The Brooklyn Rail

SEPT 2017

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SEPT 2017 Issue


Collage of parking sign advertisements by the author

In our culture we find “space” everywhere. It is prevalent as a type of background noise in our speech and writing. Space is taught in geometry, physics, architecture, and even in psychology, with terms like “personal space” and “psychological space.” The (often subliminal) purpose of adding space to terms that stand-alone is to make those terms more passive, and to give the term’s user distance from the subject. With the addition of “space” to “psychological,” consider that “psychology” suffices as a term on its own with no inherent need for the addition of space. Combining the terms adds the toughening effect of physics to the softer science of psychology. At the same time, adding space to the monolithic sounding "psychological" makes it warmer and fuzzier. Often the term space is used as an easygoing generality. For example, “narrow gap,” “narrow corridor,” or “narrow room” are all more specific than “narrow space.” With respect to storage, the term “storage space” does little more to describe the location than simply add a syllable. Other than “outer space” or “rental space,” the term is employed more often than necessary—more for effect (or affect) than for precision.

I have long held that artists can whip up a complaint for any occasion—it is perhaps the favorite sport of painters, and we gain mysterious comfort from it. Over the last decade I have become increasingly conscious of the vacuity of the term space. There are miles of art journalism where space is used to lend gravitas to minor, even silly, pictorial effects. I’ve taught in an assortment of art schools and I often hear some variation of “I’m trying to bring more space into the painting,” or “you need to give your painting more convincing space.” At this point I become the old crank and say (with complete honesty), “I don’t know what you mean by space—I don’t believe in space—please use another term.” As the student or group humors me—eliminating the term tightens the discussion—a fog of passivity lifts, and I see these painters start to think about what they commit to when they paint a picture.

Discussions about “painters’ use of space” may serve as a way of speaking about the general “feel” of a picture, its atmosphere, use of perspective or presentation of overlapping planes. These are all more precise descriptions than resorting to the grand and nebulous term space. But more insidiously, injecting space into the discussion serves to smuggle in what has become a conservative dogma of picture-making into painting. There is nothing necessarily wrong with a conservative style of painting whether it is figurative—making paintings that imitate looking through window glass—or conservative abstraction—a type of painting that imitates those effects. But this is not true for all painting.

Today pictures proliferate on surfaces intimate and large, immaterial and physical. We can virtually enter pictures animated with time and illusions of three dimensions. Increasingly, pictures are the coin of human communication. It is a creative moment in which information is substituted for experience, and image for material. Painting has a long history of picture-making that can provide a unique platform to engage present visual culture. For painting to take up this challenge it is fundamental that the history of its terms be understood and honored. Is it worth asking how essential the term space is to the history of visual art?

I confess I’m unable to form a solid definition of space, and as such this essay follows the term as a shadow, a void, or a hollow. I don’t attempt to uncover its relevance, but to track when, how and why it became the overused term in the arts that it is today. I’ve italicized space to signify that the term as I use it is empty of meaning. Instead of its meaning, this essay explores the environment of the illusive creature space—not its DNA.

Because this essay focuses on the visual arts, I begin with architecture. If architecture is the art of building volumes and voids, indeed it might hold the most solid conception of space. One might think architecture is the most ancient habitat of space. Surprisingly, using the term space to describe and conceive architecture is very recent. Because of space’s homelessness within the history of architecture and its virtual absence in centuries of discussions of painting, this essay then investigates its contested emergence at the intersection of science and philosophy in the 18th century, how the term becomes a force in 19th century mathematics and science, its induction into Modernism at the start of the 20th century, and finally to its present state.


If it seems space is an essential term for architecture—more so than for the other visual arts—and that it is an essential concept that defines architectural experience, there is a contemporary logic to that. Space has become the term used to quantify area (or emptiness itself) in order to be valued. Increasingly—as seen in educational metrics and the obsessive monetary evaluation of art—we tend to mistrust experience that isn’t quantifiable. Space as the articulation of floor area or cubic volume might seem foundational for architecture, and enshrined in its history. Area and volume are at least measurable, distinct from the indefinite concept of pictorial space as used in painting and sculpture.

Roman Fresco: Perseus and Andromeda in landscape, from the imperial villa at Boscotrecase, last decade of the 1st century B.C.

In the first century BC Roman treatise, Vitruvius’s Ten Books of Architecture, there is great discussion about architectural orders, mythical and historical precedent, proportion, symmetry, geometry, building location, building materials, and even the damaging effects of moon rays, but there is scant mention of anything even tangential to how the term space is used today. Vitruvius speaks of the technique of collating plan and elevation in only a single passage, but this is a method to create perspectival pictures of buildings, not anything that could be construed as a spatial concept. Vitruvius’s passing mention of perspectival picture-making is of far less concern than the contemporary Roman taste for fantastical paintings, to which he devotes many words and decries as decadent.

Following Vitruvius by a millenium and a half, Leon Battista Alberti’s On the Art of Building in Ten Books (1452), and still later Andrea Palladio’s Four Books on Architecture (1570), include no mention of space as an architectural priority or even an element. Palladio’s book features the Pantheon in Rome with a detailed description of its architectural structure of pilasters, architraves, capitals, etc., and includes several of his own drawings of interior and exterior elevations and plans, without mentioning that the Pantheon’s dome forms half a sphere that if completed would touch down in the center of the square floor plan. While noting that the spherical volumetric basis for the building would not necessarily indicate the conception of space as an architectural element, its complete absence would indicate no such concept was available to or was of any interest to Palladio.

Palladio from The Four Books of Architecture, 1570

Alberti systematized and promoted his fellow Florentine's idea, the architect Fillipo Brunelleschi’s storied invention of linear perspective. While we often equate perspective and space, Alberti does not relate the term to architecture. The closest he comes to discussing interior volume in his book on architecture is a short section on the appropriate proportions of the height and width of interiors to the room’s area. Area is an important term for Alberti, but it isn’t an end in itself. In his writing, area stands for a building’s plan or sometimes site—its references are to location and the purpose of its measurement is to insure good proportion. That is in keeping with the substance of the book—there is much on the practicalities of designing within the formal rhetoric of Classical types, construction techniques and anecdotes of precedents. The fundamentals that interest Alberti are tradition, proportion, and functionality—space does not figure.

Leon Battista Alberti, Della Pittura, illustration published 1804. Wikimedia Commons

As a renowned Latinist, mathematician, painter, and architect, Alberti was at the intellectual center of the Renaissance. A prolific theorist, we can compare how he thought of space in painting to its virtual absence in his conception of architecture. In De Pictura (circa 1440) Alberti writes:

At first, not knowing what aspects we behold, we see something that occupies a locus. This locus, however, is circumscribed to the space of a painter, and the designation of this regard is circumscribed to an appropriate and concrete name.

Alberti was famously precise in Latin, and while spatium does appear infrequently, it is mostly used to designate the distance between two points. While we translate (as above) spatium as space, in this case spatium is merely the flat surface on which perspectival figures are to be inscribed or painted—it is no more than a technical term. This is consistent with Alberti’s views on architecture; at the root of both arts is the intertwining of writing and drawing—the practice of inscription, description, circumscription. Although the specifics differ between painting and architecture, mathematical description is essential to achieve proper relationships within and between figures (geometric or fleshly) whether in fresco or marble. For Alberti, appropriate proportion is the common objective in the arts—space is barely considered and is certainly not a goal for painting or architecture.

Historically, virtual silence about architectural space continues through the 17th into the 19th centuries. For the influential English critic John Ruskin, the categories of his The Seven Lamps of Architecture (1849) are Sacrifice, Truth, Power, Beauty, Life, Memory, and Obedience. In the United States, Louis Sullivan and H.H. Richardson use the word space occasionally and as a gap or an open location. In the 19th century in architectural writing, space is virtually absent as an architectural concept despite the intense interest in historical architecture and the construction boom of late 19th century America and Europe.


If it seems incredible that architects did not think in terms of space until the 20th century, there are reasons why the uptake of space is so late. Even with all the perspectival studies during the Renaissance there is no concept of architectural space in the writings of Alberti because at that time there is no cogent concept of space itself. Here is the foremost authority on Physics from Antiquity through the Renaissance:

Every sensible body is in place, and the kinds or differences of place are up-down, before-behind, right-left; and these distinctions hold not only in relation to us and by arbitrary agreement, but also in the whole itself. But in the infinite body they cannot exist. In general, if it is impossible that there should be an infinite place, and if every body is in place, there cannot be an infinite body.

Gossuin de Metz, L'image Du Monde, 13th century

In this excerpt and elsewhere in his writings, Aristotle grapples with infinity, nothingness, void, body, and place in ways that never develop into a stable whole. With regard to this particular subject, clearly Aristotle lacked confidence that these terms could be consistently related to one another.

Famously, by the 17th century—over a millennium and a half later—Aristotle’s constellation of terms began to be fundamentally reconsidered, although not without resistance from authorities. Giordano Bruno and Galileo Galilei both discovered this when they questioned these concepts, and by extension the Church’s orthodoxy, intertwined as it was with Aristotle’s views.

Renes Descartes, Principia philosophiae, Illustration of plenum vortices, 1644

Descartes writes in 1644, two years after Galileo’s death:

A body’s being extended in length, breadth and depth in itself warrants the conclusion that it is a substance, since it is a complete contradiction that a particular extension should belong to nothing; and the same conclusion must be drawn with respect to a space that is supposed to be a vacuum, substance in it as well.

Despite the sizable span of time, this passage reveals Descartes’s ideas to be as related to an Aristotelian worldview as what is often referred to as Cartesian space today. Descartes’s Cartesian coordinate system is mathematical and, like Euclid’s geometry, it is not concerned with ideas of space in the ways it is thought of today.

At the same time as Descartes’s reconsideration of Aristotle, physics was becoming more physical. Ten years after Descartes published the text above, Otto von Guericke (1602-1686) demonstrated his recent invention of the vacuum pump. Before the Emperor Ferdinand III and a large crowd, Guericke assembled two twenty-inch copper hemispheres together and evacuated as much air from the center as his pump could manage. He then hitched the assembled hemispheres to two teams of fifteen horses, which could not pull them apart, demonstrating the power of an atmospheric vacuum.

Gaspar Schotts, Otto von Guericke's 'Magdeburg hemispheres' experiment, Engraving, 1657

Guericke’s talent for dramatic flourish wasn’t limited to public performance. Here is a section of his Ode to Nothing:

Nothing contains all things. It is more precious than gold, without beginning and end, more joyous than the perception of bountiful light, more noble than the blood of kings, comparable to the heavens, higher than the stars, more powerful than a stroke of lightning, perfect and blessed in every way. Nothing always inspires. Where Nothing is, there ceases the jurisdiction of all kings. Nothing is without any mischief. According to Job the earth is suspended over Nothing. Nothing is outside the world. Nothing is everywhere. They say the vacuum is Nothing; and they say that imaginary space—and space itself—is Nothing.

At the end of the 17th century, thickets of ideas and terms surrounding what we now term space—Nothingness, Void, Vacuum, Container, Quintessence, Plenum, Envelope, Aether, Cosmos, Universe, Infinity, Extension—were still thriving and prone to controversy.

In Isaac Newton’s Philosophiæ Naturalis Principia Mathematica (1687), space is a central concept, specifically “Absolute space, (which) in its own nature remains always similar and immovable.” Gottfried Liebniz, often Newton’s competitor, initiated a two-year debate (1715-16) concerning “absolute space” in an exchange of letters with Newton’s protégé, Samuel Clarke. Acrimonious at times, Leibniz calls Newton’s ideas about space “mere chimeras, and superficial imaginations....grounded upon the supposition that imaginary space is real.”

Leibniz continues: is nothing else but an order of the existence of things, observed as existing together; and therefore the fiction of a material finite universe, moving forward in an infinite empty space, cannot be admitted. It is altogether unreasonable and impracticable...These are imaginations of philosophers who have incomplete notions, who make space an absolute reality. Mere mathematicians, who are only taken up with the conceits of imagination, are apt to forge such notions, but they are destroyed by superior reasons.

The “superior reasons” which Liebniz elaborates on in this correspondence are theological and ontological, making Liebniz’s lines of thought both more ancient and more contemporary than Newton’s. Even as Newton connected his ideas about space to God, his focus wasn’t philosophy or theology, so it’s not surprising that he sloughed off this debate to his understudy. Newton’s “absolute space” isn’t absolute in a theological way; it's a plane of consistency, a functional concept to link mathematics to observable physical phenomena. Newton’s great interest was motion, and as such, his mathematics and physics are dynamic, descriptive, and practical rather than archetypal explanations of reality.

Even as notions of space were becoming more modern, it was Immanuel Kant in his Critique of Pure Reason (1781) who initiated the process of taming some of the controversies about space. Rather than debating the actuality of space, he shifted how space could be defined in regards to how humans understand both abstractions and concrete objects through representation:

Space is a necessary a priori representation, which underlies all outer intuitions. We can’t construct a representation of a state of affairs in which there isn’t any space, though we can very well have the thought of space with no objects in it. So we have to regard space as a precondition for the possibility of appearances, not as a conceptual construct out of them. Space is an a priori representation that necessarily underlies outer appearances.

Despite all evidence that humans—including architects, artists, philosophers and scientists over thousands of years—were able to imagine and construct objects and representations without space as a precondition, in this passage Kant insists that space is required for cognition. Here, Kant inserts a background to the "possibility of appearances” and the background he uncovers is space. In his Critique, Kant advances a case that, whether space as background was ever part of the state of affairs of thinking about objects and what their qualities and relations entail, it must be understood as such retroactively as well as going forward. For Kant, space not only extends infinitely in all directions, but also into the past and the future—it is an archetype that underpins all cognition whether recognized or not.

If Kant’s logic of space has come to seem sensical, it is indicative of the success of his revolutionary framing of object relations. Kant is careful to limit his conception of space to metaphysics. Space is only the prerequisite for perception of phenomena, not a phenomenon itself—thus Kant sidesteps the Newton-Leibniz debate of the existence of space. It’s not difficult to see that Kant’s careful limits could be erased so that Kant’s metaphysical considerations would give way to considering space an actual phenomenon. Still, Kant’s space is a radical invention, as indicated by the fact that this metaphysical conception required over a century for space to seep into the concepts and terminology of architects and artists.


Our relation to the world of vision consists chiefly of our perception of its spatial attributes. Without this, orientation in the outer world is absolutely impossible. We must, therefore, consider our general spatial ideas and the perception of spatial form as the most important facts in our conception of the reality of things.

While Kant himself could almost have written this, it is from a book from 1893 by the Neo-classical sculptor Adolf von Hildebrand, who studied at the venerable Academy of Fine Arts, Nuremberg in the same region where Kant lived and worked.

Hildebrand continues:

Pictorial representation, however, has for its purpose the awakening of this idea of space, and that exclusively by the factors, which the artist presents.

Here, Hildebrand transforms Kant’s theory of space as the formal necessity underpinning representations and human perception into the physical realm of representations in art. For Hildebrand, Space is the dialectic figure to Form in painting and sculpture. The German word Hildebrand uses for space is “raum”. While the Latin “spatium” conveys “interval” or “extension,” the German word has a different etymology. “Raum” comes from “räumen” which means “to clear” or “evacuate.” Hildebrand proposes his own idea “Raumganzes” translated as “total space” with an analogy of the volume of water displaced by submerged vessels, but even more vividly, one can imagine that the stone he subtracted in the process of sculpting would have been tangible to him.

Hildebrand’s The Problem of Form in Painting and Sculpture is an articulate book, and is the earliest thorough discussion of space in art I could find. It had a number of print editions and was translated into French in 1903 (and even English in 1907). However, it seems Hildebrand’s theorizing had little impact on modern art early in the 20th century. It’s likely that in Paris, then the crucible for contemporary art, Hildebrand’s conservatism would have precluded interest in his ideas.


Adolf von Hildebrand, Philoctetes, Marble, 1886 The figure is missing his left foot. According to legend, the goddess, Hera, sent a snake that bit Philoctetes’s foot.

As the late 19th century saw Kant’s theory of space as the metaphysical basis to understanding phenomena segue to being understood as a phenomenon itself, what exactly constituted space within the scientific community was being reimagined, explored and tested.

Even as Leibniz disputed Newton’s concept of “absolute space,” criticism of that theory came from a different quarter. In 1678, Christiaan Huygens, the Dutch polymath who corresponded with both Newton and Liebniz (as well as Descartes and Galileo) developed a theory that light was composed of waves as opposed to Newton’s theory that light consisted of particles moving in a straight line through “absolute space.” It can be observed that waves require a fluid medium, such as water (or air for sound). If light was composed of waves, space wasn’t absolute but a fluid medium. At the turn of the 18th to 19th century that medium was called luminiferous aether (distinct from Newton’s early theory of a gravitational aether that he later renounced). Although the idea of aether is ancient—the Greeks, Medieval Scholastics and alchemists proposed several—aethers were “invented for the planets to swim in, to constitute electric atmospheres and magnetic effluvia, to convey sensations from one part of our bodies to another…” By the early 19th century, scientists generally accepted there was a single aether that pervaded the universe. James Clerk Maxwell, the Scottish scientist and mathematician (who is quoted above) discovered that light, electricity and magnetism have a related wave structure. He wrote with some confidence in 1878:

The evidence for the existence of the luminiferous aether has accumulated as additional phenomena of light and other radiations have been discovered; and the properties of this medium, as deduced from the phenomena of light, have been found to be precisely those required to explain electromagnetic phenomena.

Although Maxwell conceded the “difficulties we may have in forming a consistent idea of the constitution of the aether,” he was able to propose it had a specific (and very low) density of ρ = 9.36 x10-19 (the density of the air we breathe is roughly ρ =1.225 ×10−3). Within ten years, tests at Case Western Reserve in Cleveland—the Michelson–Morley experiments—showed that if aether existed it had little effect on the movement of objects in the universe. This caused the aether model to be rethought though it remained an entrenched idea into the 20th century.

James Clerk Maxwell (engineer) and Thomas Sutton (photographer), First stable color photograph (of a tartan bow), 1861


In the 19th century, as scientists were grappling with electromagnetism, light, and luminiferous aether, there were a number of mathematicians who challenged the abstract foundations of Euclidean geometry. While it’s unlikely Euclid considered the background of his geometric figures space, by the 19th century space was mathematically defined as—and by—orders of dimension. The single dimension of a line, the two dimensions of a plane, three dimensional volume were the conventional orders of space. The overlap between mathematical space and observable phenomena (Newton’s physics) made mathematical space seem natural. Over the course of the 19th century this “naturalism” became increasingly dubious.

But despite the Nothing Saying word-wisdom of the metaphysicians, we know too little, even nothing at all about the true nature of space, and we may confuse something which seems unnatural to us with the Absolutely Impossible.

When the German mathematician Carl Friedrich Gauss penned this to a friend around 1824, he had already been pondering the exquisite difficulties of upending Euclid’s parallel postulate that states, "Within a two dimensional plane there is one and only one line through a given point not on the given line that will not intersect it."

While it might seem absolutely impossible, one can imagine an infinite number of lines through that given point that will not intersect the given line. The implication is that the geometric ground is curved, and thus, the lines appear curved.

With another condition, all lines through that point intersect the given line.

With this line of thought, there are three distinct geometries—Euclidean, hyperbolic and elliptic. Although these new geometries might seem unnatural, non-Euclidiean geometry was deemed possible and was defined almost simultaneously by the Russian Nikolai Lobachevsky and the Hungarian, Janos Bolyai. Unbeknowst to Bolyai, Lobachevsky proposed the hyperbolic variety first while a few years later Bolyai proposed both hyperbolic and elliptic.

Janos Bolyai the son of a mathematician friend and correspondent of Gauss, who finally overcame the difficulties of what his father warned was the “bottomless night” of the new and “unnatural” geometry. He delightedly wrote his father, “I have found things so magnificent that I was astounded….I have created a new and different world out of nothing.”

When considering two-dimensional non-Euclidean geometry in three dimensions, we picture the lines and shapes as being curved; however, within the two dimensional non-Euclidean plane the lines are straight. A basic example of a plane in elliptic geometry (as viewed from the perspective of Euclidean three dimensions) is a sphere. As distinct from a line on a Euclidean plane which extends infinitely in opposite directions, a straight line on a sphere returns to its starting point (forming a circle when looking at it from above); this is to say this type of Non-Euclidean plane is a “compact space,” i.e. does not extend infinitely. Picture the globe as this type of Non-Euclidean plane and the equator as the given line. In Euclidean geometry, every line at a right angle to the given line is parallel to other lines at that angle and do not intersect. On a globe every line perpendicular to the equator—longitudes—intersects each other, (at the north and south poles). In a sense we all inhabit a Non-Euclidean plane.

János Bolyai, Appendix Scientiam Spatii absolute veram exhibens, Tabula 2, 1832

The three mathematically distinct and internally consistent geometries Bolyai identified—Euclidean, hyperbolic and elliptic—simultaneously uncovered and invented conditions that were then available for further development. A dynamic, even inseparable relationship between discovery and construction is one of the defining characteristics of the 19th century concept of mathematical space. Another is plurality. If a curved or flat plane had a set of conditions and could be called space, that plane could also be described and studied as a geometric figure, and there are many forms and types of geometric figures. This mathematical study of the spaces on the surface of geometric figures came to be known as topology.

In Berlin in the 1850s, Gauss’s student Bernhard Riemann presented the outline for the geometry of higher dimensions with what his professor termed “a gloriously fertile originality.” The mathematical figures which he dubbed manifolds are defined as both as an object and a type of space. Reimann’s mathematics of variable dimensions constitute a seismic shift in the understanding of space. If non-Euclidean geometry was something of an “unnatural” space, Reimann’s manifold spaces, which can contain hundreds, even millions of dimensions, make any natural imagination of space impossible. It should be noted that while Reimann’s mathematics are the study of geometry in its most conventional or “pure” state, it has proved incredibly useful in the study of complex phenomena and computer science, and was key to Albert Einstein’s theory of general relativity.

Ernst Mach, Bullet in supersonic flight, Schlieren photograph, 1887

Even while artists in the 19th century were, as a whole, uninterested in space, toward the turn of the century scientists were becoming attentive to the relationship between perceptions and space—“psychophysics,” as one of the era’s physicists titled it. The Viennese physicist Ernst Mach, known for his work on sound waves and optics, was one of the great enthusiasts of space at the time. He wrote in 1901, “The perfect biological adaption of connected elementary organs among one another is thus very distinctly expressed in the perception of space.” For Mach, “geometric space” of the standard three-dimensions was both universally true and the goal the senses strove to acquire. Mach argues it is necessary to apprehend space to understand the world. Unlike Kant, who finds space essential to conceptualization, Mach finds it fundamental to experience, and sensate experience fundamental to creating a working concept of space. If the reasoning becomes a bit circular, here is Mach relating to Kant:

Kant says: “Thoughts without contents are empty, intuitions without concepts are blind.” Possibly we might more appropriately say: “Concepts without intuitions are blind, intuitions without concepts are lame.”

Henri Poincare—one of the more remarkable persons in turn of the century Paris—began his career as a provincial mining engineer, but soon made significant contributions to astrophysics, electromagnetism, qualities of algebraic equations, topology, special relativity and space-time. Poincare was also a successful public figure and was elected to several official posts. If he didn't have quite the celebrity of Gustav Eifel, he was a known man of science and was described in a 1910 popular biography as absent-minded, a bit clumsy, but possessing a prodigious intelligence and capacity to visualize complex abstractions.

In 1902, Henri Poincare published Science and Hypothesis. Although basic enough for the general reader, Einstein was said to have pored over the book with breathless excitement. His book remains fresh today, due in no small part to Poincare’s belief that “mathematical reasoning has of itself a sort of creative virtue.” In the chapter Space and Geometry, Poincare writes, “Geometry is not true, it is advantageous.” Poincare begins dividing his subject into the general headings of “Geometric Space and Perceptual Space,” casting doubt on the conventional notion that “our representations, are identical with that of the geometers….” Poincare outlines “the most essential properties of geometric space”:

1º It is continuous;

2º It is infinite;

3º It has three dimensions;

4º It is homogeneous, that is to say, all its points are identical one with another;

5º It is isotropic, that is to say, all the straights which pass through the same point are identical one with another.

He immediately contrasts this to “Perceptual space, [which] under its triple form, visual, tactile and motor, is essentially different from geometric space.” For Poincare, visual space is the two dimensional transcript of light on the retina; tactile space is the physical sensation of form, and relatedly, motor space is the sensation of movement—the sense of our bodies in movement and the muscular movements of binocular vision through which we interpret three dimensions. Because of the limitations of the human sensorium, perceptual space “is neither homogeneous, nor isotropic; one cannot even say that it has three dimensions.” Enclosed in a limited frame of sensation, perceptual space is finite and its continuity is “only an illusion.”

With this Poincare succinctly sums up Mach’s themes of physiological space and geometric space. And in the vein of Mach, Poincare goes on that while geometric space is a control through which various human perceptions can be corralled and rationalized—it also is through those perceptions of our world that geometric space is constructed.

It is at this that point Poincare’s scheme diverges form Mach’s. While Mach returns to emphasize the authority of the senses, Poincare proposes a world which “might have a geometry very different from ours.” He proposes a thought experiment—a world within a sphere where the inhabitants are largest at the center, but when they move to the outer limit their size decreases by a fixed proportion so that as the beings advance to the edge of the sphere those beings approach becoming infinitely small and so the outer limit can never be reached. To the inhabitants, their interior sphere is in all practical ways infinite. And by proposing that light rays would curve proportionally to the diminution of beings toward the outer limit, the inhabitants would live in a world where perception—visual, tactile and motor—would be consistent and feel natural. However, the intuitions necessary to define its geometry would be non-Euclidean. And from there the chapter goes on to propose techniques for picturing a four-dimensional world.


In literature of the 19th century there was a beginning of interest in science and space. Mary Shelley, Edgar Allan Poe, Jules Verne and H. G. Wells channelled interest in science in literature. In the 1880s, Charles Howard Hinton wrote an essay, “What is the Fourth Dimension?” naming the fourth dimensional cube a “tesseract,” and his fellow Englishman Edwin A. Abbott published Flatland: A Romance of Many Dimensions (1984).

Charles Howard Hinton, The Fourth Dimension, Illustration, 1904

In 1903, Marie and Pierre Curie received the Noble Prize for research in radiation. 1905 was Albert Einstein’s miraculous year when he published several papers, one which experimentally proved the existence of atoms and another which set the stage for the concept of space-time that, after a century finally excised the concept of luminiferous aether from space. Sigmund Freud’s studies of the mind were also becoming better known.

While many of these ideas took over a decade to catch fire in art, scientific ideas were employed as subjects in the studio but also to lend artworks authority in the public sphere. In the first dozen years of the 20th century in Paris, space finally made its excited and very public debut in painting and sculpture.

In his The Cubist Painters (mostly collected from 1912 essays) the wily Guillaume Apollinaire hitches the radical new developments in mathematics and science to Cubism in order to defend his circle of friends from public attacks,

Nowadays, scientists have gone beyond the three dimensions of Euclidean geometry. Painters have been led, quite naturally and one might say intuitively, to take an interest in the new possibilities for measuring space, which in the modern artist studio were simply and collectively referred to as the fourth dimension.

After the Salon des Indépendants presented the Cubist painters as a group in 1911, their paintings were fodder for public controversy, and in 1912 there was even a move to deny public funds to any venue that showed their works. Apollinaire effectively counters the criticism of Cubism being outrageous by emphasizing geometry and measurement, which were academic values. Furthermore, technology and science were popular. Another more subtle argument was implied—just as the abstruse science of the age led to admired technical achievements—that his artwork would eventually be understood and valued. And finally, even as Apollonaire distances himself from this interpretation, he notes “the fourth dimension—this utopian expression… has come to stand for the aspirations and premonitions of many young artists…”

In retrospect it can be seen how effective Apollinaire’s spinning of science and Cubism was. It provided legitimacy for his friends’ work and a legacy of ideas for artists and architects to leverage throughout the 20th century. Finally, it is through the medium of scientism that space entered the dialogue of art and took root—certainly the conditions were ripe to bring space into art. It was an extraordinary moment and it might be worthwhile looking into what ideas about space were available to Apollinaire and by what means space could have jumped platforms from the laboratory and university to the salons and studios.


In 1912, the same year of Apollinaire’s spirited defense of Cubism, his friends—the painters Albert Gleizes and Jean Metzinger—published a book weighing in on the new style. Among the observations and opinions about the history of painting, reality, form, dynamism and personality, a section on space appeared. Even as the authors cite Reimann, it is Poincare’s ideas (even his terminology—“convergence and accommodation”) that were adopted. Although Gleizes and Metzinger were regarded as the intellectuals of Cubism, they were late to the game, and while dutiful about the radical geometries of space, it doesn’t seem they had the requisite zeal to import space into art and make it stick.

Among the motley crew of Pablo Picasso’s and Georges Braque’s circle in the early years of the 20th century was a math enthusiast and insurance actuary, Maurice Princet. Some years later, Metzinger recollected, “[Princet] loved to get the artists interested in the new views on space that had been opened up….” And Marcel Duchamp who was also part of the late wave of Cubists recalled, "We weren't mathematicians at all, but we really did believe in Princet."

In 1918, the anti-modernist critic Louis Vauxcelles, who mockingly furnished both Fauvism and Cubism with their names, wrote:

Princet has studied at length non-Euclidean geometry and the theorems of Riemann, of which Gleizes and Metzinger speak rather carelessly. Now then, M. Princet one day met M. Max Jacob and confided him one or two of his discoveries relating to the fourth dimension. M. Jacob informed the ingenious M. Picasso of it, and M. Picasso saw there a possibility of new ornamental schemes. M. Picasso explained his intentions to M. Apollinaire, who hastened to write them up in formularies and codify them. The thing spread and propagated. Cubism, the child of M. Princet, was born.
Esprit Jouffret, Traité élémentaire de géométrie à quatre dimensions et introduction à la géométrie à n dimensions, 1903

There is a book that Princet is said to have presented to Picasso—Traité élémentaire de géométrie à quatre dimensions by Esprit Jouffret—that expands on Poincare’s imaginative geometry. While the late wave Cubists noisily organized shows and manifestos, Picasso was silent on the subject of space and the four dimensions. While visual connections between Picasso’s early 20th century painting and the illustrations in Jouffret’s book may be noted, it remains speculation if Picasso was thinking about space in context with his work. Whatever the particulars of how it was brought into the arts, by 1912 space was all the rage in Paris—note Umberto Boccioni’s sculpture that same year, Development of a Bottle in Space, as well as his sculpture of a figure running, titled Unique Forms of Continuity in Space, from the following year.

In the vein of Poincare’s thought experiments of imagining worlds based on different geometries see these links for a virtual experience of non-Euclidean perception:


            “Space and time are reborn to us today.” -Naum Gabo, 1920.

While it certainly wasn’t a contest, it is significant that what took hold in early 20th century art was Poincare’s mathematic flavor of space rather than Mach’s more broadly scientific (physiological and evolutionary) space or the (modified) Kantian metaphysical space of Hildebrand. In space, Kant finds an archetypal authority to support the complexities of human thought. For Mach also, space is an authority that underpins both sensate experience and higher conceptualization. For Poincare, space isn’t an authority but an opening. Even with its great complexity, Poincare’s space is light and clear, distinct from Mach’s muddy, strong prose. If the fourth dimension could represent a utopian expression—the aspirations and premonitions of young artists—Poincare’s sense of flight and possibility are within that expression. Many of the finest examples of Modernism contain the flavor of Poincare’s space, from the better works of Cubism, through Constructivism, the Bauhaus and even to Buckminster Fuller.


In the 1920s, architects begin to follow painters with the novel idea of importing space to the fine arts. This was predictable, as architects are part of the professional class (as opposed to artists who are famously unprofessional), so are more beholden to familiar conventions, and with greater responsibilities they were less available to pioneer the novel concept of space in architecture before the painters did. For both architects and artists, however, to successfully work out an unusual theory in a new medium is no simple matter. A relevant example is La Maison Cubiste, a prototype for Cubist architecture, designed by the Duchamp brothers, Gleizes, Metzinger and several of the second wave Cubists. It doesn’t appear to be much more than a slightly frisky example of bourgeois domestic architecture.

Raymond Duchamp-Villon, Model of Maison Cubiste Façade, Illustration for Les Peintres Cubistes, 1913

The Maison Cubiste may be revealingly compared to Adolf Loos’s 1930 Villa Muller, where Loos shaped three dimensional volume by taking up what Loos called in 1929 the “great revolution in architecture: the three-dimensional rendering of a ground plan”—his own neologism, “Raumplan.”

Adolf Loos, Villa Müller, Prague, completed in 1930

In 1917, the Swiss architect Charles-Édouard Jeanneret paused from his building career to paint, and teaming with Amédée Ozenfant, they developed a new brand of painting claimed as Purism. As the title of their essay Apres Le Cubisme suggests, the authors take stock of Cubism. While mathematical order is approvingly discussed, a leading objection was “aimed at the fourth dimension….which strikes only [as] the gratuitous hypotheses of the theoreticians of cubism.” By 1921, Jeanneret had changed his name to Le Corbusier and the duo published another manifesto titled Purism. In a section referring to painting composition, the four dimensions of Cubism are downsized—“space means three dimensions.”

At that time in European art there was a movement to “return to order,” and even a return to a variety of Hildebrand’s Neo-classicism. Although Purism was oriented to the future with clarity and simplicity, order was part of its stated goals. While Mach and Poincare clearly showed the distinctions between “perceptual space” and three-dimensional “geometric space,” tying the two together simplifies things. A short discussion on space appears under the heading Composition. When the painters insist, “A painting should not be a fragment….Therefore we think of the painting not as a surface but as a space,” it is a strategy to make a painting’s composition architectonically whole. While dispensing with the “absurd” fourth dimension, they also dispense with the imaginative excitement about space—it is normalized, as if it was always embedded in painting.

The year the Purist pair wrote the Apres Les Cubisme painting manifesto, Le Corbusier solely authored an architectural manifesto, Toward a New Architecture: Guiding Principles. In this text, space is joined to architecture.

            The elements of architecture are light and shade, walls and space.

That this is an early mention of space in architecture explains its slight mention and minor location within the manifesto. In the essay there’s the major heading, Three Reminders to Architects, but those are Mass, Surface and Plan. The sentence identifying space with architecture appears in a secondary subsection The illusion of plans, roughly in the middle of the essay. While Le Corbusier’s sentence is firm (like all of his writing in the 20s) space doesn’t get exactly get top billing as it's stuck at the end of a single sentence.

In 1926, toward the end of Ozenfant’s and Le Corbusier’s partnership, Le Corbusier published Five Points Toward a New Architecture. In the text, space is not mentioned. It is significant to note that Le Corbusier the painter was more concerned with space than Le Corbusier the architect.

However, by 1948 things had changed. Le Corbusier wrote yet another manifesto titled New World of Space. The essay opens with:

Taking possession of space is the first gesture of living things….The occupation of space is the first proof of existence.

The second paragraph ends with:

The essential thing that will be said here is that the release of esthetic emotion is a special function of space.

In the span of twenty-two years, Le Corbusier abandoned his rectitude about space—hush turned to gush. Le Corbusier’s revisits one of Purism’s central issues with Cubism:

Without making undue claims, I may say something about the ‘magnification’ of space that some of the artists of my generation attempted around 1910, during the wonderfully creative flights of cubism. They spoke of the fourth dimension with intuition and clairvoyance….The fourth dimension is the moment of limitless escape evoked by an exceptionally just consonance of the plastic means employed.

By the end of his major section on space, Le Corbusier writes ecstatically, even religiously, about “ineffable space.”


Although in the 20s Le Corbusier was largely silent about space, Modernist Germans, Austrians, De Stilj and Constructivist architects were warming to the term. At this time, space begins to take a new role in the academic study of art even as art history becomes a staple of scholarship in universities. Here is the second sentence of Erwin Panofsky’s Perspective as Symbolic Form (1927), an influential essay on the emergence of perspective in the Renaissance:

We shall speak of a fully “perspectival” view of space not when mere isolated objects, such as houses or furniture, are represented in “foreshortening,” but rather only when the entire picture has been transformed....into a “window,” and when we are meant to believe we are looking through this window into a space.

There is a difference between an interest in proportional relationships between figures and objects and how shapes might be inscribed on a window surface as opposed to “looking into a space.” The Renaissance masters—Alberti, Piero, Da Vinci, wrote extensively about perspective, but a “view of space” is not part of their stated considerations. In his essay, Panofsky does not make the argument that   despite these artists’ seeming obliviousness to space, it was subconsciously what they were getting at. Instead he simply applies this modern term anachronistically, but ingeniously—Panofsky uses Kant’s program of uncovering space as background not only to underwrite his entire discussion of perspective, but also to imply that looking into “a space” is, in fact, the goal of Renaissance perspective.

From the 1920s onward, space was increasingly assumed essential to painting. The evolution of painting space is similar, even as it precedes architectural space. It progressed from initial excitement through imaginative academicism to its present condition.

Over the span of the 20th century, painters began to teach and become professional academics. As they did this they took to writing about art, and space figures in their writings. Ad Reinhardt’s 1946 cartoon for PM Magazine “How to Look at Space” both embraces and lampoons the then academic fashion for speaking about painting space. Mark Rothko penned a chapter titled Space in a book of essays, likely written in the 1940s but not published until recently. Rothko primarily uses space as a platform to discuss the relationship between “tactility” and “illusory plasticity” in painting—a dialectic that he inherits from Bernhard Berenson’s 1896 publication, The Florentine Painters of the Renaissance.

Finally, in his essay Rothko posits, “Space, therefore, is the chief plastic manifestation of the artist’s attitude toward reality.” But even that is a fair distance from what Frank Stella claims, in his Norton lectures at Harvard, published as Working Space (1986):

...after all, the aim of art is to create space—space that is not compromised by decoration or illustration, space in which the subjects of painting can live.”

In forty years, space in art advanced from a manifestation to manifest goal.

#5 (112 hand through hole in aluminum ptg). From Hollis Frampton, The Secret World of Frank Stella, 1958-1962. Gelatin silver print. 9 7/16 in. x 7 7/16 in. (23.97 cm x 18.89 cm). Gift of Marion Faller, Addison Art. Courtesy of Addison Gallery of American Art, Phillips Academy, Andover, MA / Art Resource, NY

Stella’s lectures are full of art historical assertions and passionate insights. He shares his anxieties about the future [and past] of abstraction, admonitions for making great paintings, and breathless excitement for the genius of Caravaggio, but nowhere does he venture a definition for space. Space, however, gets a real workout. Overall the lectures have a zeal for space that is more like what Guericke claims for Nothing than Reinhardt’s and Rothko’s measured discussions. These lectures allow Stella the opportunity to work a room full of his admired painters, both living and dead, collapsing time to equal presentness. If early in the 20th century art historians discovered space in painting, it can be said that in Working Space the painter Stella finds space in art history.


T­he central problem with space in the arts is that it is not recognized as a convention, and a recent one at that. In ­­­­architecture, space is an efficient term to quantify area and volume. Commodifying this metric is the basis for the linkage of architecture and real estate, which is of great use. In this case, the meaning of space isn’t lofty, but is effective—its conventional nature is beside the point. Space in architecture can have use as a generality, a three-dimensional volume that awaits the rigors of specificity to become architecture.

Space in painting is a different story. Having no metric, it is pure affect. Space in painting isn’t a manifestation or aim of all painting, nor a timeless idea, but a historical Modernist convention. For example, to speak of “Renaissance space” in painting is a contemporary projection that informs more about the state of Modernism than the work and goals of 15th century painters. The question isn’t what these painters are doing with space but why our era is so insistent on totalizing the thinking of different cultures from different times with our present habits. Often “Modernist space” is discussed as something apart from other forms of space. In fact, since applying space to picture-making is a Modern invention, all space in art is essentially Modernist. An argument could be made that because space is a 20th century convention, its Modernist legacy impacts contemporary painting. But it should be addressed in its complexity as a recent expression requiring an understanding of its historical emergence. Beginning some hundred years ago, a number of painters have taken space as a theme in their work. It would be a worthwhile art historical project to investigate what exactly these spatialist ideas mean and how they are being used.

Ad Reinhardt, How to Look © 2017 Estate of Ad Reinhardt / Artists Rights Society (ARS), New York

During the middle to late 20th century, space in the visual arts was often imbued with spirituality or religion. For Le Corbusier, “ineffable space” is a sacred union between “deserving” followers and the master artist. With Rothko and Stella, it can be noted that space is an essential link from their painting to religious art, if not religion itself. However, at present there is little religious content to the term. There is Pollyannaish talk that we are in a great moment for painting. Certainly there is more effort, affection, and thought directed toward painting than ever. But the result is stacks of handsomely crafted objects and cartoonish canvases running the gamut of jokey to sincere instead of paintings that push for new ways to consider seeing, making and understanding.

Space is only one convention within the lexicon of how painting is conceived today, and its reconsideration isn’t by itself a solution to how painting can develop a more engaging connection to the world. But the intelligence of painting is constricted when a convention is unquestioned and assumed to be natural, real or eternal. Painting and its associated varieties of picture-making are far older and more real than conceptions and prescriptions for space. Fashions change. Painting endures.



James Hyde

JAMES HYDE is a Brooklyn based painter.


The Brooklyn Rail

SEPT 2017

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