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Until the 20th century, art was almost entirely representational. That is, art consisted of drawings, paintings, and sculptures that were intended to represent some aspect of the physical world -- people, animals, landscapes, or common objects. Around the beginning of the 20th century, artists began to become more free with their representations, as can be seen in the movements of Expressionism and, later, Cubism.
However, in the years preceding the first World War, a new kind of artistic movement began. This was a movement of Abstract Art, or art that was not intended by be representational. Wassily Kandinsky is generally regarded as having been the first artist to paint purely abstract pictures. This first wave of abstraction was widened by the Dada, de Stijl, and Constructivist movements.
Another kind of modern art was being developed during the time in between the world wars, the Surrealist movement. The Surrealists were not trying to remove representation from art (like the Constructivists) or to destroy art (like Dada) but to give a voice to the unconscious and irrational. Surrealism flourished in Europe in the years between World War I and World War II.
Both the Surrealists and the abstract artists seem to have had some exposure to mathematical models of surfaces. Some artists even drew direct inspiration from the mathematical models that they saw in museums. This paper will attempt to outline the contact that modern artists in the early part of the 20th century had with mathematical models and to do some informed speculation on any influence the models might have had. Along the way, we'll get to know a little about the models themselves.
Humans have been constructing approximations of polyhedra and simple geometric solids for quite a long time. Spherical forms that show all five regular polyhedra have been found in Scotland dating as far back as 2000 BC [Artman, p. 141]. However, more complicated three dimensional geometric forms do not appear until the beginning of the 19th century -- that had to wait for the development of of more sophisticated mathematical machinery.
In the mid 1600's, Descartes and Fermat were both developing coordinate geometry. Their coordinate geometry allowed points in a plane (a geometric concept) to be represented by a list of numbers. Others refined and expanded their ideas, developing the Cartesian coordinate system that we know so well today, as well as other coordinate systems. The development of plane coordinates allowed mathematicians to consider the set of solutions to an equation as a geometric object. For instance, one can consider the set of solutions of x2+y2=1 in the plane, which turns out to be a circle. We might also think of this set as the locus of zeros of the polynomial x2+y2-1. It turns out that all of the conic sections correspond to zero loci of polynomials of degree two in two variables (see below) . One can consider the shapes of the zero loci of cubic, quartic, and higher degree polynomials. This was widely investigated, and many of the investigations were illustrated with drawings of the resulting curves.
In the mid 1700's, Clairaut, Hermann, Euler, Monge and others began to study surfaces in a similar way. The zero locus of a polynomial in three variables will be a surface in three dimensional space. For instance, the zero locus of the polynomial x2+y2+z2-1 is a sphere. Some of these surfaces, like the sphere, are relatively easy to visualize. We have many objects of roughly spherical shape that occur in our day to day lives and can take these as a point of departure for our intuition about mathematical spheres. However, it might be trickier to picture other loci in your mind. What does the zero locus of x2-y2+z2-1 look like? How about the zero locus of x3+5xyz-z3?
Gaspard Monge was concerned with the representation of objects in space, and he developed descriptive geometry, which involves representing three dimensional objects in two dimensions. His ideas were developed at a school for military engineering and were a military secret until after the French Revolution. During the 1790's, Monge became an influential figure in technical education and descriptive geometry became a widely taught subject (in 1801 students at the École Polytechnique spent one fifth of their first two years studying projective geometry). Monge also studied ruled surfaces and made at least two models of surfaces which were at the Conservatoire National des Arts et Métiers in Paris.[Kidwell] According to the 9th edition of the Encyclopaedia Britannica, "As regards tridimensional figuring, the oldest known models for instruction in the higher geometry are the thread models of skew surfaces constructed about the year 1800 under the direction of G. Monge for the École Polytechnique in Paris." [Britannica] (Note: The Conservatoire still has a collection of models. See their catalogue.)
Thèodore Olivier (1793-1853) was a École Polytechnique and taught descriptive geometry at the École Centrale des Artes et Manufactures and at the Conservatoire National des Arts et Métiers. Olivier designed models of ruled surfaces, including some where were movable, to suggest how the surfaces were generated. Probably these models were displayed at the Conservatoire as well.
Around 1860, the ``golden age'' of model building begins. Many mathematicians began to building models out of a variety of materials, including plaster, cardboard, metal, and string. Most of the model builders were German, and the names of those involve include some very influential mathematicians such as Eduard Kummer, Felix Klein, and Alexander Brill. Many others were involved in model building, including non-Germans, such as Arthur Cayley.
Many of the models build were reproduced and sold by publishing houses (especially the publishing houses of Ludwig Brill and later Martin Schilling) to schools and museums around the world. By the turn of the century there were a large number of models of surfaces available, but in 1932, Martin Schilling reported that ``in the last years, no new models appeared'' and the golden age of model building was over.
Many of the models survived in universities and museums, sitting in dusty cases. They can still be seen in many places today.(See http://math.bu.edu/people/angelav/projects/models/locations.html for a list of places which have models.) Certainly in the first half of the 20th century, there were a number of museums and schools that had collections that could be viewed by the public.
The models certainly provide visual and artistic, as well as mathematical, inspiration. Browsing through the first Gerd Fischer's Mathematical Models: From the Collections of Universities and Museums, one is struck by their beauty and elegance, and if caught reading the book by a non-mathematician, you may be hard pressed to convince anyone that you are, in fact, looking at mathematics instead of abstract sculpture. It is with this in mind that we turn to the artists.
Ring Cyclide: 
Kuen's Surface: 
Constructivism is an artistic and architectural movement that began in Russia in the 1910's. For the Constructivists, content and form are not separate, in contrast to older forms of art, where content ruled form. According to Naum Gabo, one of the founders of the movement, ``It [Constructivism] has revealed an universal law that the elements of a visual art such as lines colours, shapes, possess their own forces of expression independent of any association with the external aspects of the world.'' [Gabo, p. 7]
It is clear that the Constructivists felt that science and art had a deep and important relationship. In his article "The Constructivist Idea in Art" in The Circle, Gabo argues that science and art come from the same creative spirit, but act in different ways upon the world. The layman has very little access to science (because it is technical and intimidating), but a greater access to art. "The force of Science lies in its authoritative reason. The force of Art lies in its immediate influence on human psychology and in its active contagiousness." [Gabo, p. 8-9]
At least some of the artists associated with the Constructivist movement had some exposure the mathematical models.
Naum Gabo (1890-1977) was a sculptor and a major architect of the Constructivist movement. In Constructing Modernity: The Art and Career of Naum Gabo, Martin Hammer and Christina Lodder discuss the impact that mathematical models had on Gabo's work. Gabo studied physics and engineering at the University of Munich, and Hammer and Lodder speculate that he probably encountered mathematical models during his studies, as they were in wide use at that time. Indeed, Gabo's early figurative constructions, such as Head No. 2 (1916), are reminiscent of of cardboard models of second order surfaces made with interlocking cross sections. [Hammer, p. 50]
Naum Gabo, Head No. 2, 1916, enlarged version 1964. 
A direct link to mathematical models is provided by a series of drawings made by Gabo around 1933. These drawings seem to be sketches of mathematical models and were probably made at the Institut Henri Poincaré in Paris. Later, in 1936, Gabo made a drawing Study for Construction in Space: Crystal which is a tracing of the illustration of `Cubic Space Curves: Tangent Surface of Cubic Ellipse' from an article on mathematical models in the fourteenth edition (1929) of the Encyclopaedia Britannica. [Hammer, p. 389-390]
Antoine Pevsner, Developable Column, 1942. 
Along with his brother, Naum Gabo, Antoine Pevsner(1886-1962) helped to develop the constructivist movement (he co-authored the Realist Manifesto with Gabo in 1920). In 1923 Pevsner moved to France, where he lived for the rest of his life. Some of his sculpture seems to have been influenced by mathematical models. According to Anthony Hill, "Although he always denied it Pevsner based his Developable Surfaces on a concept found in certain mathematical models. " [Hill, p. 144]
Henry Moore, Stringed Figure No. 1, 1937. 
Henry Moore (1893-1986) used string in many of his sculptures for a short period of time, and this was influenced by stringed models that he had seen at the Science Museum in London. As Moore himself says:
Undoubtedly the source of my stringed figures was the Science Museum...I was fascinated by the mathematical models I saw there, which had been made to illustrate the difference of the form that is halfway between a square and a circle. One model had a square at one end with twenty holes along each side, making eighty holes in all. Through these holes strings were threaded and lead to a circle with the same number of holes at the other end. A plane interposed through the middle shows the form that is halfway between a square and a circle. One end could be twisted to produce forms that would be terribly difficult to draw on a flat surface. It wasn't the scientific study of these models but the ability to look through the strings as with a bird cage and see one form within the other which excited me. [Moore, p. 105]
Barbara Hepworth (1903-1975) is also known for the use of string in her sculpture, and upon viewing some of her pieces, one is reminded of certain mathematical models.
Barbara Hepworth, Sculpture with Colour (Deep Blue and Red) 1940.
Hepworth was an abstract sculptor, and had some involvement with the Constructivist movement, particularly from the mid 1930's to the mid 1940's, when Naum Gabo was living nearby. The effect of mathematical models on Hepworth is less clear than it was on Henry Moore. It is clear, however, that Hepworth knew about mathematical models. Martin Hammer and Christine Lodder quote a December 1935 letter from Hepworth to her husband Ben Nicholson: "John Summerson says there are some marvelous things in a mathematical school in Oxford -- sculptural working out of mathematical equations -- hidden away in a cupboard -- I think I shall go to Oxford as soon as I get back from Leeds." [Hammer2, p. 115-116]
Barabara Hepworth also had a close and productive relationship with J.D. Bernal, a crystallographer who was interested in the relationship between art and science. Apparently, Hepworth enjoyed visits from Bernal, during which he would discuss with her the mathematics and geometry in her works [Barlow]. Hepworth had an interest in "higher geometry" and she probably viewed the models at the school in Oxford herself, but the exact influence of the models in unclear.
At least some of the surrealists had an admiration for mathematical models. The front cover of the catalogue of the 1936 surrealist exhibition at New Burlington Galleries in London is a drawing of a statue of a man (with a monster's head) holding four mathematical models, and with another model at his feet. The main figure associated with the models is Man Ray.
One day I was told about some mathematical objects at the Institut Poincaré in Paris. These were built by the tutors [Note: This is from the English translation, given by the narrator. A better translation would be "professors".] to explain algebraic equations. I went to see them, although I am not particularly interested in mathematics. I didn't understand a thing, but the shapes were so unusual, as revolutionary as anything that is being done today in painting or in sculpture. And I spent several days photographing and sketching them with the intention of doing a series of painting influenced and inspired by these objects.
Man Ray, Mathematical Object: Ruled Surface, 1936. 
That series was completed in 1948 and was called Shakespearean Equations. Each of the paintings was given the name of one of Shakespeare's plays. The paintings were exhibited in William Copley's gallery in Beverly Hills.
Man Ray, King Lear, 1948. 
Right now, I may or may not be focusing on the things found here.
See also my complete list of references.
[Artman] B. Artmann. ``Symmetry Through the Ages: Highlights from the History of Regular Polyhedra,'' 139-148 In Eves' Circles, Joby Milo, ed. Mathematical Association of America, 1994.
[Baldwin] N. Baldwin. Man Ray: American Artist. Clarkson N. Potter, Inc., New York, 1988.
[Barlow] A. Barlow. "Barbara Hepworth and Science," Barabara Hepworth Reconsidered, D. Thistlewood, ed. 95-107.
[Gabo] N. Gabo. ``The Constructive Idea in Art,'' Circle: International Survey of Constructive Art, J.L. Martin, B. Nicholson, N. Gabo, eds. Faber and Faber, Ltd., London, 1937.
[Hammer] M. Hammer, C. Lodder. Constructing Modernity: the Art and Career of Naum Gabo. Yale University Press, London, 2000.
[Hammer2] M. Hammer. "Hepworth and Gabo: a Constructivist Dialogue," Barabara Hepworth Reconsidered, 109-133. Liverpool University Press, Liverpool, 1996.
[Hill] A. Hill. "Constructivism -- the European Phenomenon," Studio International, V 171 April 1966, 140-147.
[Kidwell] P. A. Kidwell. ``American Mathematics Viewed Objectively: The Case of Geometric Models,'' Vita Mathematica, R. Calinger, ed.
[Moore] H. Moore and J. Hedgecoe. Henry Spencer Moore, Simon and Schuster, New York, 1968.
[Britannica] ``Mathematical Drawing and Modelling'' in Volume XV of The Encylopaedia Brittanica: A Dictionary of Arts, Sciences, and General Literature, 9th Edition, p. 628-629.
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