The direct view of the baptistery is hidden, but its reflection in the mirror reveals its painted image on the reverse side of the panel-and the viewer is convinced that what he or she

Serge Salat (artist), 16, rue de la Glaciere, 75013 Paris, France.

Francoise Labbe (artist), 16, rue de la Glaciere, 75013 Paris, France.

Manuscript solicited b Jacques Mandelbrojt.

Received 19July 1993.

y--/ "

s.J A A

LEONARDO, Vol. 27, No. 3, pp. 241-248, 1994 241

------------

? 1994 ISAST



sees is reality. The painting reveals the real-it substitutes itself for the building with such precision that the viewer cannot tell whether it is a painting or the baptistery itself that he or she is looking at. The device aims not so much to "show" a painting as to "demonstrate" it. With this experiment, Brunelleschi unfolded the principles of an art that enabled the construction of a three-dimensional volume on a flat surface: perspectiva artificialis rather than perspectiva naturalis, which, at the time, designated the science of optics.

The transparent, crystalline space of

perspective was the result of a compli- cated theoretical manipulation in which the eye, the infinite and a mirror were attributed precise roles and a fixed location. All of the elements were linked to-

gether by the hole pierced in the back of the painting and the two mirrors. Our aim with the Fractal Cube has been to rethink Brunelleschi's device, with its

cloud-reflecting mirror, its eye at the back of the painting and the infinity hole in the centre of the representation.

THE CHANGING OF THE PARADIGM A paradigm is an object or a device that functions as a model for thought, regu- lating it and playing variations on it in the most diverse fields. A paradigm is an artefact comparable to the manifest es-

ri U ..... Fig. 2. Velazquez, * _ "r Las Meninas (The

Royal Family), oil painting, 318 x 276 cm, 1656. (Prado Museum, Madrid)

1 E1,1... i

sence of the object, real or ideal, that

thought attempts to grasp. A paradigm enables us to show, exhibit, indicate, represent and compare. Brunelleschi's device functioned for more than 4 centu- ries as the major paradigm of representation in the Western world. His conception of space as a cubic box was

systematized by Descartes in the form of an unlimited system of cubic coordi- nates. Brunelleschi invented what is known as "central" perspective-that is to say, one that supposes a centre, or a single viewpoint where the viewer must

place him or herself so as to see exactly what the painter saw. The eye of the viewer must be exactly opposite a precise point in the painting-the "principal point" of the composition. His experiment made it possible to demonstrate the compulsory concordance between the viewer's viewpoint and the principal point of the picture, since the two points merged together and were folded into one another by the artifice of the mirror. What is more, by inscribing the trace of the eye in the form of a hole at the very heart of the image, the experiment revealed the central place that, from that time on, humankind attributed to itself in the representation of space.

It is this paradigm that twentieth-cen-

tury thought has progressively destroyed in art, science and literature. We hope to show how the Brunelleschian paradigm is now being replaced by a new

paradigm derived from G6del's logic, contemporary theories of complexity, chaos theory and fractal geometry. The Fractal Cube that we built and exhibited in 1992 at the Pompidou Centre is an

attempt to give concrete form to the new paradigm. It uses the same elements as Brunelleschi's device (eye, door, cubic grid, mirror) but combines them in a different configuration, one that disperses the centrality of the viewer and sweeps away any coincidence between representation and "reality."

Brunelleschi's genius resides in his

having given a semblance of theoretical

consistency to a device for representation whose shortcomings are nonethe- less obvious. The reduction of reality to forms describable by linear contours, the immobilization of the viewer's

(single) eye at a point coinciding with the projected infinite, the self-reference that insinuates itself into the system due to the duplication of the representation in the mirror's reflection-these limita- tions present difficulties that had to be

rethought in the space of contemporary mathematics as elaborated by G6del and Mandelbrot.

Indeed, the time is ripe for a new

paradigm. We believe the challenge consists now of building a space with suffi- cient complexity to adequately reflect the complexity of the real.

From Platonic Volumes to Fractals Leonardo da Vinci appears to have been the first to have seen the limits of de-

composing the world with the aid of Pla- tonic volumes. In the last years of his life, he seems to have been haunted by billowing images of the deluge, an ob- session that took form in the admirable series of drawings now conserved at Windsor Castle, in which we see the

metamorphosis of a universe into wave and cloud.

The crests of ocean waves, furling over towards their troughs, beat and rub at the hollows of their surface; subjected to this friction, the falling water breaks into small particles and changes itself into a thick fog that is taken up by the winds like curling smoke or swirling mist; then, as it rises into the air, it turns into cloud [2].

Out of this waving and sinuous geometry of clouds, this tempest driving clouds and waves towards the vortex of the world, Leibniz was to formulate a

general model of the continuous:

The dividing up of the "continuous" should not be considered as being like

242 Salat and Labbe, The Fractal Cube and the Paradigm Shift



i ' --

(3b) (3a)

Fig. 3a. Fractal Cube, mixed-media installation, 9 x 9 x 9 ft, Centre

Georges Pompidou, Paris, 1992. The Fractal Cube is a frozen instant in a complex metamorphosis between the real and the virtual, the figure and the ground, the unchanging and the changing, form and flux, Euiclidean geometry and fractal geometry.

Fig. 3b. Fractal Cube, mixed-media installation, 9 x 9 x 9 ft, Centre

Georges Pompidou, Paris, 1992. The layered space of the Fractal Cube.

Fig. 3c. Fractal Cube, mixed-media installation, 9 x 9 x 9 ft, Centre

Georges Pompidou, Paris, 1992. In the Fractal Cube, two geometrical paradigms-fractal space and Euclidean space-coexist simultaneously.

(3c)

sand breaking into grains, but like a folded sheet of paper or a pleated tu- nic, in which there are an infinity of folds growing smaller and smaller, without the body itself dissolving into points or minima components [3].

For Leibniz, in incessantly dividing themselves, the components of matter form whirlwinds within whirlwinds that

gradually grow smaller, with minute whirlwinds again forming in the concave intervals between other whirlwinds. Mat- ter presents an infinitely porous texture that is spongelike or cavernous but de- void of empty space, as there is always a cavern within every cavern. Each body,

no matter how small, contains a world, veined with irregular passages, sur- rounded and penetrated by a fluid that

grows ever more subtle.

Today, the theory of fractals expands the Baroque theme of the fold and re- discovers the intuitions of Leibniz. Benoit Mandelbrot has shown that fractal geometry [4]-according to which every fragment of an object has the same structure as the whole-can

explain the structure of most of the ob-

jects that make up our universe: galax- ies, the moon, the earth, the atmo-

sphere, the ocean, waves and clouds. Fractal objects are built like living

trees, by successive repetitions. And yet, unlike the trees with which we are famil- iar, mathematical fractal objects iterate

indefinitely. For if there were not an in-

finity of iterations, the branchings of the final iteration would not subdivide themselves, and thus they would not re- semble the entire tree. An infinite com-

plexity is folded into each fragment of a fractal object, no matter how small. As successive enlargements of the border of the Mandelbrot set show, there is no limit to the envelopment of smaller and smaller folds within other folds, of smaller and smaller whirlwinds within other whirlwinds in a fractal object.

Salat and Labbe, The Fractal Cube and the Paradigm Shift 243



From a Centered World to the Endless Interlocking of Metasystems Perspective composition shows what cannot be seen. Infinity cannot belong to the surface of a painting.

Caught between two parallel lines, the eye will never see them at a distance sufficiently great so that they meet in a point. What is this thing that never reveals itself, and which, if it did, would cease to exist? It is infinity, that which would be limited and finite if it revealed itself, for that which may not be revealed is that which may not be de- limited [5].

In Brunelleschi's device, the point of the eye and the vanishing point come

together in the hole at the back of the

painting. The prime function of Brunelleschi's

device was to reveal this coincidence of the location of the gaze with infinity. He

pierced a hole at this point and invited the viewer to regard the scene back-to- front, that is to say, literally from infinity, in a mirror. Relegated to infinity, the viewer no longer belongs to the space that the representation opens.

The entire history of central perspective has attempted to show the eye of the

painter or of the viewer on the plane of the picture. This originating point, which is often confused with the main

vanishing point, and sometimes placed outside the painting, or even made into

multiple points, has been called by various writers the "subject point," the "eye point" or the "transposed viewpoint."

As Philippe Comar writes:

Paradoxically, in mathematics, the eye (in other words the centre of projection) is the only point whose image is not defined on the plane of the picture. Thus the foundation of perspective is based on an uncertainty: the place from which we should view a painting can never be shown in the painting itself. Unless we have recourse to an artifice-such as the mirror in Van Eyck's Portrait of the Arnolfini- the place of the painter or of the viewer is in its essence an invisible place [6].

This "non-place" is a hole that forms a

spot in the centre of the image. "Unlike

perception itself," says psychoanalyst Jacques Lacan:

in a painting there is always something whose absence we notice. It is the middle field of vision, where, in actual seeing, the separating power of the eye exerts itself to the full. In every painting, it can only be absent and replaced by a hole-the reflection, as it were, of the pupil-behind which is the gaze. Consequently, and insofar as painting enters into a relationship with desire, the position of a central screen is always marked, and it is precisely by this that, in front of the painting, I am elided as the subject of the projection plane [7].

At the beginning of the twentieth cen-

tury, Godel reassessed the question of the closure of any system of signs in the

space of mathematics [8]. Turning his mind to the space of arithmetic, he stud- ied a problem that was analogous to the blind spot in perspective painting. G6del's theorem (1931) shows that el-

ementary arithmetic is incomplete (in the sense that in arithmetic the real goes

; e Fig. 4. Aleph I, mixed-media installation, 9 x 9 x 9 ft, Fifth World Bien- nial, Sofia, Bul- garia, 1989. In the Fractal Cube, the cleft by which the viewer gains access is the system's major break in symmetry, its fundamental incompleteness.

beyond what is demonstrated) and that its supposed consistency cannot be

proven without having recourse to methods that are beyond its control.

Beyond mathematics, G6del's theorem points to the vanity of any attempt to build complete axiomatic systems. It reveals the fundamental impossibility of

any self-referential discourse concerning a system. It casts all systems of signs headlong into an infinite cascade of in-

terlocking metasystems. In painting, Van Eyck and Velazquez

had attempted long before G6del to pose the question of self-reference. In the dual portrait of the Arnolfini, Van Eyck placed-right in the "eyehole"-a con- vex mirror that arrests the vanishing point and reflects for the viewer, along with an image of the space seen in a reverse shot, the open door through which the painter himself looks at the scene. By a strange, looping device, the exterior

space of painting (metapainting) was en- folded into painting.

In Las Meninas (Fig. 2), Velazquez represents himself in the act of painting the king and queen, who are supposedly standing in front of the painting the viewer is looking at, and are thus not in the picture. The canvas on which

Velazquez is painting faces away from the viewer, who sees only the back of it.

Velazquez has represented himself

standing in front of the canvas, brush in hand. In the foreground are the Infantas, the dwarves and the ladies-in-

waiting, who are all looking at a point in front of the painting that the viewer seems to occupy. In the background is the mirror in which appears the reflection of the painting on which Velazquez is working, representing the royal couple; also in the background is an

open doorway in which a second male

figure seems to be on the point of going through. We thus have the door and the mirror of both Brunelleschi's device and Van Eyck's painting, but here they are not encased in one another but offset, disconnected.

In fact, in Las Meninas there are two

major systems that have been offset. The first is defined by the triangle constituted by the painter as he has represented himself in the composition, the

royal couple standing in front of the picture, and the reflection of the painting in the mirror. This system, in which a

picture facing away from the viewer is reflected in a mirror, corresponds in ev-

ery point to Brunelleschi's device. The second system is constituted by the viewer looking at the painting and the




figure standing in the open doorway at the vanishing point of the composition, opposite the viewer. This vanishing point is not located (as Michel Foucault

thought it was) in the mirror, but in the hand of the figure in the doorway. In re-

ality, it is perpendicular to this point that Velazquez placed himself to paint the real picture, Las Meninas. It is inter-

esting to note that the figure in the doorway, Don Nieto Velazquez, bears the same name as the painter. In a way, it is as if Velazquez has represented himself twice in this work, once in a fictive

position where he is supposedly painting the king and queen, and once again in his real position, in the form of his "double" Don Nieto Velazquez. This schism was made necessary because of

Velazquez's purpose: that of representing the operation of painting within the

painting itself. Velazquez does not so much represent Brunelleschi's device as he deconstructs it by separating the door and the mirror, which the Floren- tine architect had encased one in the other.

Hubert Damish writes that

the indefinitely suspended gesture of the figure at the door echoes the other side of the painting, that of the hand that held the brush.... Las Meninas is an auto-referential work. We shall never know which painting the painter is busy doing; but the canvas turned inside out is the emblem of the inaugural operation of modern painting, the very thing which made Brunelleschi pierce a hole in his panel, and turn it about to look at it in a mirror.... The picture is reflected in his operation. This relates to nothing less than the position of the subject, to the emergence of a science, and to the status of representation [9].

Fig. 5. A fragment of the Fractal Cube, computer drawing, 1992. In this varia- tion of the Sier- pinski Cube, full and empty parts are identified either by self-similarity or by inversion of form and ground.

Fig. 6. Aleph II, mixed-media installation, 18 x 18 x 9 ft, Grand Palais, Paris, 1989. In the artists' works, the opening of the system to allow the spectator to enter into it echoes the well-known device of the door and the mirror, which was used by Brunel- leschi when he invented perspective.

The Fractal Cube: Building the Space of the New Paradigm for Representation The separation of the subject and the world, of the inside and the outside, is the ordering paradigm of the classical world. Throughout the twentieth cen- tury, in science, art, philosophy and literature, this separation has been effaced by the emergence of a new paradigm. In the new paradigm, the subject cannot be

separated from the world. The limits between the subject and the world, between beings and things, and between

things themselves, are no longer the clear-cut separations of the Renaissance.

They have become complex borders

comparable to those of fractal objects. The Euclidean paradigm of the space of representation is being replaced by a fractal paradigm.

The Fractal Cube is an attempt to build the space of the new paradigm. Like Brunelleschi's device, the Fractal Cube is an object, a device, a machine for seeing. To date, nine cubes have been built, all of them different [10]. All have the same dimensions: 9 x 9 x 9 ft. The most elaborate Fractal Cube was exhibited at the Pompidou Centre in 1992 (Fig. 3). It is a steel structure with mir- rored surfaces, adjustable throughout, which enables the setting of six perfectly flat, internal faces that are parallel and

highly reflective. The high precision of the device makes it possible to obtain a virtual view of 125,000 cubes that are identical or symmetrical to the initial cube. Inside this reflecting system, fragments of cubic figures have been built in

plexiglass and blue fluorescent alu- minium. These fragments become com-

plete through reflection, forming the virtual overall figure: a huge generating form with 18-ft sides. This huge figure is a complex sculpture derived from

Sierpinski's cube, whose mode of gen- eration we will describe elsewhere in this article. It is made of thousands of square faces perforated by hundreds of thousands of holes cut by a digitally con- trolled laser beam.

The entire set constituted by the re-

flecting cubic system and the large cubic

sculptures contained inside it forms a

strange environment into which the viewer may physically enter. In this

space, viewers may see themselves reflected identically 125,000 times. This

multiplication is comparable to the mathematic iteration that gives rise to the Mandelbrot set. Because of the

slight irregularities in the reflecting sur-




faces, the iterated function is never a

perfect symmetry, but a nonlinear function that, by successive iteration, gives rise to a high degree of chaos. Viewers

experience fractal space vertiginously suspended between order and chaos.

They also experience the loss of a privi- leged position in the centre of space, because they see their own reflection reflected identically in 125,000 identical cubes. This figure is a good approxima- tion of the infinite, and viewers thus have a real experience of the dispersion of their image in the infinity of vertigi- nous space, which threatens their feel-

ing of unity and identity. Because of this, the Fractal Cube is

not a mathematical demonstration, nor is it the application of a theory. It was not generated by an equation. It is the result of complex architectural and

sculptural work based on notions of rep- etition, identity, difference, orientation, symmetry, thresholds and closures, borders and edges, the finite and the infinite, form and the absence of form, the

position of the eye and the conscious- ness of the body and space. In its first versions, those presented at Sofia in Bul-

garia and at the Grand Palais in Paris, its main aim was to destabilize all the spatial references of the viewer and to threaten his or her feeling of identity. It

sought to provoke an intense feeling of

"body loss" through the multiplication of the image and the ambiguity of the

spatial situation. It is precisely because of this unusual combination of sculptural work and contemporary science

that the Fractal Cube is for us an inter-

esting device. Its paradigmatic aim, similar to that of Brunelleschi's device, goes beyond the simple demonstration of the

pertinence of one geometry or another in describing reality. The Fractal Cube is no more an illustration of fractal theory than Brunelleschi's device was a simple application of Euclid's geometry. The Fractal Cube directly addresses the con-

temporary conception of the subject: a

subject that is split-exploded, in psy- choanalytic terms.

The Fractal Cube confronts the problem of the origin of space in the same manner as Brunelleschi confronted it with the hole pierced in the back of his

painting. In the Fractal Cube, the rift into which the viewer penetrates takes the place of Brunelleschi's hole. Unlike Brunelleschi's device, which was centred and placed the viewer in the centre of

space, the Fractal Cube, similar to con-

temporary philosophy and sciences, confronts the problem of the loss of ori-

gin and the sense of an infinite recur- sion in which logical systems are encased endlessly in one another without our ever being able to reach their

founding origin. Spectators literally enter a vertical

cleft in the Fractal Cube (Fig. 4) that appears to be infinite, where they gradually discover the "layered" structure of a

paradoxical space. An infinite cubic grid in the three dimensions of space enfolds a complex, luminous and transparent cubic sculpture. This sculpture begins a

process of repetitive division of space to-

Fig. 7. The Spiral and the Fractal Cube, mixed-media installation, 36 x 36 x 9 ft, axonometric drawing, Eighteenth Milan Triennial, 1992. To give form to the impossible threshold between this world and infinity, the artists have implemented strategies aimed at making spatial limits ambiguous, turning them into themselves in strange loons.

ward the infinitely small. The sculpture dissolves into a cascade of cubes, an in- cessant subdividing.

Each face of the Fractal Cube is a

Sierpinski carpet, a simple version of which can be created as follows: By sub-

tracting from a square a central ninth

part of it and then subtracting again a central ninth part of each of the remain-

ing 8/9ths, and so on, we arrive-after an infinite number of subtractions of smaller and smaller squares-at a figure whose surface tends towards zero but whose contour tends towards the infinite. The same manipulation carried out on a three-dimensional figure gives rise to Menger's sponge, a cube of evanes- cent volume within an infinitely folded

envelope, a surface growing out of scale to envelop a reality that is vanishing, an

infinity of folds on the surface of an in-

finitely complex void. The cube, which was once the arche-

type of three-dimensional space, is cast into an intermediate state in which its volume disappears while its envelope grows more complex as it extends without limit. The cube has become a cloud.

In reality, in the sculpture we have built, we propose only an eighth part of

Sierpinski's cube, the remaining seven-

eighths being obtained by reflection in a trihedron of mirrors. The overall figure has a strong geometrical coherence and

appears as an "object" floating in space. The next step consists of creating a

negative of the preceding form by build-

ing only the central empty parts of

Sierpinski's cube and filling them with small replicas of the cube itself. This new figure has an added interest: In it, full and empty parts are identified either by self-similarity, change of scale or inversion of form and ground (Fig. 5). The projections onto the reference

planes of the trihedron that organizes space are either Sierpinski's squares or their negatives, according to the scale at which we look at the sculpture. If we now enclose this sculpture in a reflect-

ing cube, we obtain a complex cubic

representation of the infinite. At all scales, the Fractal Cube is an in-

finite cubic grid that holds spaces in the same way as did the grid of the Renais- sance, but with a far greater degree of

complexity. Each fragment of infinite space contains a mirror image of the en- tirety of infinite space.

In the actual Fractal Cube, a whirl- wind of shapes disappears into the depth of darkness. Space does not un- fold as far as the eye can travel. Broken and dislocated in each of its compo-

246 Salat and Labb, The Fractal Cube and the Paradigm Shift

! J.



nents, it seems to explode, swirling in an

increasingly chaotic vortex before plunging into thick shadow. In fact, we have made space symmetrical around the six faces of the cube. The cascade of symmetries between two parallel mirrors resembles a series of folds incessantly enfolding the two symmetrical halves of

space into one another. We have folded and folded again, repeatedly, fold over fold, the real and the virtual around the same planes of symmetry, creating a border that is an infinitely layered fold where the real and the virtual are super- posed to form a highly complex figure.

It is precisely thus that-amidst this

too-perfect order, infinitely folded over itself-chaos and its yawning abyss is born. The slightest irregularity of the folded surface, amplified to infinity, leads to a chaotic fracturing of the figure. The level of order decreases with the number of folds and with the appar- ent distance effect. The part of space nearest where we are is thus more or- dered. If we turn our regard into the distance, we create a three-dimensional border, fluctuating and ambiguous, between order and chaos.

The Fractal Cube is a frozen instant in a complex metamorphosis between entire dimensions and fractal dimensions, the real and the virtual, the unchanging and the changing, form and flux. It is the simultaneous coexistence of two geometrical paradigms. It shows how we can create fractal space as a strictly Eu- clidean grid and how we can generate chaos from order. The space of Euclid and that of Mandelbrot are not op- posed; they coexist in a constant ex- change of their properties. The Fractal Cube is the metamorphosis of Euclidean

space into fractal space. It is at once a cube dissolving into a cloud and a cloud crystallizing into a cube.

In the Fractal Cube, the dissolving of lines, out of which traces of the initial order emerge intermittently, creates a slight vertigo: Light lines seem to tremble slightly, like the space that they evoke, slipping away in a vanishing movement where things are and are not, contain and do not contain. The cube is a fragile fragment of order in a sea of disorder, a window opening onto chaos. The dissolution of form constitutes the work as much as near order does.

The opening of the system to allow the spectator to enter into it echoes the well-known device of the door and the mirror, which was used by Brunelleschi when he invented perspective and by Velazquez when he tried to represent

Fig. 8. Vanishing Cubes, sculpture,- 9 x 9 x 9 ft, axonometric drawing, Lousiana Museum of Modern Art, Denmark, 1993. At once a compact block and a transparent cage, an object that is both shut and open, the . Fractal Cube unfolds a network of contradictory spatialities.

the device that founds all painting inside the painting itself.

We have made many attempts-all different-to give form to the impossible threshold of the Fractal Cube at exhibi- tions in Sofia, Paris, Milan and Copen- hagen. The strategies we implemented aimed at making spatial limits ambiguous, turning them into themselves in

strange loops, creating disjunctions or violent paradoxes. In the first version, shown at Sofia, two cubes intersect to define an ambiguous threshold, a place torn between two contradictory, simultaneous spaces. The first cube is Euclid- ean, finite, opaque. The mirrors that constitute its faces are covered with a

transparent, matte, nonreflective film. The second cube's reflecting surfaces are infinitely iterated. The cube is di- vided by plexiglass edges that contain fluorescent light tubes. The doorway in which the viewer stands is a rectangle of

space cut simultaneously in the two co- existing but incompatible universes, on

one side the infinite and the abyss, on the other, closure and limitation.

The second version of the Fractal Cube, shown at the Grand Palais in Paris (Fig. 6 and Color Plate A No. 1) has sides measuring 9 ft and is enclosed in an 18-ft black cube. Viewers penetrate the complete darkness of a black corri- dor into the invisible. Suddenly they see a column of blue light that seems infinite-similar to a kind of world axis. This column is the axis of a first 18-ft- diameter sphere built of luminous plexiglass, into which viewers can also enter. There they discover that the column is not the sole axis of a single world, but that it proliferates in a veritable forest of blue columns stretching to infinity and

containing 125,000 spheres of light that enclose 125,000 identical images of themselves. The second cube is also par- tially reflective, which creates an infinity of the second order (aleph 2 in Cantor's

theory of transfinites). This infinity of the second order contains an infinite




number of rooms infinite in the first order (see Fig. 1). The infinity of the second order infinitely multiplies the threshold of the infinity of the first order. The viewer is simultaneously envel-

oped in the 125,000 spheres of the infin-

ity of the first order and is dislocated by the interminable duplication of the in-

finity of the first order in the echo chamber of the infinity of the second order. He or she wanders in a labyrinth of labyrinths, in a paradoxical superposi- tion of limitation and dispersion, cen-

trality and dissemination. In the third version of the Fractal

Cube, shown at the Eighteenth Milan Triennale (Fig. 7), a spiral wall 120 ft

long x 9 ft high was built around the Fractal Cube. In reality, each of the forms is doubled: around the Fractal Cube proper, a cubic grid twice its size creates a transparent spatial envelope materialized by the blue fluorescent

edges of the cube; inside the spiral wall, a cylinder creates an extra internal layer. The ambiguous encasement of these different forms creates a paradoxical itiner-

ary in which a cylinder contained in a cube is metamorphosed into a cube contained in a cylinder, without any break in the continuous flow that furls the forms one into the other. Exterior and interior, form and content, finite and infinite, closure and opening, circle in

square and square in circle turn about in infinite play.

In the Fractal Cube shown at the Louisiana Museum of Modern Art in

Copenhagen (Fig. 8), all of these contra- dictions are pushed to their limit. A Eu- clidean grid envelops the Fractal Cube but is in turn enveloped by it, recreated in its interior, its obscure entrails. The external Fractal Cube also contains itself on the inside, at the same scale, thanks to the interplay of reflections. What the viewer sees inside-a grid around a cube-is exactly what he or she sees outside, defying all notion of an interior or an exterior of our bodies.

In the final analysis, these strategies

were all attempts that were bound to fail, because since G6del we know that the question of the threshold has to be

ceaselessly deferred in the infinite construction of a limitless labyrinth where fractal cubes of increasing size interlock without end.

In the Fractal Cube, the cleft by which the viewer gains access is the system's major break in symmetry, its fundamental incompleteness. To re-establish this broken symmetry, the system must be

enveloped in another system, creating a

system of systems, a metasystem, con-

taining an infinity of entries similar to the entry where the viewer stands, opening to an infinity of spaces that are themselves infinite.

It would be necessary to ceaselessly in- sert the system into metasystems, without ever compensating the system's initial incompleteness. This series of encased cubes would become an infinite

labyrinth of embranchments. Each pro- gression of the series from one cube to another would bring to light a new infin-

ity of embranchments, themselves enclosed in each embranchment of the

preceding space.

A CRYSTAL OF ABSENCE At once a compact block and a transparent cage, an object that is both shut and

open, set in constant transition between order and chaos, infinitely large and in-

finitely small, the Fractal Cube unfolds a network of contradictory spatialities. It is an object and a space whose dimensions are impossible. It is the impossible crystallization of a space that is contra-

dictory because of its conflicting knots. The cube encloses nothing but a void.

Were we to pursue the receding opera- tions that engender it to infinity, there would be nothing more than an infinity of folds on the surface of nothingness. The cube represents nothing. It is noth-

ing more than a device for general and local self-reference, an encasing of borders, some of which have been turned

inside out like a glove. It is a dizzying amplification of Las Meninas on a stage that has been deserted by the Infantas, where there is no longer any canvas, where the gesture of the painter is no

longer suspended in a congealed mo- ment of time and where the mirrors no

longer reflect the image of the sover-

eigns. There is no longer either subject or scene, and there is no unique viewing point. The identity of the viewer and of the person viewed on all sides by his or her own reflection is threatened. "I look at one point alone," wrote Lacan, con-

cerning perspective, "but in the world I am looked at from everywhere" [11]. In the new paradigm, the subject has disap- peared and a multiplied and off-centred

gaze wanders endlessly in the labyrinth.

References and Notes

1. Manetti, Vita di Filippe Brunelleschi, 1475 (Milan: 1976).

2. See Leonardo da Vinci, C.A., 132 r.b., Les Carnets de Leonard de Vinci (Paris: 1942) T II, p. 243.

3. Leibniz, Pacidius Philalethi, pp. 614-615; see also Gilles Deleuze, Le Pli, Leibniz et le Baroque (Paris: Les Editions de Minuit, 1988).

4. See Benoit Mandelbrot, The Fractal Geometry of Nature (New York: W.H. Freeman, 1977).

5. See da Vinci [2] T II, p. 300.

6. Philippe Comar, La perspective en jeu (Paris: Gallimard, 1992) p. 35.

7. Jacques Lacan, Le Seminaire, Livre XI, Les quatre concepts fondamentaux de la psychanalyse (Paris: Edi- tions de Seuil, 1973) pp. 99-100.

8. For more information, see Ernest Nagel and James R. Newman, G6del's Proof (New York: New York Univ. Press, 1958); and Soloman Feferman, John W. Dawson, Jr., Stephen C. Kleene, Gregory H. Moore, Robert M. Soloway and Jean van Heijenoort, eds., Kurt Gddel, Collected Works, Vol. 1: Publications, 1929-1936 (Oxford: Oxford Univ. Press, 1986).

9. Hubert Damish, L'origine de la perspective (Paris: Flammarion, 1987) p. 401.

10. The Fractal Cube was exhibited during 1992 and 1993 at the Georges Pompidou Centre in Paris, at the Eighteenth Milan Triennial, at the Louisiana Museum of Modern Art in Copenhagen, at the P3 Art and Environment Museum in Tokyo, at the Art Museum of the Seoul Art Center in Seoul, and at the Graham Foundation in Chicago.

11. See Lacan [7] p. 84.




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