From the Exhibition:

The Islands of Benoît Mandelbrot Fractals, Chaos, and the Materiality of Thinking

Focusing primarily on the work of Benoît Mandelbrot (1924–2010), one of the most notable mathematicians of the twentieth century, The Islands of Benoît Mandelbrot: Fractals, Chaos, and the Materiality of Thinking explores the role of images in scientific thinking. With their capacity to generate and shape knowledge, images are at the very core of scientific investigation: charts, graphs, notebooks, instrument readings, technological representations, even mental abstractions–all make up the essential stuff of which it is made.

The exhibition The Islands of Benoît Mandelbrot: Fractals, Chaos, and the Materiality of Thinking explores the role of images in scientific reasoning. By examining a sketch by Edward N. Lorenz from the early 1960s, this text analyzes the specific role played by drawing in Lorenz’s attempts to understand chaotic forms.

A line rises from the central axis of a piece of graph paper, unfolding arcade-like towards the right and left while also gesturing outwards towards three-dimensional space. This spatial effect is primarily generated by adjacent loops and hooks, which the eye perceives as voluminous bulges; their curves open a new perspective into the background, as if the skeleton of a tent unfolds.

On the left, the figure is framed by a large elliptical segment, which the eye automatically completes into a circle. This shape reaches beyond the boundaries of the printed pattern and establishes a connection between the different pictorial dimensions, relating figure, paper, and background to one another. The resulting overall circular structure echoes throughout the entire drawing: Suggestive of parabolic shapes, it covers the sheet as if with lamellae, pressing it into the surface.

However, the flatness of the shape is challenged by the drawn lines that surround it: One structure, located in the middle, vehemently pushes into the optical foreground, producing—along with heavy adjacent arches—the impression that one is looking at a Gothic groined vault. In order to produce these lines, Lorenz had to make bold decisions and forcefully retrace the lines’ paths; yet these lines also seem to be in an intimate dialogue with completely different lines on the sheet—with lines that have been erased, gently drawn, dotted, or fragmented—that together create a subtle network documenting a tentative search for a not yet clearly defined shape on the threshold between two- and three-dimensionality. The eye cannot decide if it perceives a skeleton or a scaffold, a theater tent lying on the floor, a shell, or perhaps a stretched wing.

Amid this ambivalent construction, a roundish structure stands out, situated in the upper left center. Here, two- and three-dimensionality are combined in a single form, which sometimes appears like a bulging nub, sometimes like a hole. The eye oscillates between surface and space. This tendency is carried to extremes by the draftsman: To the right and left of the symmetry axis, the structure seems to form two opposing masses that look like organic bodies vaguely reminiscent of bird heads (as on the eagle statues jutting out of the side of the Chrysler building in New York City, for example). This back and forth between seeing lines and seeing volumes causes the perception of the spectator to vibrate. Owing to drawing as a medium of thinking and discovery, the shape initiates a subtle interplay between figure and ground, surface and volume, as well as between architecture, organics, and mathematics.

The drawing was produced in the early 1960s by the American mathematician and meteorologist Edward N. Lorenz, who worked at MIT until his death in 2008. It was found in one of the twenty loosely arranged boxes of Lorenz’s unprocessed archive material housed at the Library of Congress. Together with Benoît Mandelbrot, Lorenz was one of the first and most famous investigators into “deterministic chaos.” Today, both are popularly associated with glossy computer pictures: Mandelbrot with fractals, Lorenz with one of the most popular icons of chaos theory, the so-called “Lorenz attractor”—a shape commonly represented as consisting of two overlapping wings composed of connected, arbitrarily colored spirals.

Beginning in the early 1960s, Lorenz started using computers to experimentally confirm the existence of chaotic behavior in dynamical systems—one of the first scientists to do so. In 1963, he published his discoveries in the Journal of the Atmospheric Sciences, a seminal article that is represented in “The Islands of Benoît Mandelbrot.” A comparison between the published images and the newly discovered sketch reveals remarkable differences. In the published version, the subtlety of the preliminary study is completely lost: The composition is symmetrical; there is no optical combination of nub and hole; the spatial organization is lacking in all ambivalence; and the viewer’s perception does not oscillate between different dimensions. Only the previously unknown working drafts have the power to make the characteristic geometric property of Lorenz’s “chaotic attractors” come alive. It becomes immediately clear that the shapes depicted in these sketches have as their essential characteristic that they are both surface and space at the same time, constituting an infinitely thin, two-dimensional plane that is nevertheless spatially vaulted and infinitely folded. Such a shape breaks with the conventions of Euclidean geometry; it is “in-between” dimensions—or, in other words, it has a “fractal dimension.” Although it is difficult to imagine these geometric properties, they are nevertheless responsible for the practical impossibility of long-term prediction-making concerning the behavior of dynamic systems, and in this early rendering of his discovery, Lorenz creates an image that allows the viewer insight into how Lorenz’s reliance on the practice of making sketches led him to understand this “fractal dimension” and how it might take on form.

Lorenz’s newly discovered preliminary studies enable us to reconstruct the graphic means he employed to explore and understand his discoveries in the realm of chaotic dynamics. These sheets are particularly revelatory in regard to the ability of drawing to inspire the process of thinking in specific areas. In this case, the discovered sketches helped to confirm a basic thesis of the BGC exhibition as a whole: that shapes of chaos are shapes of visual thinking on and with the paper. However, these knowledge-generating epistemic shapes have been hidden from the public for decades because of the overwhelming flood of popularized glossy imagery in the field—imagery that has been aseptically cleaned, all trace of the thinking human hand removed.The Islands of Benoît Mandelbrot attempts to focus on that hand—and on the critical role it plays in the process of developing perception and moving toward formulations of scientific laws, theories, and discoveries.

Nina Samuel was a Visiting Assistant Professor at the BGC.