by Craig Adcock
With Desargues' terminology such as "tree," "trunk," "branch," and "limb" in mind, these works look positively geometrical. In Network of Stoppages, for example, the pattern of lines resemble branches, especially if the painting is rotated ninety degrees clockwise. In the background, the nude woman in "Young Man and Girl in Spring," the first layer of Network of Stoppages, is then centered in the boughs of the tree. From this perspective, she becomes a precursor for the Bride as an "arbor-type." In the Bottlerack, the prongs appear to be rotated around a central axis (an arbre) and suggest reiterated line segments (rameaux or branches). That these interpretations can be taken seriously is reinforced by an interesting photograph of Duchamp taken in 1942 showing him standing in front of a tree that has been provided with prongs so that it can act as a bottle dryer (Fig. 20). A number of bottles, which have been hung upon this "arbre-séchoir," can be seen behind Duchamp, and he has a network of linear shadows, which have been cast from the branches of the tree, falling across his face.(57)
On the right-hand side of the canvas, there is an irregular, open-sided rectangular "solid." The left side of this solid is a white surface that recedes into the space of the canvas according to one-point perspective. From each corner of the white surface, two lines, drawn with the templates of the Three Standard Stoppages, extend at more or less right angles toward the right. One of each of these is black and the other red. The black lines at all four edges are drawn with the same template. Each set of lines at the upper boundary of the solid cross one another at two points, and each set are drawn in the same way. The two lines at the lower edges of the solid do not cross one another, and they are rotated and inverted with respect to one another.
There are also a series of color bands (twenty-four in all) extending orthogonally back into the space of the "solid," or into its virtual shape. They seem to continue on behind it. These bands are connected to the curved line segments that comprise the ambiguous edges of the transparent solid, a volume we could think of as a 3-space with fluctuant, transparent faces. Each of the color bands is surrounded by a number of concentric circles that also recede back into the painting's virtual space according to one-point perspective. The vanishing point coincides with the bottom edge of the canvas just to the right of center below the indexical hand, which, incidentally, is a hand-painted readymade element executed by a certain A. Klang, a sign painter Duchamp hired to carry out this task. Klang's minuscule signature is visible near the sleeve.
Duchamp's complex geometrical arrangement is made even more complex by the shadow of the Hat Rack, which occupies the same region of the canvas as the "solid." On one level, the Hat Rack resembles a tree, and the shadows cast from its multiple branches suggest yet another "arbor-type." We know that the Bride is based, in part, on the idea of the cast shadow, "as if it were the projection of a four-dimensional object."(61) The way the Hat Rack interacts with the "solid" is indicative of the complexities that would be involved in such spaces: The lines and color bands seem to overlay the shadow, but the shadow seems to overlay the white rectangle at the left side of the "solid." The shadow can thus be read as both in front of and behind the chunk of space outlined and bounded by the elements of Duchamp's design.
The spatial complexities of Tu m' can also be seen in the recession of its orthogonals. They plunge backward in a way that is comparable to the convergence of orthogonals in the Large Glass. In the former, the lines come together just at the lower edge of the painting, in the latter, just at the upper boundary of the Bachelors' domain. In Tu m', the vanishing point is where the "solid" (and also its edges drawn with the Three Standard Stoppages) would disappear. In the Large Glass, the point is at the center of the three plates of glass running across the Bride's horizon. It is where these "lines" would disappear, if rotated ninety degrees. The Bride's garments, when thus folded up, can be taken as orthogonals to a point of intersection--the intersection of parallel lines at infinity.
In Euclidean geometry, parallel lines do not intersect. The mathematical convention that they do intersect at infinity was one of Desargues' important contributions. (Parallel lines do seem to intersect at the vanishing point of a perspective system, which may have given Desargues his idea.) Thinking of parallel lines as meeting at infinity eventually contributed to the development of non-Euclidean geometries in the nineteenth century.(62) The conceptual point where parallel lines meet cannot be seen, any more than the curvature of space can be perceived directly. If the curved lines in the Three Standard Stoppages are taken as references to non-Euclidean lines of sight, then they are fundamentally hidden in "garments" of the Bride, just as the vanishing point in Tu m' seems to disappear off the edge of its hyperspatial expanse.
The left side of Tu m' is also complicated. In addition to the shadows of the Bicycle Wheel and the Corkscrew, lines drawn with the templates of the Three Standard Stoppages are placed at the lower left-hand side of the canvas. Each of these line segments is at the edge of three curved surfaces that seem to fall back into the space of the canvas. If these irregular planes are thought of as a "pencil of surfaces" (Desargues uses the term "ordonnance de plans"), they would withdraw downward at more or less right angles to the space of the canvas toward a line of intersection located at an infinite distance. (Desargues says that a sheaf of parallel planes can be imagined converging at an "essieu," an "axle," just as an "ordinance of lines" can be imagined intersecting at a "point à une distance infinie.")(63) The edge of the upper member of this pencil of planes is black, and it is drawn with the same "stoppage" that was used at each edge of the rectangular "solid" on the right side of the canvas. The edge of the line segment in the middle register was used as the other line at the edges of the upper boundary, and the edge of the line segment in the lower register was used as the other line at the edges of the lower boundary of the "solid." The shadow of the Bicycle Wheel seems to overlay this arrangement of superposed curved surfaces. There is also a sequence of flat color squares receding according to a plunging perspective back from the center of the canvas into an infinite space at the upper left corner of the canvas. This arrangement of color squares seems to overlay the shadow of the Bicycle Wheel. In contrast, the shadow of the Corkscrew, which seems to spiral out from the axle of the wheel, overlays the color squares. Reading the shadows as riding on the surface of the actual canvas is thus complicated by their relationships with objects occupying the virtual space depicted "inside" the canvas. Duchamp further emphasizes the spatial oddities of his picture by using various forms of "intersection." The corkscrew intersects the canvas by seeming to spiral into it; the safety pins pierce the surface of the canvas; and the bottle brush and the bolt go through the front side of the picture and are fastened to it from behind.
Duchamp is obviously playing with real and represented objects and with real and represented space in Tu m'. To further complicate the issues, he paints a trompe l'oeil tear in the surface of the canvas, which is held together by the real safety pins. In addition to these ready-made elements, the bottle brush juts out from the tear at right angles to the canvas. As an actual object, a readymade, the bottle brush casts actual shadows that can be contrasted with the virtual shadows of the Bicycle Wheel, the Corkscrew, and the Hat Rack, which Duchamp traced onto the surface with pencil. In terms of its geometry, the bottle brush is really only visible when we look at Tu m' from the side, at an oblique angle (Fig. 22). When we view the canvas straight on, all we see is the end of the brush. Looking at the canvas from the side also allows us to see the other elements of the painting, and they seem less stretched out, less constrained by the plunging perspective. The shift is particularly apparent in the sequence of color squares at the upper left side of the canvas. In fact, we now notice that these shapes are not really squares, but parallelograms that look more "natural" from the side than from the front.
Duchamp probably learned something about these kinds of anamorphic effects during the period he was working at the Bibliothèque Sainte-Geneviève in Paris. One of his notes for the Large Glass, which he wrote at this time, suggests consulting the library's collection: "Perspective. See the catalogue of the Bibliothèque Sainte-Geneviève. The whole section on perspective: Nicéron (Father J.-F.), Thaumaturgus opticus."(64) Many of the books on perspective available to Duchamp at the library deal with the unusual, or "aberrant," systems used in anamorphosis. These include works by Father Jean-François Nicéron, whom Duchamp mentions by name in his note.(65) One of Nicéron's images from Thaumaturgus opticus (Fig. 23) is evocative of Tu m', especially if the sketch is fully extended (the left-hand side of the upper part continues at the right-hand side of the lower part).(66) Thus reconnected, the long, narrow dimensions of the image approximate those of Tu m'. Duchamp may also have seen a similarity here between the string held by the assistant in the left-hand part of the drawing and the segments of string in Three Standard Stoppages. In Nicéron's illustration, as in perspective drawings generally, the curling end of the line is meant to indicate that it is a thread used in the construction of the image, rather than being an integral element of the imagery.
Duchamp's thread is more complex. The strings in the Three Standard Stoppages are themselves spaces, one-dimensional spaces, and they are intended to indicate a more difficult geometry than the one Nicéron had in mind. But Duchamp's manner of taking an oblique view and his interest in observing a scene through a visual system rotated away from normal space, is very similar to the way Nicéron turns his outstretched images onto the wall. Duchamp's (and Nicéron's) procedure is also reminiscent of Hans Holbein's famous portrait, The French Ambassadors (Fig. 24), in which a distended skull crosses the picture plane at more or less right-angles to the orthogonals of the perspective system used to construct the painting.(67) The French Ambassadors is a favorite image among postmodernists, primarily because it brings together two different ways of looking at objects in one picture.(68) The primary visual order, the three-dimensional space of the scientific perspective, is undermined by the anomalous skull falling across it. The abnormal space of the death's head interpenetrates the normal space where the ambassadors live, casting a shadow across their existence. It also displaces the dominant viewing subject from a position in front of the painting to one at the side--to a position that is essentially outside the picture's frame of reference.(69) As the skull comes into adjustment, the painting becomes distorted, and vice versa.
Jean Clair has discussed Tu m' in terms comparable to those
just used to describe Holbein's painting. He points out that, when
looked at obliquely, "the shadows of the readymades and the design
of the parallelepiped straighten up."(70)
He also notices the way in which the bottle brush seems to rotate
out from the surface of the canvas, changing from a "dot,"
or point, into "no more than a line." According to Clair,
the function of the bottle brush is similar to that of the skull in
Holbein's picture: namely, "to expose the vanity of the painting.
But this time of all paintings."(71)
We can amplify Clair's remarks by pointing out that, as we move to the side of Tu m', the surface of the picture is visually rotated. If we were able to continue on around the picture in order to look at it edge on, the surface would be reduced to a line segment, from which the "line segment" of the bottle brush would extend at a right angle. The bottle brush is a readymade, a counterpart of an orthogonal, one that comes out into our space rather than receding into the space of the painting. The sequence of color squares, apparently attached to the surface of the canvas with the bolt, would presumably be receding in the opposite direction along the axis of the shaft (the axle) of the bolt back into the space of the canvas, which as we move to the side, is not only flattened into a two-dimensional surface, but further reduced to a one-dimensional line segment. Clair's statement that as the "painting vanishes, the readymade makes its appearance," is quite true. We could also say that the actual readymade (the bottle brush) makes its appearance as the virtual readymades and their shadows disappear. And vice versa: as the real elements of the work vanish, the virtual elements reappear.
A similar language could be used to describe the intersection of the strings with the glass plates of the Three Standard Stoppages. They trail off at right-angles, as it were, along lines that are orthogonal to the canvas strips, as if they had been rotated out of the virtual space of the "Prussian blue" into the actual space of the canvases. If the strings are analogous to "lines of sight," they are like threads lying "in" the surface of the perspectival plane, as we have seen in Desargues' perspective renderings (Figs. 13 and 14) or in Nicéron's illustration (Fig. 23). In this sense, the strings can be taken as anamorphic lines crossing the representational space of the sheets of glass. Recall what Duchamp's space was intended to show: his glass has "neither front, nor back; neither top, nor bottom," and it can be used as a "three-dimensional physical medium" in the construction of a "four-dimensional perspective." In the Large Glass and the Three Standard Stoppages, Duchamp was both literally and figuratively boxing and encasing the geometrical elements of his iconography--inside glass and inside an n-dimensional projective system. With Tu m', he was also enclosing the basic elements of his own working method, and, indeed, the basic elements of painting as a general practice, inside a complex pictorial space, one with unusual curvatures.
Duchamp's works such as the ones I have discussed in this paper,
with their various projections and intersections, each in their turn
folding up into the next, suggest that he was thinking about different
kinds of geometries. Henri Poincaré, among the artist's most likely
mathematical sources, often discusses the interrelationships of geometries.(72)
Projective geometry, which was prefigured in Renaissance perspective
and initially elaborated in the work of such seventeenth-century mathematicians
as Desargues and Blaise Pascal,(73)
was later, during the nineteenth century, recognized as being central
to mathematics in general. By the end of the century, both Euclidean
and non-Euclidean geometry had been subsumed under the principles
of projective geometry.(74)
Projective geometry deals with properties of geometrical figures that remain invariant under transformation. It studies mappings of one figure onto another brought about by projection and section, and it tries to find qualities that remain fixed during these procedures (Desargues' Theorem and Pascal's Theorem describe famous examples). Twentieth-century mathematicians have invented methods of transformation that are even more general than projection and section. One of the most important of these approaches, topology, considers geometrical properties of figures that are unchanged while these figures undergo deformations such as stretching and bending. Especially in the context of the present discussion, Poincaré can be thought of as the "father of modern topology,"(75) a subject that he referred to as analysis situs (Latin for "analysis of the site"; "topology" coming from the Greek equivalent for "study of the place"). He points out that this geometry "gives rise to a series of theorems just as closely interconnected as those of Euclid."(76)
Duchamp's Tu m' can very nearly serve as an illustration for Poincaré's arguments. As pointed out earlier, the elongated shadows can be taken as anamorphic deformations, and thus as references to topological transformations with four-dimensional, or more generally, n-dimensional ramifications (branchings), particularly insofar as anamorphic projections seem to intersect normal space at oblique angles. In ways that are like Holbein's famous skull, the cast shadows in Tu m' seem to traverse the space of the picture and, in this sense, they are orthogonal to it (shadows are literally orthogonal to the surfaces on which they are cast). From the perspective of the fourth dimension, the strings in Three Standard Stoppages can also be interpreted as falling away from normal space along perpendicular lines, at least insofar as they plummet toward the horizon of the Bride. Duchamp's cast shadows, and perhaps his cast segments of strings, are projective analogies for higher-dimensional spaces. His general approach can be seen in the following note:
For an ordinary eye, a point in a three-dimensional space hides, conceals the fourth direction of the continuum--which is to say that this eye can try to perceive physically this fourth direction by going around the said point. From whatever angle it looks at the point, this point will always be the border line of the fourth direction--just as an ordinary eye going around a mirror will never be able to perceive anything but the reflected three-dimensional image and nothing from behind.(77)
Looked at "edge-on," in the sense of being seen undergoing an n-dimensional rotation, the individual "stoppages" can be taken as trailing off into the fourth direction of what Duchamp calls the "étendue."(78) From such a perspective, they would be perceived as points. The viewer equipped with a four-dimensional visual system, to use Duchamp's words, would be able to ascertain that a "point" is always a "border line" of this "fourth direction." At the center of the Bride's garments, the Stoppages recede anamorphically into the labyrinth of the fourth dimension, a space that is orthogonal to normal space. Duchamp was probably aware that in descriptions of n-dimensional geometry, when n is greater than 3, the convention is to say that planes intersect at points, unlike what happens in three-dimensional space where, of course, they intersect along lines.(79) The curvature of the string does not really affect this n-dimensional argument since curvature depends upon whether or not the space is Euclidean, non-Euclidean, or whatever.(80) We can, in a sense, choose the space to have any curvature we want.(81)
In Tu m', readymades cast shadows onto the surface of the painting, but these shadows do more than ride on the surface. As we have seen, they are interlocked in curious ways with the entities depicted in the space of the picture, convolutions that indicate Duchamp was interested in the readymades and their shadows as geometrical objects. The shadows themselves have perspectival implications and topological associations; and they are obviously seen differently under changing angles of view. As we walk "around" the picture, it presents shifting aspects. In Tu m', and, indeed, in most of his works, Duchamp was interested in exploring both actual viewpoint and philosophical point of view, as well as the effects of the two acting together.
Such consequences were apparently on Duchamp's mind when he chose readymades: bicycle wheels, corkscrews, and hat racks were works of art depending upon how they were perceived. He was involved with a discourse of surface (and reflective surface) in many of his works (often using glass and mirror in their construction). Because projective analogies such as shadows and falling pieces of string can be related to several different geometries, not just to n-dimensional Euclidean, or for that matter n-dimensional non-Euclidean geometry, Duchamp can entail other regimes of meaning into his system. Within any given framework, one which might, say, be used to interpret the Three Standard Stoppages, Network of Stoppages, Tu m', the Large Glass, Nine Malic Molds, or the readymades, Duchamp understood that the implications of choosing one standpoint over another were manifold (and the etymological associations of this last term are germane here).(82) Duchamp believed that, just as how we use a particular geometry to interpret the shape of the world is largely a matter of discretion, as Poincaré argued, so too is our choice of the interpretive frameworks that we use in making our aesthetic judgments. As an artist, Duchamp was engaged in self-referential, contemplative activities. He tried to look at himself seeing, and by so doing, to dislocate himself from the center of his own perspective.
57. This photograph appears in Robert Lebel, "Dernière soirée avec Marcel Duchamp," L'Oeil (Paris) no. 167 (November 1968): 18-21; also reproduced in the supplement "Marcel Duchamp et Robert Lebel" in the facsimile edition of Sur Marcel Duchamp (cited n. 56); see also Gough-Cooper and Caumont (cited n. 12), under their entry for April 29, 1942. The photograph was taken just before Duchamp left France for the United States. Mirroring the famous movie script, he sailed from Marseilles to Casablanca, and from there to Lisbon and then to New York, arriving on June 25. See Jennifer Gough-Cooper and Jacques Caumont, Plan pour ecrire une vie de Marcel Duchamp, vol. 1, Marcel Duchamp catalogue (Paris: Centre National d'Art et de Culture Georges Pompidou, Musée National d'Art Moderne, 1977) 23.
58. Bonk (cited n. 21) 218, argues that Duchamp fashioned the templates for the Three Standard Stoppages in 1918 when he was working on Tu m' and needed to draw their curvatures several times. This chronology would mean that he used something other than the templates, perhaps tracing paper or some other means, to draw the lines in Network of Stoppages in 1914. See also Duchamp's correspondence with Katherine S. Dreier in Affectionately, Marcel (cited n. 21) 199-207.
59. For a more detailed discussion of Duchamp's use of shadows on Tu m', see Adcock, Marcel Duchamp's Notes (cited n. 10) 41-49. For a more traditional approach, but nonetheless interesting for Duchamp's work, see Thomas Da Costa Kaufmann, "The Perspective of Shadows: The History of the Theory of Shadow Projection," Journal of the Warburg and Courtauld Institutes 38 (1975): 258-87.
62. Kemp (cited n. 36) 123-24, points out that Desargues' discussion of conic sections "helped sow the seeds of non-Euclidian geometry, but was only to be fully taken up by Poncelet in the nineteenth century. Vital steps in the development of new postulates appear to have been taken independently by Kepler and Desargues. The new geometry challenged central assumptions of Euclidian theory. Straight lines came to be interpreted as equivalent to circles which possess radiuses of infinite length, and parallel lines regarded as meeting at infinity." For the contributions of Poncelet and Kepler alluded to here by Kemp, see Jean-Victor Poncelet, Traité des propriétés projectives des figures (Paris, 1822); Johannes Kepler, Ad Vitellionem paralipomena quibus astronomiae pars optica traditur (1604); a translation of this last work is included in an appendix, "Kepler's Invention of Points at Infinity," in Field and Gray (cited n. 35) 185-88.
65. Jean-François Nicéron, Thaumaturgus opticus (Paris, 1646); Stephen Jay Gould and Rhonda Roland Shearer, "Drawing the Maxim from the Minim: The Unrecognized Source of Nicéron's Influence upon Duchamp," Tout-Fait: The Marcel Duchamp Studies Online Journal 1, no. 3 (December 2000), argue that Duchamp is very likely to have also used Nicéron's earlier French edition, which contains material not included in the Latin edition; see Jean-François Nicéron, La Perspective curieuse, ou magie artificielle des effets merveilleux (Paris, 1638) News <http://www.toutfait.com/issues/issue_3/News/stephen/stephen.html>. For an interesting discussion of the ways in which epistemological perspective can affect the interpretation of data, see David Magnus, "Down the Primrose Path: Competing Epistemologies in Early Twentieth-Century Biology," in Biology and Epistemology, ed. Richard Creath and Jane Maienschein (Cambridge: Cambridge University Press, 2000) 91-121.
66. For a discussion of Nicéron's image, see Martin Kemp, The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat (New Haven and London: Yale University Press, 1990) 210-11; Kemp does not mention Duchamp or Tu m'. Nicéron's illustration was also included in La perspective curieuse, pl. 33; see Kim H. Veltman, in collaboration with Kenneth D. Keele, Linear Perspective and the Visual Dimensions of Science and Art, Studies on Leonardo da Vinci I (Munich: Deutscher Kunstverlag, 1986) 164-65.
67. For a discussion of this painting, see Jurgis Baltruaitis, Anamorphic Art, trans. W. J. Strachan (New York: Abrams, 1977) 91-114; for interesting analyses of anamorphosis, see Fred Leeman, Hidden Images: Games of Perception, Anamorphic Art and Illusion from the Renaissance to the Present, trans. Ellyn Childs Allison and Margaret L. Kaplan (New York: Abrams, 1976); see also Kim H. Veltman, "Perspective, Anamorphosis, and Vision," Marburger Jahrbuch für Kunstwissenshaft 21 (1986): 93-117.
68. Holbein's painting is discussed by Jacques Lacan, The Seminar of Jacques Lacan, Book XI: The Four Fundamental Concepts of Psychoanalysis, ed. Jacques-Alain Miller; trans. Alan Sheridan (New York and London: W. W. Norton, 1981) 88; see also Martin Jay, Downcast Eyes: The Denigration of Vision in Twentieth-Century French Thought (Berkeley, Los Angeles, and London: University of California Press, 1994) 48, 362-64; and Tom Conley, "The Wit of the Letter: Holbein's Lacan," in Vision in Context: Historical and Contemporary Perspectives on Sight, ed. Teresa Brennan and Martin Jay (New York and London: Routledge, 1996) 45-60.
69. Dalia Judovitz, in a discussion of René Descartes's interests in both "normal" and "aberrant" perspective systems, makes a similar point about Holbein's image; see her essay "Vision, Representation, and Technology in Descartes," in Modernity and the Hegemony of Vision, ed. David Michael Levin (Berkeley, Los Angeles, and London: University of California Press, 1993) 66-67. Judovitz discusses Tu m' in her book Unpacking Duchamp: Art in Transit (Berkeley, Los Angeles, and London: University of California Press, 1995) 221-26, but does not discuss the painting's anamorphic characteristics.
71. Ibid., emphasis in the original; see also Clair's essay, "Marcel Duchamp et la tradition des perspecteurs," in Abécédaire, vol. 3, Marcel Duchamp catalogue (Paris: Centre National d'Art et de Culture Georges Pompidou, Musée National d'Art Moderne, 1977) 52-59.
73. For an early discussion of these mathematicians in the context of art history, see William M. Ivins, Jr., "Desargues and Pascal," chap. 8 in Art & Geometry: A Study in Space Intuitions (Cambridge: Harvard University Press, 1946).
74. Morris Kline, Mathematical Thought from Ancient to Modern Times (New York: Oxford University Press, 1972) 285-301, 834-60; see also idem, "Projective Geometry," in Mathematics in the Modern World, Readings from Scientific American, ed. Morris Kline (San Francisco and London: W. H. Freeman, 1968) 122-27.
75. For an accessible source that refers to Poincaré in these terms, see Albert W. Tucker and Herbert S. Bailey, Jr., "Topology," in Mathematics in the Modern World, Readings from Scientific American, ed. Morris Kline (San Francisco and London: W. H. Freeman, 1968) 134-40.
78. The complexities of the four-dimensional continuum are suggested by the following passage from the only note in the Green Box with a specific reference to a higher space (Duchamp's term is "étendue 4" in the original French): "As there is gradually less differentiation from axis to axis, i.e., as all the axes gradually disappear in a fading verticality, the front and the back, the reverse and the obverse acquire a circular significance: the right and the left, which are the four arms of the front and the back, melt along the verticals. The interior and exterior (in a four-dimensional continuum) can receive a similar identification." See Duchamp, Salt Seller (cited n. 2) 29; idem, Duchamp du Signe (cited n. 18) 45.
79. A modern way of putting this matter would be to say: "Two planes having a common point have at least one more common point. If this is satisfied, the space must be three-dimensional; if it is not satisfied, so that there are two planes with a unique common point, then the space is at least four-dimensional." Encyclopaedia of Mathematics, s.v. "Higher-Dimensional Geometry," by A. D. Aleksandrov. For a more sophisticated definition, see H. S. M. Coxeter, Introduction to Geometry (New York and London: John Wiley & Sons, 1961) 185-86.
80. There are a large number of possibilities. One of the textbooks that I have on my shelves begins with the following statement: "From the beginnings of geometry until well into the nineteenth century it was almost universally accepted that the geometry of the space we live in is the only geometry conceivable by man. This point of view was most eloquently formulated by the German philosopher Immanuel Kant (1724-1804). Ironically, shortly after Kant's death the discovery of non-Euclidean geometry by Gauss, Lobachevski, and Bolyai made his position untenable. Today, we study in mathematics not just one geometry, or two geometries, but an infinity of geometries." Albrecht Beutelspacher and Ute Rosenbaum, Projective Geometry: From Foundations to Applications (Cambridge: Cambridge University Press, 1998) 1.
82. A generalized mathematical "surface" is a "manifold" and can have any number of dimensions. It can also have any number of curvatures. This important way of thinking about geometrical configurations is due to Bernhard Riemann (1826-1866) and is customarily referred to as "Riemannian Geometry." This sense of "Riemannian Geometry" can be distinguished from the sense used to refer to his prior invention of a specific (ungeneralized) non-Euclidean geometry with constant positive curvature, customarily referred to as "Riemann Geometry" or elliptical geometry; see Peter Petersen, Riemannian Geometry (New York: Springer-Verlag, 1998). In a questionaire about the Three Standard Stoppages in the archives of the Museum of Modern Art in New York dated 1953 (the year the work entered their collection), Duchamp said that the assemblage was "a humorous application of Riemann's post-Euclidean geometry which was devoid of straight lines" (see Naumann, cited n. 12, p. 170). That Duchamp used the term "post-Euclidean," rather than simply "non-Euclidean," indicates that he may very well have been sophisticated enough to have understood the distinctions under discussion here.