Dedekind Cut
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Antonio Sanvico (1925-2000)
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Antonio Sanvico (1925-2000) |
Dedekind Cut
Milan Golob; Paintings and Duchamp (Fart & Languidness, 10,5×21 cm), 1994, Gallery Škuc, Ljubljana.
Linda Dalrymple Henderson
Marcel Duchamp and the New Geometries (Part 4)*
Duchamp and the Laws of Four-Dimensional Geometry
Under "The Continuum" are blended Duchamp's notes on four-dimensional geometry (including four-dimensional vision) and his later thoughts on the way the fourth dimension might be represented. Two notes in the section on "Perspective" have already been singled out as belonging properly with the notes on four-dimensional geometry.[85] On the other hand, the final drawing and two notes at the conclusion of "The Continuum" are largely independent of the rest of the serious thoughts on geometry expressed in this section. Lacking a mathematical basis, they are more typical of the "playful physics" of the Green Box or of earlier notes in A l'infinitif.[86] Since Duchamp's ultimate solution for representing" the fourth dimension was to depend largely upon mirrors and shadows, a temporary disregarding of the notes under "The Continuum" that deal with mirrors (as well as those previously withdrawn for inclusion in earlier phases of his research) results in a coherent section on four-dimensional geometry.
During the course of his study of n-dimensional geometry Duchamp quickly began interpreting it for his own uses, so that only a few of the notes are simple records of theorems of four-dimensional geometry. Those that do fall into this category of straight, forward geometry deal with "Angles."[87] His interest in the "hinge plane" created by the intersection of two spaces in a four-dimensional angle also led Duchamp to consider the generation of four-dimensional hypersolids by rotating a space about such a hinge plane, just as a plane rotating about a line produces a three-dimensional- space.[88]
While these notes might have been drawn from any text on four-dimensional geometry, Duchamp's speculations on the intersections of geometrical figures are much more personal and a number of them are not geometrically true as written. However, certain of these statements appear to reflect Duchamp's attempt to establish the way in which four-dimensional vision would operate. When these notes are read in the light of the first note of "The Continuum" in which Duchamp sets out these "rules," they gain a conditional validity. Duchamp's scheme for four-dimensional vision equates the four-dimensional eye's perception of a geometric figure with the human eye's perception of the figure of the next lower degree. He writes in the first note under "The Continuum,"
On the 4-dim'l vision.
In the 4-dim'l continuum the plane is always seen as a line. It has no more
perspective development.
The line is seen as a point.
Express how a volume is seen. (Define this perception of the whole.)
The 3-dim'l body when seen in the 4-dim'l continuum is seen as a whole.
(does it have a reverse and a front side like the plane seen in space?)[89]
Thus, Duchamp's notes on the figures produced by intersections must be prefaced by the condition, "As seen by the four-dimensional eye." Only then can such statements as the following be accepted as true: "In a continuum, 2 straight lines intersecting do not determine a plane-they merge (parallel lines also merge). In a continuum - the plane reduces to its own section: the straight line."[90] Without this qualification, Duchamp would seem to be making the mistaken suggestion that the geometry of the four-dimensional continuum is necessarily non-Euclidean. Although new definitions of perpendicularity and parallelism are necessary once a fourth dimension is added to space, four-dimensional geometry remains basically Euclidean and parallel lines do not merge.[91]
The source for Duchamp's notion of the dimension, reducing characteristic of four, dimensional vision was undoubtedly Jouffret. In the Traite elementaire Jouffret had explained the way in which different dimensions interrelate: ". . . Our space is only an elementary slice out of the four-dimensional continuum that surrounds it on every side. From the point of view of the fourth dimension, it is infinitely thin and absolutely flat, and this is true for every entity it contains."[92] Jouffret was, of course, speaking only figuratively: three-dimensional space remains three-dimensional within the four, dimensional continuum. However, the addition of a fourth perpendicular allows a single, all-encompassing view of the space, analogous to the sphere's total perception from his vantage point above Flatland. Duchamp, however, seems to have taken Jouffret's analysis literally.
Another aspect of Duchamp's individualized view of the four-dimensional continuum is his fascination with the sphere as the basic element of the "four-dimensional line." In this case, he is reversing his approach of reducing the number of dimensions of an object in order to imagine a four-dimensional view of it. By thinking of a point as a sphere, he has raised its number of dimensions to three. Duchamp's idea of a "four, dimensional line" produced by a series of nesting spheres of increasing radii[93] may have been derived from some popular contemporary source, but he recognizes its futility in another note: "This straight 4-dim'l line = 3-dim'l space and does not get out of that space."[94]
Actually, Duchamp's notion of a "four' dimensional line" is faulty, for to a geometer a line in the four, dimensional continuum is the same line that we know from our plane or solid geometry. Nevertheless, the artist's speculation on the repetition of three, dimensional spheres along a line into the fourth dimension did suggest a means of producing a four, dimensional continuum. This idea was to figure in Duchamp's final formulation of the nature of the continuum, which involved the elemental parallelism of virtual images, and the concept of the Dedekind cut. As Duchamp writes, "Starting with a sphere (having a radius of 10 cm. for ex.) This sphere, 3-dim'l continuum, is a Dedekind cut in a n-dim'l continuum composed of the virtual images of that sphere repeated to infinity, i.e. composed of the positions in a 3-dim'l continuum which this sphere can occupy, and of all the unities that."[95]
Duchamp had explored the process of generating geometrical figures of the next higher degree in his motion studies of 1911 and 1912. Now in the notes for the Large Glass he returned to the notion in a purely geometrical context, as a variant on the rotative generation that hinges could produce. He writes,
Elemental parallelism: repetition of a line equivalent to an elemental line (in the sense of similar at any point) in order to generate the surface. Same parallelism when passing from plane to volume: Sort of parallel multiplication of the n-dim'l continuum, to 'form the n + 1 dim'l continuum. . . .
Will the passage from volume to 4-dim'l figure be produced through parallelism? Yes. But this elemental parallelism being a geometric process requires an intuitive knowledge of the 4-dim'l continuum. . . . A representation of the 4-dim'l continuum will be realized by a multiplication of closed volumes evolving by elemental parallelism along the 4th dimension. Of course one still has to define by intuitive knowledge the "direction" of this 4th dim. Hence the mirror?[96]
If elemental parallelism had failed Duchamp in works such as Nude Descending a Staircase in 1912, in its purer, geometrical context it became the key to his visualization of the four, dimensional continuum. Poincare provided the scholarly theory for Du, champ's synthesis with his discussions of Dedekind cuts in La Science et l'hypothese and of cuts and continua in a chapter of Dernieres Pensees, "Pourquoi l'espace a trois dimensions." This final book of Poincare was published in 1913, after his death, but "Pourquoi l'espace a trois dimensions" had also appeared in the Revue de Metaphysique et de Morale in 1912.[97]
The notion of cuts was essential to Poincare's final discussion of the number of dimensions of space, which, in contrast to his earlier views, affirmed the three-dimensionality of space. However, the Dernieres Pensees chapter on dimensions offered more than just the conclusion that because our space can be completely divided or cut by a two-dimensional surface, it must have three dimensions. From this discovery Poincare had arrived at a successful way of defining the n-dimensional continuum, a definition Duchamp was to utilize in his own researches. According to Poincare,
We know now what a continuum of n dimensions is. A continuum has n-dimensions when it is possible to divide it into many regions by means of one or more cuts which are themselves continua of n-1 dimensions. The continuum of n dimensions is thus defined by the continuum of n-1 dimensions.[98]
Duchamp's note involving the three-dimensional sphere as a Dedekind cut of the n-dimensional continuum was quoted above. Two other notes also relate to Poincare and to Dedekind cuts. The first reasons about cuts in two, three, and four dimensions:
In the plane, the 2-dim'l native is either on one side of an infinite straight line or on the other side. Therefore this line is for him a Dedekind cut (Poincare) creating 2 distinct plane fields.
-In the 3-dim'l space we must cross a plane in order to know its 2 sides.
-In the 4-dim'l continuum a 4-dim'l body will be such that if sectioned by a median space3 the 2 separate 4-dim'l parts will be symmetrical about this median space.[99]
Duchamp's last statement involving Poincare and Dedekind cuts sums up his geo, metrical exploration and demonstrates again the important role that "virtual images" were coming to play in his thoughts on the fourth dimension:
Also the 4-dim'l continuum: Poincare's explanation about n-dim'l continuums by means of the Dedekind cut of the n -1 continuum is not in error. It is on the contrary confirmed and it is even by basing oneself on this explanation that one can justify the name of 4th dimension given to the continuum of virtual images in which the Dedekind cut could only be obtained by means of the 3-dim'l prototype object considered in its geometric infinity.[100]
The Final Solution: The Mirror of the Fourth Dimension
From his highly theoretical notion of the repetition of the virtual images of an object to form a four-dimensional continuum, Duchamp returned in the end to the more practical problem of representing a four-dimensional Bride. The idea of a mirror mentioned but not developed in his early "Use transparent glass and mirror" note was now to playa central role. One of the last notes of "The Continuum" reveals this new connection of "virtual image" with mirrors:
Virtuality as 4th dimension. Not the Reality in its sensorial appearance, but the virtual representation of a volume (analogous to the reflection in a mirror).[101]
During the course of his exploration of geometry and four, dimensional vision, Duchamp had kept in mind how these higher dimensional figures would appear to a three-dimensional eye as well. One of his notes on the "four' dimensional line" had continued on the subject of human vision:
For the ordinary eye in a space any point is the end of a line (whether straight or not) coming from a continuum4. The eye could go endlessly around the point (in the 3-dim'l), it will never be able to perceive any part of this 4-dim'l line other than the point where it meets the 3-dim'l medium.
In a subsequent note, he writes,
It is certain that every point of space conceals, hides, is the end of the line of the continuum. One would like to go around this point and perceive this 4th direction which comes (at this point) into contact with space3-
A line of space also conceals a plane; it is like the section of this plane, the only one visible to the eye.[102]
For Duchamp, it was the mirror that offered this same curious property of being the "border line of the 4th direction." As he writes in one of the three notes which form his final statement in A l'infinitif on the representation of the fourth dimension,
Analogy between:
Reflection in a plane mirror and the 3-dim'l section of a 4-dim'l body by a 3-dim'l
space-
For an ordinary eye, a point in space hides, conceals the 4th direction of the continuum-which is to say that this eye can try to perceive physically the 4th direction by going around the said point. From whatever angle it looks at this point, this point will always be the border line of the 4th direction.
-Just as the ordinary eye going around a mirror will never be able to perceive
anything but the reflected 3-dim'l virtual image and nothing from behind-[103]
The next two notes of "The Continuum" are perhaps the most difficult of all of Duchamp's notes to decipher. However, they hold the key to his interest in the strangely self, sufficient space within a mirror:
The plane of the mirror is a convenient way of giving the idea of 3-dim'l infinite space. It is at this plane that the 3-dim'l infinity stops. (There is no contradiction in putting it this way since it is only to familiarize the mind with the ideal representation of the 4-dim'l continuum).
Incorrectly speaking, the line which seems to stop at the mirror's plane should simply cross through and continue to infinity in its own 3-dim'I continuum. It would not enter the 4-dim'l continuum which would contain the line without being cut by it (as a point is contained in a plane without cutting it).Comparison. From: A point does not cut a 3-dim'l continuum; a line does.
By analogy, given a cube-its reflection in a mirror--one could say that a straight line perpendicular to the plane of the mirror wilt not cut the cube's image (will not hide the cube's image). Because the eye goes around the line without thick, ness. This line will stop at the mirror's plane.--On the contrary, a plane or any opaque surface touching the mirror will cut or hide from the observer's eyes a part of or the whole image of the cube in the mirror. The 4-dim'l continuum is essentially the mirror of the 3-dim'l continuum.[104]
Duchamp was to use the mirror in two distinct ways. Here he concludes, "The 4- dim'l continuum is essentially the mirror of the 3-dim'l continuum." And yet, just as we normally think. of a mirror image as having one less dimension than the object reflected, Duchamp in "Appearance and Apparition" treated the mirror image of an object as its "mold" in n - 1 dimensions.[105] White these interpretations are exactly opposite, the magic of the mirror as the symbol of the four-dimensional continuum must have carried over to Duchamp's more frequent use of it as the two-dimensional bearer of a shadow with three virtual dimensions.
In these last two notes of A l'infinitif Duchamp again demonstrates that he had put away his geometry texts in favor of personal geometrical reasoning by the process of analogy. This is the context in which mathematical errors creep into Duchamp's notes. Just as his ideas on the figures that result from geometrical intersections could only be accepted if interpreted in terms of four-dimensional vision, these last two notes contain statements that are not strictly true in geometry.
The problem in this case stems from Duchamp's use of the word couper. In his notes based on Poincare's discussion of Dedekind cuts, Duchamp demonstrates his correct understanding of a cut as the n – 1-dimensional form that can completely divide a form of n dimensions. Thus, a line "cuts" a plane, a plane "cuts" a three-dimensional space, and a three-dimensional space "cuts" the four-dimensional continuum.
The second of the two notes just quoted, however, contradicts this mathematical truth: "A point does not cut a three-dimensional continuum; a line does." While the first half of his assertion is correct, the second half is not. A line is not one but two dimensions less than a three-dimensional space. Duchamp repeats this error in a note included in A l'infinitif under "Perspective," in which only his first assertion is correct: "-Just as a point cuts a line and does not cut a plane, so a line of infinite length or surface element cuts a volume and does not cut a 4-dim'l 'solid.' But the plane or surface cuts this 4-dim'l solid."[106] Although it is true that a line does not cut a four-dimensional "solid," that same line does not cut a volume, nor does a plane cut a four-dimensional solid. Once again, Duchamp is dealing with forms in the relation n:n - 2 instead of n:n - 1.
It is hard to believe that Duchamp was confused on this relatively straightforward issue, for which he had Poincare's clear explanation. Instead, he seems to have been using the word couper in a slightly different sense.[107] Couper in the quotations above implies the notion "to cross" in the sense of entering and exiting from a higher dimensional figure. It was the interaction of a line and a four-dimensional figure that most concerned Duchamp. As a result of his literal reading of Jouffret's dimension-reducing explanation of four-dimensional vision, he seems to have thought of the four-dimensional continuum as a space that would virtually swallow up the one-dimensional line we know. Thus, in Duchamp's thinking a line would be totally surrounded, like a point in a plane or in a three-dimensional space. A plane, on the other hand, could (in Duchamp's view) traverse the four-dimensional continuum. Duchamp's speculation on the need for a "four-dimensional line" of greater volume than the ordinary one-dimensional line stems from this personal reinterpretation of geometry.
With the qualification that the geometry involved in these later notes bears the stamp of Duchamp's own thinking, these passages do represent an ingenious attempt to prove the four-dimensionality of a mirror. The reasoning that the interior of a mirror is four-dimensional is clearer in the second of the two notes. Duchamp begins by setting up his condition that a point does not cut [cross] a three-dimensional continuum whereas a line does. By analogy, he reasons, a line does not cut [cross] a four-dimensional continuum but a plane does.[108] The "experimental proof" to demonstrate the number of dimensions within the mirror is carried out with the image of a cube in the mirror. For this situation Duchamp defines "to cut" as "to hide the cube's image," with the result that a line does not hide the image while a plane does. Therefore, according to the artist's personal brand of logic, the image that is not cut by a line must be four-dimensional.
The purpose of the first of the two notes was to emphasize the independence of the space within a mirror. Here Duchamp's rule that a line does not cut a four-dimensional continuum is again central to his argument, for it alone can explain the way in which a line perpendicular to the surface of the mirror cannot enter the mirror space. Instead, the line "slips through" the mirror, as if it were a point, and continues on in its own three-dimensional space.
* Linda Dalrymple Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art (Princeton: Princeton University Press, 1983), Chapter 3, "Marcel Duchamp and the New Geometries," pp. 145-163.
Marcel Duchamp and the New Geometries (Part 5)
Notes:
[85] The first of these is the five-line note beginning "-Just as a point cuts a line ..." (Salt Seller, p. 89). The second note commences, "By analogy the flat being2 has a length" (ibid., p. 90).
[86] For the issue of gravity, mentioned in this note, see n. 122 below.
[87] See Salt Seller, note starting at the bottom of p. 92 and continuing at the top of page 93. Also, see the short note beginning "Angles4:" on p. 96.
[88] Ibid., the bottom half of p. 96 and the top half of p. 97.
[89] Duchamp, A l'infinitif, in Salt Seller, p. 90.
[90] Ibid., p. 93. See also the continuation of these notes on intersections on the top half of p. 94.
[91] The consideration of points, lines and planes "at infinity" leads to a Riemannian (elliptic) non-Euclidean geometry. This geometry at infinity can also be employed in higher dimensions, as Manning demonstrates in Sections 131-133 of Geometry of Four Dimensions. However, it is doubtful that Duchamp was deeply enough involved in n-dimensional geometry to have discovered this possible non-Euclidean variant on it. Rather, his non-Euclidean sounding statements seem to stem from his personal ideas on how four-dimensional vision might operate.
[92] Jouffret, Traite elementaire, pp. 183-84. Adcock first noted the relevance of Jouffret's views on such slices [tranches] in "Marcel Duchamp's Notes," ch. 3.
[93] See the note that begins "sph sph sph sph," in A l'infinitif (Salt Seller, p. 91).
[94] Duchamp, A l'infinitif, in Salt Seller, p. 95. Duchamp's discussion seems related to Boucher's suggestion in his Essai sur l'hyperespace that "the section of a 4-dimensional sphere of radius R by our space is evidently a sphere whose apparent radius varies from o to R" (p. 137).
[95] Duchamp, A l'infinitif, in Salt Seller, p. 97. This note ends abruptly in this way.
[96] Ibid., p. 92. "Elementary parallelism" is mentioned in Duchamp's note from "Appearance and Apparition" quoted above (ibid., p. 85), and in the Green Box note closely associated with it (ibid., p. 70).
[97] Poincare does not actually use Dedekind's name for the cuts he discusses in his topological analysis of continua in Dernieres Pensees. However, in La Science et l'hypothese he had already established the connection of such cuts with the work of the mathematicians of the Berlin school, Dedekind and Kronecker, in number theory (La Science et l'hypothese, ch. 2).
[98] Poincare, Dernieres Pensees, p. 67; Last Essays, p. 29.
[99] Duchamp, A l'infinitif, in Salt Seller, p. 94.
[100] Ibid.,p. 98. In La Gueridon et /a virgule (pp. 58- 59) Suquet has restated his theory that the three-dimensional mirrored strips of glass that originally separated the upper and lower halves of the Large Glass represented a three-dimensional "cut" of two four-dimensional spaces. Duchamp, however, was seeking to make only the Bride's domain four-dimensional.
[101] Duchamp, A l'infinitif, in Salt Seller, p. 99.
[102] Ibid., p. 95.
[103] Ibid., p. 91.
[104] Ibid., pp. 91-92;, couper and, ligne translated' as "cut" and '~line" instead of Salt Seller's "intersect" and "curve."
[105] Ibid., p. 85.
[106] Ibid., p. 89, with variation in translation, described in n. 104 above.
[107] That Duchamp's interpretation of couper was different is confirmed by the way in which he and Cleve Gray chose to translate the word in the printed text for A l'infinitif. Throughout the passages cited in nn. 104 and 106 above and n. 109 below, couper is translated as "to intersect." Actually, the use of "intersect" in these passages causes even greater mathematical problems. "Intersection" as such can occur between figures of any dimensions (including a point and an n-dimensional continuum), since the term simply means that the figures have a point (or points) in common.
[108] Duchamp contradicts this notion for the case of four-dimensional vision: "The 4-dim'l eye is such that an infinite plane surface (for ex.) does not cut the continuum" (A l'infinitif, in Salt Seller, p. 94, with translation variation described in n. 104 above).