VII.14 Mathematics and Art

Florence Fasanelli

1 Introduction

This article focuses on the relationship between the history of mathematics and the history of art in twentieth-century France, England, and the United States. The effect of mathematics on artists and the direct interactions between artists and mathematicians have both been extensively studied. These studies show that knowledge of mathematics has had a significant influence on many artists, as well as on musicians and writers. In particular, the increasingly wide acceptance, during the nineteenth century, of mathematical ideas that had once been revolutionary contributed strongly to what is now called modern art. At the end of the nineteenth century and the beginning of the twentieth, artists expressed on canvas and in sculpture their understanding of the fourth dimension and of NON-EUCLIDEAN GEOMETRY [II.2 §§6–10]. In doing so, they left behind their earlier training and heritage, which had been heavily based on a mathematical perspective derived from EUCLID [VI.2]. Their new ideas reflected the progress that had been made in mathematics, and many of the artists who formed new schools of thought were also engaged in interpreting these new mathematical developments.

The connection between mathematics and art is rich, complex, and informative. This is evident in some of the artistic styles and the philosophies that developed under the influence of new mathematics (and science), and in the creation of mathematics to fulfill artistic needs. Some examples include the paintings (with their often-studied geometries) of Italian mathematician Piero della Francesca (ca. 1412–92), who, having made a transcription of Jacopo of Cremona’s Latin translation of Archimedes’ Codex A, wrote out his own mathematical theories of perspective; Hans Holbein’s (1497–1543) Ambassadors (1533), which illustrates how an artist can use a distorted variation on mathematical perspective to fool the eye (anamorphosis); Artemesia Gentileschi’s (1593–1652) deliberate correction of a smattering of blood in her first version of Judith Beheading Holofernes (1612–13) to a parabolic arc of blood in the second version (1620) to match a sketch that her friend, scientist, court mathematician, and amateur painter Galileo Galilei (1564–1642) had made as a study for his as yet unpublished law of projectile motion; various works of the Dutch portrait painter Johannes Vermeer (1632–75) using the camera obscura; Johann Hummel’s (1769–1852) paintings of the making of the great granite bowl in Berlin, which used Gaspard Monge’s (1746–1818) Géométrie Descriptive (1799); sculpture by Naum Gabo¹ (1890–1977) and his brother Antoine Pevsner (1886–1962) following their youthful academic study of solid geometry; and the mathematically understandable but physically implausible scenes by Maurits Cornelis Escher (1898–1972).

This article begins with a brief history of the development of perspective in art, because it is necessary to understand this in order to understand the rebellion against it that had such a decisive impact on modern art. This is followed by a short summary of the changing course of geometry in the nineteenth century through the development of non-Euclidean geometry and n-dimensional geometry. We then move on to the activities of artists, beginning in France in the early twentieth century and continuing with the works of representative artists in other countries, all the while keeping in mind the mathematics that provoked their artistic responses.

2 Development of Perspective

During the fifteenth century artists were still primarily employed to produce images of sacred subjects, but there was an increased interest in having pictures match aspects of the physical world. Lacking any precursors, artists had to devise their own axioms of linear perspective. At the beginning of the sixteenth century these early ideas of mathematical perspective were spread by books that contained visual representations. Mathematics that was previously known only in writing or orally now took a visual form, which was copied in engravings and spread across Europe.

The first writings on perspective were by Leon Battista Alberti (1404–72) and Piero della Francesca, while the ideas of Filippo Brunelleschi (1377–1446), the Florentine architect and engineer who was in fact the first to consider a mathematical theory of perspective, were captured by his biographer Antonio Manetti (1423–97). Artists and mathematicians continued to develop the rules of perspective while looking for ways to best represent space and distance. Among the mathematicians, Federico Commandino (1509–75), renowned for his Latin editions of the works of Greek mathematicians such as Euclid, ARCHIMEDES [VI.3], and APOLLONIUS [VI.4], was the first to write about perspective for the benefit of mathematicians rather than artists. His student Guidobaldi del Monte (1505–1647) published the influential book Perspectivae libri sex in 1600, in which he showed that any set of parallel lines not parallel to the plane of the picture will converge to a vanishing point.

Great artists, notably Leonardo da Vinci (1452–1519) and Albrecht Dürer (1471–1528), were now portraying mathematics in a visual form. Mathematician Luca Pacioli’s (1445–1517) De Divina Proportione (1509) includes Leonardo’s unsurpassed woodcuts of polyhedra (among them the first published illustration of a rhombicuboctohedron), and Dürer’s Unterweysung der Messung (1525) contains the first illustration of nets for models of polyhedra. Dürer’s own new knowledge of perspective, whose secrets he had learned on a trip to Italy from Germany, inspired him to create his famous illustrations of how to draw a picture in which all the elements are in one-point perspective (see figure 1).

Figure 1 An artist using Dürer’s perspective machine. © Copyright: The Trustees of the British Museum.

In the seventeenth century, Girard Desargues (1591–1661), a French engineer and architect who wrote on practical subjects, continued the study of perspective that had been begun by the Renaissance artists. In doing so he invented a new, “non-Greek” way of doing geometry, which he published in his Brouillon Project d’une Atteinte aux Événemens des Rencontres du Cône avec un Plan (1639). In this essay, he attempted to unify the theory of conic sections through the use of projective techniques. This new PROJECTIVE GEOMETRY [I.3 §6.7] was based on his earlier realization than an artist can construct a perspective image without using a point from outside the picture field. However, of the original fifty printed copies of the Brouillon Project only one survives and his work, including his “perspective theorem,” was made known through the publications of other mathematicians. Abraham Bosse (1602–76), a friend of Desargues who ran a famous atelier where the art of engraving was taught, was responsible for publishing much of Desargues’s work, including that on the theory of perspective. But Bosse’s promotion of Desargues’s innovative ideas created controversy in art circles and seriously damaged his professional reputation. However, in the twentieth century, when engraving was revived as an important art form, a replica of Bosse’s studio was built in Paris.

In the early eighteenth century, the mathematician and amateur painter BROOK TAYLOR [VI.16] published Linear Perspective: Or a New Method of Representing Justly All Manner of Objects as They Appear to the Eye in All Situations (1715), the first book on perspective to give a general treatment of vanishing points. As Taylor wrote on the title page, the book was “a work necessary for painters, architects, etc., to judge of, and regulate designs by.” Taylor invented the phrase “linear perspective” and stressed the importance of what is now described as the main theorem of perspective: given any direction not parallel to the plane of the picture there is a “vanishing point” through which the representations of all lines in that direction must pass.

Since ancient times the axioms of Euclid’s Elements have provided the basis for the understanding of two- and three-dimensional figures, and in the fifteenth century they provided the foundations for the study of perspective. But during the nineteenth century the long-standing debate about whether to accept Euclid’s fifth axiom (the “parallel postulate”) was resolved in a way that was to provoke a radical change in the perception of geometry: it was demonstrated by several mathematicians—notably LOBACHEVSKII [VI.31] in 1829, BOLYAI [VI.34] in 1832, and RIEMANN [VI.49] in 1854—that a consistent “non-Euclidean geometry” was possible in which the fifth axiom no longer held.

The mathematician and expositor HENRI POINCARÉ [VI.61] provided popular accounts of these new ideas in his books La Science et l’Hypothèse (1902) and Dernières Pensées (1913), which were widely read in France and elsewhere. Poincaré’s works provoked the highly influential French (and later American) artist Marcel Duchamp (1887–1968) to attach new meanings to the concepts of space and measurement. Duchamp famously discussed and used Poincaré’s essays “Mathematical magnitude and experiment” and “Why space has three dimensions” to create artistic works of a completely new kind. (Duchamp’s ideas have been explored by the art historian Linda Dalrymple Henderson, who has used Duchamp’s extensive notes to analyze his understanding of four-dimensional and non-Euclidean geometries.)

3 Four-Dimensional Geometry

The modern movement known as cubism was greatly influenced by ideas of the fourth dimension. One of the ways cubists came into contact with these ideas, as well as with non-Euclidean geometry, was through their reading of popular science fiction. In Gestes et Opinions du Docteur Faustroll (1911), French author Alfred Jarry (1873–1907), a close friend of Spanish artist Pablo Picasso (1881–1973), attracted by the novelty of higher-dimensional geometries, wrote about the work of the British mathematician ARTHUR CAVLEY [VI.46]. In 1843, Cayley published “Chapters in the analytic geometry of n dimensions” in the Cambridge Mathematical Journal. This work, along with Hermann Grassmann’s (1809–77) Die Lineale Ausdehnungslehre, published in German a year earlier, was of interest not only to mathematicians but also to the general public, who recognized that in spaces of higher than three dimensions basic concepts had to be redefined and generalized.

In 1880 Washington Irving Stringham (1847–1909), in another influential article, “Regular figures in n-dimensional space,” published in the American Journal of Mathematics, extended EULER’S FORMULA [I.4 §2.2] for polyhedra to new objects called “polyhedroids” in which polyhedra are joined by their faces so as to enclose a hyperspace. This article, which included illustrations of four-dimensional figures created by Stringham, was cited for the next twenty years in the most important mathematical texts on four-dimensional geometry. Stringham’s figures intrigued several artists during the first decade of the twentieth century: Albert Gleizes’s (1881–1953) painting Woman with Phlox (1910) has flowers that are similar to Stringham’s “ikosatetrahedroid”; while in Henri Victor Gabriel Le Fauconnier’s (1881–1946) Abundance (1910–11) Stringham’s “hekatonikosihedroid” appears.

Art forms evolved as artists found new ways of responding visually to the world around them. This was particularly true of cubism, in which the artist depicted objects from several viewpoints at once. In order to make sense of a cubist painting, the viewer was invited to construct a single (elusive) object from an array of different perspective “facets” laid out across the picture’s surface.

The n-dimensional geometries influenced not just the visual arts but also literature, including works by Rudyard Kipling and H. G. Wells, and music, for example Edgard Varese’s “Hyperprism” (1923). Some mathematicians used this new mathematics for humorous purposes: two examples were Charles Dodgson in his Through the Looking Glass of 1872 and Edwin Abbott in Flatland: A Romance of Many Dimensions of 1884. The latter in particular was read by French artists and was referred to in other mathematics books that they read, such as those by Esprit Pascal Jouffret (1837–1907).

4 Formal Protests against Euclid

In the early twentieth century, informed by Poincaré’s exposition of “the fourth dimension” and by knowledge of non-Euclidean geometry, a group of artists, including Gleizes and Jean Metzinger (1883–1956), explicitly attempted to liberate themselves from the geometry of three-dimensional Euclidean space. In an essay titled “Du Cubisme” they stated, “If we wished to tie the painter’s space to a particular geometry, we should have to refer to the non-Euclidean scholars; we should have to study, at some length, certain of Riemann’s theorems.” Here they appear to be referring to RIEMANNIAN GEOMETRY [I.3 §6.10], in which the notion of shape is less rigid than it is in Euclidean geometry. They go on to say, “An object does not have one absolute shape, it has several, it has as many as it has planes in a range of meaning.” It is likely that they are referring here to Poincaré’s “Les géométries non euclidiennes” in La Science et l’Hypothèse. The title of a (lost) 1913 painting of Metzinger’s, Nature morte (4^me dimension), gives a good indication of his interest in representing three and four dimensions on a two-dimensional surface. Both Riemann’s geometry and the fourth dimension lay behind what these artists were trying to accomplish; they referred to both, however, as “non-Euclidean.”

In 1918, enraged by the destruction wrought by World War I, a dozen artists, including Jean (Hans) Arp (1886–1986) and Francis Picabia (1879–1953), signed the Dada Manifesto. In it, they explicitly stated their belief that “all objects, sentiments, obscurities, apparitions and the precise clash of parallel lines are weapons for the fight [against conformity].” By the 1930s, more and more artists were using their knowledge of mathematics to change, in a radical way, the appearance of sculpture and painting.

5 Paris at the Center

During the last decade of the nineteenth century and the years before the outbreak of World War I, artists were profoundly influenced not just by mathematics, but also by the extraordinary developments and discoveries in science and technology. For instance, motion pictures (1880s), radios (1890s), airplanes, cars, X-rays (1895), and the discovery of electrons (1897) all had an impact on the work of artists. The pioneering painter Wassily Kandinsky (1866–1944) wrote that an artist’s block he was experiencing disappeared when he learned of what was new in science; his old world collapsed and he could begin painting again.

While it is not entirely clear how knowledge of scientific and mathematical thinking came to working artists in the early twentieth century, it is nevertheless evident that many artists were familiar with articles about mathematics written for the general public. There was also at least one tutor with whom they explored mathematics in depth. In 1911, in Paris, the mathematician and actuary Maurice Princet (1875–1971) gave informal lectures on four-dimensional geometry, using mathematician Esprit Pascal Jouffret’s Traité Élémentaire de Géométrie à Quatre Dimensions et Introduction à la Géométrie à n Dimensions (1903). Jouffret’s Traité, which makes reference to Flatland, contains ways to present four dimensions on paper, the diagrams by Stringham of polyhedroids in four-dimensional space, and clear presentations of the ideas and theories of Poincaré. A second book, Mélanges de Géométrie à Quatre Dimensions (1906), emphasizes similar points.

Princet’s audience was the Puteaux cubist group (which was sometimes called the “Section d’Or”). The central figures in this group were the three brothers Raymond Duchamp-Villon (1876–1918), Duchamp, and Jacques Villon (born Gaston Emile Duchamp) (1875–1963). Princet’s involvement with the artists continued, even after his divorce from Alice Géry (1884–1975), who shared a bohemian life with the best man at their wedding, Pablo Picasso, and who later married André Derain (1880–1954). Géry had introduced Princet to the artists. An avid reader, she may have been the sitter for Seated Woman with a Book (1910), an early cubist painting by Picasso.

Together, in Paris, Princet and Duchamp privately studied Poincaré and Riemann, who were two important sources for Duchamp’s work, as we have already seen. Duchamp’s own notes, written a decade later as he created his famous painting The Bride Stripped Bare by her Bachelors, Even (The Large Glass) (1915–23), document his increasing interest in and understanding of four-dimensional and non-Euclidean geometries. Referring to Jouffret’s book, which explained how a three-dimensional projection of a four-dimensional figure can be considered as a sort of “shadow,” Duchamp told friends that the bride in his picture was a three-dimensional projection of a four-dimensional object recorded in two-dimensional form. He also refers to the fact, which fascinated him, that electrons were known to exist but could not be directly observed, claiming that his picture contained elements that were not directly represented. These notes, and others containing speculations on mathematics, were published in À l’Infinitif (1966). Working in a field hitherto dominated by fifteenth-century Renaissance perspective and its dependence on a Euclidean framework, Duchamp and other artists learned with excitement that many mathematicians no longer felt it necessary to subject themselves to Euclidean restrictions, and art was dramatically changed.

Rather surprisingly, Riemann and Poincaré were even part of the original inspiration for Duchamp’s famous “readymades,” found objects presented as art. As the artist Rhonda Shearer described in the New York Academy of Science newsletter in 1997, Duchamp was very taken with Poincaré’s description of the creative process in Science and Method. Poincaré reported on his accidental discovery of the so-called Fuchsian functions. After days of “unfruitful” conscious work spent trying to prove that the functions do not exist, he changed his habit and one evening drank black coffee late at night. The next morning, “fruitful” ideas came into his conscious mind. From these he selected “tout fait” (readymade) ideas and saw, surprisingly, a way to prove the existence of the mathematical functions whose existence he had previously doubted. Duchamp used the term “readymade” (and “tout fait”) in 1915. The examples he selected, titled, and signed are ordinary manufactured objects such as a urinal turned upside down, Fountain (1917), and a bottle drying rack, Bottle Rack (1914), thought to be the first readymade.

6 Constructivism

In Russia in 1920, the artists Naum Gabo and Antoine Pevsner wrote that they had turned to mathematics in order to rethink their work. As they put it: “We construct our work as the universe constructs its men; as the engineer constructs his bridges; as the mathematician his formulas of the orbits.” Gabo began to use a stereometric system that he had studied in engineering, creating sculptures such as “Head No. 2” (see figure 2). The subject of stereometry goes back at least as far as 1579, where it is listed in the “Groundplat” of John Dee’s celebrated Mathematicall Praeface to Billingsley’s edition of Euclid. It concerns the measurement of properties of solids, and was widely taught at universities in the nineteenth and twentieth centuries: indeed, it is still taught today in some European countries. Gabo and Pevsner constructed their sculptures out of planar parts, so space, rather than mass, became the sculptural element. Density was no longer important, with the result that the subtraction techniques used in classical sculpture (where material is carved away from a solid block leaving the artist’s work as the solid) were no longer necessary. Sculpture became airy; surfaces became less significant and have remained so, at least within the tradition that became known as constructivism.

Figure 2 Gabo’s Head No. 2, COR-TEN steel, 1916 (enlarged version 1964). The works of Naum Gabo: © Nina Williams.

This tradition was first formalized in the Russian Realistic Manifesto (1920), written and signed by Gabo and Pevsner. There they argued that “The material formation of the object is to be substituted for its aesthetic combination. The object is to be treated as a whole … a product of an industrial order like a car.” Gabo took constructivism to the Bauhaus in Germany and then to France and England in the 1930s, where he worked alongside the British artists Barbara Hepworth (1903–75) and her husband Ben Nicholson (1894–1982). Gabo and Nicholson (with Leslie Martin) edited Circle: International Survey of Constructive Art (1937), which contained articles by themselves as well as ones by Hepworth, Piet Mondrian (1872–1944), and the critic Herbert Read (1893–1968), among others. In Circle, Gabo, referring back to the seventeen-year-old Realistic Manifesto, spells out what is meant by constructivism by guiding the reader to see how two cubes (shown in the photograph in figure 3) can illustrate the distinction between two kinds of representation of the same object: carving and construction. The cubes have different methods of execution and different centers of interest: one is mass and the other makes visible the space in which mass exists. Constructivism created an artistic context in which a mathematically understood space became a sculptural element. As Gabo wrote: “The stereometrical method in which [the right-hand cube] is executed shows elementarily the constructive principle of a sculptural space expression.”

Figure 3 Gabo’s two cubes: carving and construction. Image courtesy of the Library of Congress.

These artists studied mathematical models in museums and catalogues. These models, designed by mathematicians for teaching about surfaces, were made of string, cardboard, metal, and plaster. The same artists also studied photographs produced by the surrealist Man Ray showing strings and striations of surface lines on a model that had been found by another surrealist artist, Max Ernst, at the Institut Henri Poincaré in Paris. Ray portrayed these models with impressionistic patterns of light and shadow (see figure 4); he was interested in the “elegance”—the aesthetic persuasiveness—of the model, though aware that the original model-maker had sought to give visual form to an elegance inherent in the mathematical equations themselves. Other artists too, such as Hepworth and Gabo, stated that it was not mathematics itself but the beauty of the mathematical models that provided the inspiration for their work. Hepworth studied mathematical models that were on display in Oxford, considering them to be “sculptural working out of mathematical equations.” They inspired her to add strings to her own work. However, she wrote that her inspiration was not the mathematics exhibited by the strings, but rather their power: “the tension I felt between myself and the sea, the winds, and the hills.”

Figure 4 Man Ray’s Allure de la Fonction Elliptique, 1936. Image courtesy of the National Gallery of Art.

Figure 5 Olivier’s Intersection of Two Hyperbolic Paraboloids, 1830. Image courtesy of the Union College Permanent Collection, Schenectady, NY.

A close friend of both Gabo and Hepworth was the renowned sculptor Henry Moore (1898–1986). Moore too spoke and wrote about the influence of mathematical models on his work. He had seen stringed figures of Theodore Olivier (see figure 5) and after making many of his own mathematical models introduced strings into his sculpture in 1938, later considering it to have been the most abstract of his work. He said he “had gone to the Science Museum in South Kensington and had been greatly intrigued by some of the mathematical models … hyperboloids and groins … developed by [Fabre de] Lagrange in Paris, that have geometric figures at the ends with colored threads from one to the other to show what the form between would be. I saw the sculptural possibilities of them, and I did some.” Moore recognized that the use of strings connecting protrusions actually created a barrier between the solid sculpture and the space around the sculpture (see figure 6). The string barrier made it possible to see the captured space. Moore and Gabo made different uses of the mathematical models. As Moore later put it, Gabo “developed this string idea so that his structure always became space itself, whereas I liked the contrast between the solid and the strings . . . I was making an outside shape a sculpture in its own right (Interior/Exterior forms), yet one which was not completed until each part was connected to the other.”

Figure 6 Moore’s Stringed Figure No. 1, cherry wood and string, on oak base, 1937. Image (taken by Lee Stalsworth) courtesy of the Hirshhorn Museum and Sculpture Garden, Smithsonian Institution, Joseph H. Hirshhorn Purchase Fund (1989).

7 Other Countries, Other Times, Other Artists

7.1 Switzerland and Max Bill

In the mid 1930s the Swiss designer and artist Max Bill (1908–94) became intrigued by a one-sided surface, unaware that it had been published in 1865 by the German mathematician and astronomer AUGUST FERDINAND MÖBIUS [VI.30]. Bill, when he needed a design for a sculpture to hang in a stairwell, independently invented his own MÖBIUS STRIP [IV.7 §2.3], by dangling a long narrow rectangle of flexible material and then attaching the corners appropriately (1935).

Having been informed some years later of the connection between his sculpture and its mathematical forerunner, Bill, who liked the simplicity of geometric forms, continued to earn commissions by making sculptures based on topological problems and single-sided surfaces (see figure 7). In a 1955 essay on the mathematical approach in contemporary art, he wrote that mathematics, by giving all phenomena a meaningful arrangement, is an essential method to understand the world. For Bill, when mathematical relationships are given form they “emanate undeniable aesthetic appeal, such as goes out from space-models, as, for instance, those that stand in the Musée Poincaré in Paris.”

Figure 7 Bill’s Eindeloze Kronkel, bronze, 1953–56. Image courtesy of Mary Ann Sullivan, Bluffton University.

7.2 Holland and Escher

From the second half of the twentieth century onward there has been a groundswell of interest in the relationship between mathematics and art, particularly since 1992 when artists and mathematicians from around the world began holding joint annual conferences to explore old and new ideas about the connections between their disciplines. The popularity in the West of this interdisciplinary study is in no small part due to the unusual drawings and prints made by Maurits Cornelis Escher (1898–1972), a Dutch graphic artist—or “craftsman,” as he wished to be known. Escher was deeply interested in tessellations and “impossible” objects that are not constructible in three dimensions but that can nevertheless be portrayed in two dimensions. While his oeuvre is not thought of as an integral part of twentieth century art, he is greatly appreciated by mathematicians and also by the general public. Among his best-known works are pictures based on Penrose triangles and on the Möbius strip.

He was inspired by knowing and learning from mathematicians including Georg Pólya (1887–1985), Roger Penrose (1931–), and Harold Scott MacDonald “Donald” Coxeter (1907–2003). Escher was introduced to the international mathematics community in 1954 when the organizing committee for the Amsterdam meeting of the International Congress of Mathematicians inaugurated an exhibition of his work at the Stedelijk Museum. After Penrose viewed Escher’s 1953 print Relativity at this exhibition, he and his father, geneticist Lionel Penrose (1898–1972), were inspired to create impossible figures: the Penrose tribar and the Penrose staircase published in the British Journal of Psychology in 1958—the Penroses sent Escher an offprint of the article. Escher subsequently used these in two well-known lithographs: Waterfall (1961), in which water runs in perpetual motion from the base of a waterfall to the top of the waterfall; and Ascending and Descending (1960), which features a building with an impossible staircase which constantly rises or falls (depending on the direction you go around it) but returns to the same level. Coxeter’s field was symmetry in the Euclidean and hyperbolic planes, but he also took pleasure in analyzing the works of artists from a mathematical point of view. Escher began a correspondence with him shortly after the congress, at which they met, and it lasted until his death in 1972. In 1957 Coxeter requested the use of two of Escher’s drawings to illustrate planar symmetry in “Crystal Symmetry and Its Generalizations,” his presidential address to the Royal Society of Canada—in this way Escher’s work spread among the mathematical community. In 1958, Coxeter sent Escher a letter containing a reprint of his address. The response was a request: “Could [you] give me a simple explanation how to construct the following circles, whose centers approach gradually from the outside till they reach the limit?” Coxeter’s reply, meant to be helpful, gave Escher one small piece of useful information; the rest of the lengthy letter was unintelligible to the artist. But from the pictures and his own keen geometric intuition, Escher was able to construct the circles he required, and by 1958 he was the first graphic artist to have used the three main geometries in his works: Euclidean, spherical, and hyperbolic. Coxeter was astounded that an artist, untrained in mathematics, could produce such accurate “equidistant curves” as he did in his 1958 woodcut Circle Limit III. Escher always claimed that he knew little mathematics, but many of his prints are a direct result of using mathematics. Mathematician Doris Schattschneider has said that Escher was really a “secret mathematician,” since much of his work depended on his pursuit of mathematical questions that arose from his interests and his interaction with mathematicians, which he referred to as “Coxetering.” He did, however, write that he preferred to find solutions and understanding by himself.

As well as his artistic and mathematical legacy, Escher had an important influence on crystallographers, who have used his symmetry drawings for analysis. Crystallographer Caroline MacGillavry has pointed out that Escher began a deep study of color symmetry and created a classification system in 1941–42, which was some time before crystallographers became interested in this field of study, which has become very active. The International Union of Crystallography subsequently commissioned Escher to illustrate MacGillavry’s Symmetry Aspects of M. C. Escher’s Periodic Drawings, first published in 1965. Its purpose was to interest “students in the laws which underlie repeating designs and their colorings.”

7.3 Spain and Dalí

As we have seen, some artists were influenced by their own knowledge of mathematics, others by a less direct appreciation of mathematical thinking, and still others by the appeal of mathematical models. Another kind of connection is illustrated by the example of the surrealist artist Salvador Dalí (1904–89) and his relationship with the mathematician and graphic artist Thomas Banchoff (1938–). Banchoff is a professor of mathematics at Brown University, known for his research in differential geometry in three and four dimensions. Since the late 1960s, he has also been involved in the development of computer graphics. Dalí’s 1954 painting of Christ crucified on a hypercube was reproduced in a 1975 article about Banchoff’s pioneering work, which used computer animation to illustrate geometry beyond the third dimension. This led to a series of meetings between Banchoff and Dalí over the next decade, at which hyper-cubes and other aspects of geometry and art were discussed. One joint project was the design for a giant sculpture of a horse that would appear realistic from only one viewing position. Dalí eventually envisioned a horse with its head in front of the viewer and its rump somewhere on the moon—clearly a project solely of the imagination. Dalí created works using anamorphoses, as other artists, beginning with Leonardo, had done. He prized his interactions with scientists and mathematicians, later stating, “Scientists give me everything, even the immortality of the soul.” Dalí also met the French mathematician René Thom (1923–2002) to discuss catastrophe theory, which, in 1983, he sought to represent in what turned out to be his last series of paintings.

7.4 Other Recent Developments: The United States and Helaman Ferguson

So far we have seen how mathematics has influenced art. Occasionally, artists have actually created mathematics, for instance to produce sculpture by means of carefully chosen mathematical equations. The noted American sculptor/mathematician Helaman Ferguson (1943–) divides his time equally between mathematics and the interpretation of mathematics in his art. As a mathematician he designs algorithms for operating machinery and for scientific visualization. In 1979 he found a method for finding integer relationships between more than two real or complex numbers—this was later named one of the top ten algorithms of the twentieth century. As an artist, he carves in stone. In 1994, he asked mathematician Alfred Gray (1939–98) to develop equations for a Costa surface (named after the graduate student who invented equations for describing a minimal surface with holes), so that he could sculpt the surface (see figure 8). Gray developed the equations in terms of the Weierstrass zeta function. This could be used with Mathematica, which made it possible for Ferguson to create a stone sculpture. Ferguson sees his art as deriving from applied mathematics that has been developed over the course of the last two centuries:

Start with physical observations about soap films in nature (Plateau), write down a differential equation model describing minimizing surfaces (Euler–Lagrange), define a minimal surface geometrically in terms of curvature (Gauss), discover a minimal surface with non-trivial topology (Costa), draw computer images of the surface (Hoffman–Hoffman), recognize symmetry and prove the surface has not self intersections (Hoffman–Meeks), discover fast parametric equations for the surface (Gray), and finally return to nature with a sculpture, a solid form of a “soap film” big enough to touch and climb on.

7.5 The United States and Tony Robbin

The development of n-dimensional geometry also had a powerful effect on many other European and American artists, and this continued into the late twentieth century. Interest was boosted in the 1970s with the development of computer graphics by mathematicians and artists. Examples can be found in the work of American artist Tony Robbin (1943–), who has explored concepts of dimension in painting, prints, and sculpture (see figure 9). In late 1979, Robbin, who had also been a student of mathematics, was working on Banchoff’s parallel processor computer and managed to visualize for the first time a four-dimensional cube, an event which radically changed his art, and which led him to develop two-dimensional works that portrayed the spatial fourth dimension. Writing in his book Fourfield: Computers, Art & the 4th Dimension (1992), Robbin tells us, “When the fourth dimension becomes part of our intuition our understanding will soar.” Some of Robbin’s constructions, paintings, and prints show figures in independent planes: that is, in overlapping spaces that cannot be fully seen in three dimensions. If the viewer wants to see two structures in the same place at the same time and rotating with respect to one another (as though projected from four-dimensional space), then looking at one of Robbin’s wall-relief sculptures lit by red and blue light while wearing 3D glasses (one red and one blue lens) will create a full stereoscopic effect of the four-dimensional figure. In digital prints it is Robbin’s lines and polyhedra that imply four dimensions, with the two-dimensional picture being a shadow of the higher-order object.

Figure 8 Ferguson’s Invisible Handshake II: a triply punctured torus with negative Gaussian curvature. Image courtesy of the artist.

Figure 9 Robbin’s Lobofour, acrylic on canvas with metal rods, 1982, collection of the artist.

7.6 Hayter and Atelier 17

In 1927, the British surrealist and printmaker Stanley William Hayter (1901–88) decided to revive the almost lost skill of intaglio printing and established an experimental studio, “Atelier 17,” in Paris. This was followed by another in New York from 1940 to 1950 before he returned to Paris. Hayter was aware that many of the artists who used his facilities were working with a “different space from that seen through the classical window of Renaissance representation” that had existed when engraving flourished a hundred years earlier. The founding of Atelier 17 was central to the revival of the print as an autonomous art form, and Hayter’s sensitivity to the significance of mathematics in the experimental techniques of printmaking (which had been evolving since the nineteenth century) is quite apparent: “Man’s increasing consciousness of and power over space (in physics and mathematics) have been reflected in new and unorthodox methods of demonstrating space and time graphically,” so that “many properties of matter and space, which had been represented diagrammatically only by the scientists, found their expression in graphic and affective forms.” A printmaker in the twentieth century could use an arrangement of transparent webs to define planes above the picture plane. Specifically, by hollowing out spaces in the plate being engraved—possibly even gouging all the way to the bottom of the plate—the artist could make a projection in front of the plane of the picture. Although artists could have used this technique much earlier, it became important only at the end of the nineteenth century when the representational aspect of intaglio had been challenged by photography. They therefore used the gouge to create the third dimension. Hayter also describes in About Prints (1962) how Abraham Bosse’s seventeenth-century atelier was organized and reconstructed in Paris in the twentieth century.

In World War II Hayter’s interest in mathematics revealed itself in a more practical way, when, in collaboration with artist and patron of art Roland Penrose and others, he set up a camouflage unit and, as Art News reported in 1941, constructed

an apparatus which can duplicate the angle of the sun and the consequent length of cast shadows at any time of day, and day of the year, at any given latitude. This complex of turntables, discs inscribed with a scale of weeks, allowances for seasonal declination, and so on is just the kind of working mathematics he really delights in.

8 Conclusion

There has been a complex and fruitful relationship between Western art and mathematics in the twentieth century. Gabo, Moore, Bill, Dalí, and Duchamp are notable artists who have been influenced by mathematics, and Poincaré, Banchoff, Penrose, and Coxeter are among the mathematicians who influenced them. In the other direction, twentieth-century mathematicians, like their forebears in the fifteenth and sixteenth centuries, often turned to art to explore and exhibit, or even just to explain more expressively, the meaning of their mathematics. They have also likened their creative processes to those of artists. As the French mathematician ANDRÉ WELL [VI.93] wrote to his sister, author Simone Weil (1909–43), from military prison in 1940, “When I invented (I say invented, and not discovered) uniform spaces, I did not have the impression of working with resistant material, but rather the impression that a professional sculptor must have when he plays by making a snowman.”

Further Reading

Andersen, K. 2007. The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge. New York: Springer.

Field, J. V. 2005. Piero della Francesca: A Mathematician’s Art. Oxford: Oxford University Press.

Gould, S. J., and R. R. Shearer. 1999. Boats and deckchairs. Natural History Magazine 10:32–44.

Hammer, M., and C. Lodder. 2000. Constructing Modernity: The Art and Career of Naum Gabo. New Haven, CT: Yale University Press.

Henderson, L. 1983. The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Princeton, NJ: Princeton University Press.

Henderson, L. 1998. Duchamp in Context: Science and Technology in the Large Glass and Related Works. Princeton, NJ: Princeton University Press.

Jouffret, E. 1903. Traité Élémentaire de Géométrie à Quatre Dimensions et Introduction à la Géométrie à n Dimensions. Paris: Gauthier-Villars. (A digital reproduction of this work is available at www.mathematik.uni-bielefeld.de/-rehmann/DML/dml_links_title_T.html.)

Robbin, T. 2006. Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought. New Haven, CT: Yale University Press.

Schattschneider, D. 2006. Coxeter and the artists: two-way inspiration. In The Coxeter Legacy: Reflections and Projections, edited by C. Davis and E. Ellers, pp. 255–80. Providence, RI: American Mathematical Society/Fields Institute.

1. Gabo was born Naum Neemia Pevsner but changed his name to distinguish himself from his painter brother.