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Fourth Dimension


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Fourth Dimension

Alice Laurence Ives (1874-1880)
2012, oil on canvas, 31×31 cm

Fourth Dimension

Alice Laurence Ives (1874-1880)

Fourth Dimension




Linda Dalrymple Henderson
Marcel Duchamp and the New Geometries (Part 1)*

Marcel Duchamp's statements late in life on the degree of his involvement with the fourth dimension and non- Euclidean geometry have left a curiously contradictory record. Robert Lebel wrote in his 1959 monograph on Duchamp, "Marcel, on his part, had drawn his own conclusions from the works of Lobachevsky and Riemann."[1] Yet, the attitude Duchamp expressed in subsequent interviews in the 1960s was hardly that of one who had seriously studied the new geometries.[2] For example, in Pierre Cabanne's 1967 Entretiens avec Marcel Duchamp, Duchamp portrays himself as a mere dabbler in higher dimensions and does not even mention non- Euclidean geometry:

Cabanne: On this scientific side, you have considerable knowledge. …

Duchamp: Very little. I never was the scientific type.

Cabanne: So little? Your mathematical abilities are astonishing, especially since you didn't have a scientific upbringing.

Duchamp: No, not at all. What we were interested in at the time was the fourth dimension. In the "Green Box"[3] there are heaps of notes on the fourth dimension.

Do you remember someone called, I think, Povolowski? He was a publisher, in the rue Bonaparte. I don't remember his name exactly. He had written some articles in a magazine popularizing the fourth dimension, to explain that there are flat beings who have only two dimensions, etc. It was very amusing, appearing at the same time as Cubism and Princet.

Cabanne: Princet was a fake mathematician-he too practiced irony. . . .

Duchamp: Exactly. We weren't mathematicians at all, but we really did believe in Princet. He gave the illusion of knowing a lot of things. Now, I think he was a high-school math teacher. Or in a public school.

In any case, at the time I had tried to read things by Povolowski, who explained measurements, straight lines, curves, etc. That was working in my head while I worked, although I almost never put any calculations into the "Large Glass." Simply, I thought of the idea of a projection, of an invisible fourth dimension, something you couldn't see with your eyes.[4]

Distance from the idealistic Cubist era in which the fourth dimension and the new geometries had been such major concerns, as well as simple lapses of memory on Duchamp's part, may explain the tone of his discussion with Cabanne. Duchamp, the close friend of Princet,[5] certainly knew that he was an insurance actuary, but here remembers him as a "high-school math teacher." It is Cabanne, influenced by later literature on Cubism, who suggests that Princet was a "fake mathematician." Nevertheless, Duchamp does agree with Cabanne that Princet "practiced irony," and in an earlier section of their interview Duchamp had described Princet as "an extraordinary being" who "played at being a man who knew the fourth dimension by heart."[6] As previous chapters have demonstrated, however, such a wide variety of notions were encompassed by "the fourth dimension" that Princet could have been a legitimate mathematician and still have only been able to "play at being a man who knew the fourth dimension by heart." Most importantly, Duchamp affirms that "we really did believe in Princet."

Over the years Duchamp's final, more simplified explanation of the fourth dimension in terms of a three-dimensional projection or shadow may have lessened in his mind the significance of his own geometrical researches during that period, as well as those of Princet and the Cubists. In the section of the conversation quoted above, Duchamp credits himself with hardly any mathematical ability and seems almost to be "practicing irony" on his part. Discussing the popular articles by "Povolowski," he humbly states, "I had tried to read things by Povolowski, who explained measurements, straight lines, curves, etc."

The author whom Duchamp remembered as "Povolowski" was none other than Gaston de Pawlowski, author of Voyage au pays de la quatrieme dimension and the single major source of literature on the fourth dimension in Paris during 1910, 1911, and 1912. Pawlowski's importance for the Puteaux Cubists, established in Chapter 2, confirms this identification. Furthermore, Jean Clair in Marcel Duchamp ou le grand fictif has established a number of convincing analogies between aspects of Pawlowski's science fiction and Duchamp's terminology in his notes for the Large Glass.[7] The similarity of the Frenchman Pawlowski's name to that of Jacques Povolozky, a Russian publisher established at 13 rue Bonaparte, must have caused Duchamp's confusion.[8]

Even when his "Povolowski" is identified as Gaston de Pawlowski, however, several discrepancies remain in Duchamp's discussion. The idea that "there are flat beings who have only two dimensions," in the tradition of Abbott and Hinton, does not playa major role in Pawlowski's science-fiction tales of the fourth dimension. Basing himself on Poincare, Pawlowski merely mentions in passing the two-dimensional sphere dwellers posited by Helmholtz.[9] Nor does Pawlowski explain "measurements, straight lines, curves, etc.," as Duchamp recalled.

The answer perhaps is that Duchamp, remembering Pawlowski's importance for the Cubist painters, linked his name with subject matter that he himself derived from other sources, particularly the writings of Jouffret and Poincare. Jouffret's Traite elementaire de geometrie a quatre dimensions gives a detailed exposition of a two-dimensional world, citing both Hinton and Abbott's Flatland.[10] Poincare discusses the case of two dimensions in a variety of contexts, including Helmholtz's arguments about the geometry that would be established by two-dimensional beings on a sphere, and the topological question of continua and cuts.[11] And, in fact, the notes that Duchamp began making in 1912 for The Bride Stripped Bare by Her Bachelors, Even (The Large Glass) establish conclusively that he did read Jouffret and Poincare.

In one of his notes for the Large Glass Duchamp cites Jouffret in connection with an idea basic to his final conception of the Bride of the Large Glass: "The shadow cast by a 4-dim'l figure on our space is a 3-dim'l shadow (see Jouffret 'Geom. a 4 dim.' page 186, last 3 lines)."[12] His references to Poincare are more complex and suggest the more scholarly, geometrical reasoning to be found in a number of the notes for the Large Glass. On the subject of "Dedekind cuts," Duchamp asserts of a two-dimensional being in a plane, "Therefore this line is for him a Dedekind cut (Poincare) creating 2 distinct plane fields." Of the analogous situation in a four-dimensional continuum, Duchamp notes that "Poincare's explanation about n-dim'l continuums by means of the Dedekind cut of the n - 1 continuum is not in error.[13]

Duchamp's references to Jouffret and Poincare occur in a group of notes for the Large Glass that were rediscovered by Duchamp only in 1964 and were published in 1966 as A l'infinitif (The White Box).[14] Before the publication of A l'infinitif Duchamp's reputation had been based largely upon an earlier collection of notes, the Green Box of 1934, which contained few references to geometry. Even the final collection of Duchamp' s Notes, published in 1980 does not equal A l'infinitif in geometrical interest.[15] Indeed, A l'infinitif reveals a Duchamp quite different from his traditional image as a Dadaist practitioner of black humor. It is not at all difficult to visualize this scholarly Duchamp grappling with the ideas of Lobachevsky and Riemann.

The product of the union of Duchamp's ironical outlook and his scholarship was a "playful physics" created by "slightly distending the laws of physics and chemistry."[16] As Duchamp himself explained his dual role to Cabanne, "I was interested in introducing the precise and exact aspect of science, which hadn't often been done, or at least hadn't been talked about very much. It wasn't for love of science that I did this; on the contrary, it was rather in order to discredit it, mildly, lightly, unimportantly. But irony was present."[17]

The parallel between Duchamp's "playful physics" and Jarry's pataphysics is unmistakable: an irreverent artist for whom nothing is sacred delves into avantgarde mathematics and science in secret so as to discredit longstanding beliefs still held by the majority of the public. The new geometries were an ideal tool for "slightly distending" the laws of science, for although they had actually existed for over fifty years, serious information about the n-dimensional and non-Euclidean geometries and their possible scientific ramifications was only gradually becoming available to nonspecialists in this period. In the minds of most people, conditioned by the overriding positivism of the nineteenth century, the world was still clearly three-dimensional and Euclidean. How revolutionary and subversive, then, the fourth dimension and non- Euclidean geometry must have seemed to Duchamp and Jarry. Yet both men retained their ironical distance and neither professed absolute loyalty to the new geometry and science, knowing that its truth, too, was relative and would someday have to be "slightly distended" as well.

Duchamp's admiration for Jarry is well known,[18] and the latter's Gestes et opinions du docteur Faustroll must have had a profound impact upon Duchamp on its first appearance in 1911. However, Duchamp was to go much deeper into the new geometries than had Jarry. While n-dimensional geometry figures in only one chapter of Gestes et opinions and Jarry's discussion of non- Euclidean geometry in his "Commentaire pour servir a la construction pratique de la machine a explorer Ie temps" is relatively brief, Duchamp has left a detailed record of his ponderings on the new geometries. From the evidence available, it seems that Duchamp spent much more time than Jarry patiently working out the applications of the new geometries for his art. Jarry, on the contrary, explored a wider range of contemporary science than Duchamp, but he then quickly turned his initial findings to humorous purposes.

Duchamp's intellect is laid bare in the sections of A l'infinitif on ·"Perspective" and "The Continuum." In these notes Duchamp was attempting to work out the mechanics of portraying the fourth dimension, and his seemingly ever-present ironical stance disappears temporarily. Here is the scholar Duchamp, laboring in private to do the necessary background research and reasoning for a work that would astound and mystify the public. Decades later, the notes themselves possess an air of mystery and require of the reader a preparation similar to Duchamp's in order to penetrate their forbidding exterior.

Duchamp's Introduction to n-Dimensional and Non-Euclidean Geometry

While the fourth dimension may have been brought to Duchamp's attention by popular literature like that of Pawlowski and perhaps even by his brothers' neighbor at Puteaux, Kupka, the connection of the fourth dimension with Cubism would only have been discovered once Duchamp met members of the Cubist circle around Met~ zinger and Gleizes. For the more mathematical side of the new geometries and an introduction to the writings of Jouffret and Poincare, Duchamp, like the Cubists themselves, was probably indebted to Princet, or, initially, to secondhand accounts of his ideas on Cubism and n-dimensional geometry. Once the fourth dimension became an artistic interest at Puteaux, however, the hyperspace philosophy with which Kupka may have already been acquainted through Theosophical literature would also have taken on a new relevance for Duchamp.

In the spring of 1911 Duchamp, along with his brothers, Jacques Villon and Raymond Duchamp-Villon, and certain other members of the Societe Normande de Peinture Moderne, had seen the first public manifestation of Cubism in salle 41 of the Salon des Independants. Although Duchamp-Villon was a member of the Salon d' Automne hanging committee that placed works by members of both groups together in salle 8, few of the painters had actually met those in the other group before the Salon d' Automne itself. Gleizes, for instance, states in his memoirs that he did not know Duchamp and Villon until the conclusion of the Salon d' Automne. Duchamp confirmed Gleizes's view in the course of his conversations with Cabanne, asserting, "At the end of 1911 I met Gleizes, Metzinger, and Leger, who moved in the same circle."[19]

After the closing of the Salon d'Automne on 8 November, the two groups began to meet jointly for their discussions at Puteaux and Courbevoie. With such nonartist friends of the Gleizes and Metzinger circle as Princet and Apollinaire invited to the gatherings, Duchamp would soon have met Princet. The fourth dimension must have quickly become a shared concern, for at the subsequent joint exhibition at the Galerie d'Art Ancien et d'Art Contemporain (20 November to 16 December 1911) Apollinaire gave his first lecture discussing the new art in relation to a fourth dimension.

Although Gleizes has indicated that he did not meet Duchamp and Villon until after the Salon closed in early November, it is probable nevertheless that others in the two groups were introduced to one another during the course of a Salon in which their works hung together. In addition, La Fresnaye and others may also have served during the summer of 1911 as channels for the exchange of ideas between the Mercereau circle and the Duchamp family. By October 1911 Duchamp had very likely heard from one of Princet's artist friends about Jouffret' s Traite elementaire de geometrie a quatre dimensions. In that month he began making drawings for his highly subjective Portrait of Chess Players and the metaphor of chess had been employed by Jouffret himself in his introductory discussion of the possible reality of the fourth dimension. If Duchamp did not know of Jouffret when he commenced his studies, he surely did so when he painted the final version in December 1911. By that time the parallels with Jouffret would have been obvious to Princet and to geometry-oriented Cubists like Metzinger, who had read Jouffret.

Marcel Duchamp; Portrait of Chess Players, 1911, oil on canvas, 101 x 108 cm

Marcel Duchamp; Portrait of Chess Players, 1911, oil on canvas, 101 x 108 cm

In his introduction to the Traite elementaire, Jouffret explains his opinion that, despite the great mathematical value of n-dimensional geometry, it is virtually impossible to "see" in one's mind the n-dimensional bodies that are its subject matter. He nevertheless recounts Hinton's assertion to the contrary and compares this marvelous faculty to that of a blindfold chess player. Jouffret begins with a quotation from Hinton:

There is really no more difficulty in conceiving four-dimensional shapes, when we go about it in the right way, than in conceiving the idea of solid shapes, nor is there any mystery at all about it.

When the faculty is acquired, or rather when it is brought into consciousness, for it exists in every one in imperfect form-a new horizon opens. The mind acquires a development of power, and in this use of ampler space as a mode of thought, a path is opened by using that very truth which, when first stated by Kant, seemed to close the mind within such fast limits. Our perception is subject to the condition of being in space; but space is not limited as we at first think.

Jouffret himself then continues,

Some rare chess players have the ability to conduct several games simultaneously without looking. Without even a piece of paper to make notes, the player who accomplishes this tour de force is placed so as not to see his opponents, who themselves are placed before as many chessboards; an intermediary makes the moves on the chessboard which are ordered by the single player and tells him the responses ....

. . . How does his mind work? To this question some respond that it is a matter of memory and method; almost all say that they visualize a chessboard with players, drawn in their thought as in an interior mirror, according to the expression of Taine; they will make you a sketch of it, if you desire. . . .

M. Poincare has said, no doubt ironically: "One who devoted his life to it could perhaps eventually be able to picture the fourth dimension."

For our part, we have already stated our opinion. It is that the reader should not hope to objectify, as does the blindfolded chess player with the pieces on his mental chessboard, either the four-dimensional beings which are the object of this study, or the movements that we impart to them; he would exhaust his intelligence in vain efforts seeking to break through the infinitely thin plane [la tranche infinitesimale], which extends between those beings and himself. If there are really four dimensions, our mind is confined in the first three.[20]

The Cubists had adopted Poincare's opinion on this matter rather than Jouffret's more pessimistic view. For Duchamp, who was an enthusiastic chess player himself and who in 1910 had already painted a quite naturalistic Chess Game (Philadelphia Museum of Art, The Louise and Walter Arensberg Collection), a reinterpretation of the chess theme in a four-dimensional Cubist idiom would have been an exciting challenge. Because Princet, too, was a chess player, Duchamp's effort may have quickly established a bond between the two men and a grounding for their developing friend, ship.[21]

The six preparatory drawings, executed in October of 1911, and the painted sketch of November, December 1911 (Musee Nationale d'Art Moderne, Paris) contribute important clues to the iconography of the final Portrait of Chess Players.[22] While the definitive version shows chessmen located only on a plane between the heads of the two players and grasped in one player's hand, the earlier drawings and the painted sketch are pervaded by a multitude of chessboards, chessmen, and even additional heads seemingly representing the "minds" of the players at various stages of the game. The subject of Portrait of Chess Players is, in fact, the mental processes involved in a chess game, the succession of psychological states of the players. Although it is not a blindfold match that is being played, Duchamp has succeeded in portraying the "mental chessboard" and "interior mirror" of the chess player described by Jouffret.[23]

* 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. 118-126.

Marcel Duchamp and the New Geometries (Part 2)

Marcel Duchamp and the New Geometries (Part 3)

Marcel Duchamp and the New Geometries (Part 4)

Marcel Duchamp and the New Geometries (Part 5)


[1] Lebel, Marcel Duchamp, trans. George Heard Hamilton (New York: Grove Press, 1959), p. 27. Although Duchamp gave the impression that he had actually read the books of Lobachevsky and Riemann in an interview with William Camfield on 4 Apr. 1961 (Camfield, "La Section d'Or," p. 141), he was more specific (and self-deprecatory) the next month in a letter to Serge Stauffer. Writing on 28 May 1961 in answer to Stauffer's question on this subject, Duchamp stated, "Naturally I never read seriously the works of Riemann because I would have been incapable of it" (Serge Stauffer, "100 Questions a Marcel Duchamp et 19 questions annexes; correspondance inedites, 1957- 1967" [Unpublished typescript, Musee National d'Art Moderne, Paris, n. pag.]). Stauffer's text is discussed in Craig Adcock, "Marcel Duchamp's Notes for La Mariee mise a nu par ses celibataires, memes: An n-Dimensional Prospectus for a Life's Work" (Ph.D. dissertation, Cornell University, 1981). Duchamp could have gained a working knowledge of the ideas of Lobachevsky and Riemann simply by reading Poincare and other secondary sources available in Paris.
[2] Dore Ashton, "An Interview with Marcel Duchamp," Studio International, CLXXI (June 1966), 245; and Jeanne Siegel, "Some Late Thoughts of Marcel Duchamp," Arts Magazine, XLIII (Dec. 1968-Jan. 1969), 22. See the conclusion of the present chapter for Duchamp's statements on the new geometries in these interviews.
[3] Duchamp misspoke here, for nearly all of his notes dealing with the fourth dimension are collected in A l'infinitif (The White Box) (New York: Cordier & Ekstrom, 1966) and not in the Green Box.
[4] Cabanne, Dialogues with Marcel Duchamp, pp. 39-40.
[5] Golding, Cubism, p. 31.
[6] Duchamp, as quoted in Cabanne, Dialogues, pp. 23-24.
[7] Before the publication of Clair's Marcel Duchamp ou Ie grand fictif, an excerpt of Clair's text was printed in L'Arc, LIX (1974), 44-57, as "Les Vapeurs de la mariee." Although many interesting comparisons are made in Clair's book (e.g., between Duchamp's terminology of his automobilelike Bride and Pawlowski's humanoid automobiles), it must be read with caution. Unfortunately, Clair used the 1923 edition of Pawlowski's Voyage au pays de la quatrieme dimension, which contains a lengthy introductory section not present in 1912, as well as added chapters and alterations in certain original chapters. Pawlowski's introductory "Examen critique" (pp. 9-53 of the 1971 Editions Denoel reprint, which Clair employed) was written in the wake of the popularization of Einstein and Relativity Theory in the early 1920s and thus frequently differs from ideas expressed in the 1912 edition.
A more subtle change in Pawlowski's book between 1912 and 1923 was a new emphasis on the relationship of humor to the fourth dimension, an idea to which Clair gives more attention than it is due. Clair (pp. 71-73) quotes from a section of the later "Examen critique" on the subject as well as from two chapters of the 1923 edition entitled "La Naissance de l'humour" and "La Revolte des singes." These two chapters, however, were revised versions of two 1912 chapters, "Le Flot qui l'apporta" and "Les Quatre Dimensions de l'esprit." In contrast to the 1923 emphasis on humor, "Les Quatre Dimensions de l'esprit" in 1912 had presented a lofty argument for the fourth dimension as the quality that distinguishes the human mind from all other forms of being. On the problems in Clair's text relating to the timing of the Comoedia publication of episodes of "Voyage," see nn. 31, 33 below.
[8] The Russian Jacques Povolozky (1881-1943) arrived in Paris around 1909, where he began publishing Russian-language books as well as translating two novels by Artsybachev. A prominent member of the Russian community in Paris, Povolozky came to know an increasing number of non-Russian artists, including Picasso, Gleizes, Severini, and Picabia. After World War I his 13 rue Bonaparte address was frequently used as a gallery, and Picabia's works were exhibited at his Galerie "La Cible" in 1920. Povolozky also continued to publish books under the name "Editions 'La Cible,' " including Severini's Du Cubisme au classicisme and numerous works by Gleizes. In no phase of his career, however, was Povolozky active as a popularizer of the fourth dimension (Information courtesy M. Michel Kaplan, Paris).
[9] Pawlowski, Voyage au pays de la quatrieme dimension, p. 28.
[10] See Jouffret, Traite elementaire, pp. 186-87.
[11] See, e.g., Poincare, La Science et l'hypothese, pp. 52-53; The Foundations of Science, p. 57. See also Poincare, Dernieres Pensees, pp. 65-66; Mathematics and Science: Last Essays, p. 29.
[12] See Duchamp, A l'infinitif, in Salt Seller: The Writings of Marcel Duchamp (Marchand du sel), ed. Michel Sanouillet and Elmer Peterson (New York: Oxford. University Press, 1973), p. 89. The Jouffret quotation was first reprinted by Arturo Schwarz, who included it in his Notes and Projects for the Large Glass, trans. George Heard Hamilton, Cleve Gray, and Schwarz (London: Thames & Hudson, 1969), p. 36. Schwarz also discussed this text in his Complete Works of Marcel Duchamp (New York: Harry N. Abrams, 1969), p. 23. More recently, the text has been commented upon briefly by Nicolas Calas in Mirrors of the Mind (New York: Multiples, Inc. and Castelli Graphics, 1975), pp. 11-13.
[13] Duchamp, A l'infinitif, in Salt Seller, pp. 94, 98.
[14] The Cordier & Ekstrom edition of A l'infinitif consisted of 150 boxed sets of notes in facsimile. However, the publication in 1973 of Salt Seller, an enlarged and translated version of Michel Sanouillet's 1958 anthology of Duchamp's writings, Marchand du sel, made the A l'infinitif notes available on a wide scale. A French-language version of Salt Seller is now available as Duchamp du signe, ed. Michel Sanouillet (Paris: Flammarion, 1975).
Since the first discussions of Duchamp and the fourth dimension (by the author and Jean Clair) appeared in 1974-1975, a number of other texts on this subject have been published. These include the essays by Clair and Ulf Linde in the four-volume catalogue L'Oeuvre de Marcel Duchamp (Musee National d'Art Moderne, Centre National d'Art et de Culture Georges Pompidou, Paris, 31 Jan.-2 May 1977), Clair's book Duchamp et la photographie (Paris: Editions du Chene, 1977), and his article, "L'Echiquier, les modernes et la quatrieme dimension," Revue de l'Art, no. 39 (June 1978), pp. 59-68. In addition to Craig Adcock's 1981 dissertation on "Marcel Duchamp's Notes," another important contribution to the study of Duchamp and the fourth dimension is John Dee's 1977 essay, "Ce Façonnement symetrique." Dee's study of Hinton has suggested important parallels with Duchamp, which were previously overlooked. Dee does overstate the case for Hinton's importance in this era somewhat, however, when he compares the Englishman's contribution to the early twentieth century with that of Marsilio Ficinos' to the Renaissance. Since Hinton's books were not even owned by the Bibliotheque Nationale or the Bibliotheque Sainte-Genevieve, his influence must be recognized as limited to a relatively small audience in Paris.
[15] See Marcel Duchamp: Notes, ed. and trans. Paul Matisse (Paris: Centre National d'Art et de Culture Georges Pompidou, 1980). The new notes, which introduce Duchamp's notion of "inframince" (discussed below), are analyzed in depth by Adcock in "Marcel Duchamp's Notes," ch. 3.
[16] Duchamp, Green Box, in Salt Seller, pp. 49, 71.
[17] Duchamp, as quoted in Cabanne, Dialogues, p. 39.
[18] See Schwarz, Complete Works, p. 9. Duchamp joined the Jarryesque College de 'Pataphysique in 1959 with the rank of "Transcendent Satrap." See the chronology included in the catalogue for the exhibition Marcel Duchamp, ed. Anne d'Harnoncourt and Kynaston McShine, ex. cat. (Philadelphia Museum of Art and The Museum of Modem Art, New York, 1973), p. 28.
[19] Gleizes, Souvenirs: Le Cubisme 1908-1914, p. 28; Duchamp, as quoted in Cabanne, Dialogues, p. 24.
[20] Jouffret, Traite elementaire, pp. xiv-xvi. Hinton himself had compared his system for learning to visualize the four-dimensional tesseract to his method for teaching boys to play blindfold chess in three or four lessons. See Charles Howard Hinton, Scientific Romances, 2nd ser. (London: Swan Sonnenschein & Co., 1896), p. 18.
[21] Clair has dealt with the higher dimensional associations of chess and chessboard configurations in early twentieth-century art in "L'Echiquier, les modernes, et la quatrieme dimension." The painter Camoin refers to his own chess games with Princet in Crespelle, Montmartre vivant, p. 111.
[22] The six drawings and the oil sketch are illustrated in Schwarz, Complete Works, illus. 41-47 (cat. nos. 165-170 and 176). Schwarz includes as illus. 45 (cat. no. 169) a drawing known only through a photograph in the Box of 1914.
[23] Duchamp himself never played blindfold chess, but later he did become a friend of George Koltanowski, who was at one time a blindfold chess champion. See Schwarz, Complete Works, pp. 57-70, for Duchamp's chess career. Dee has noted an additional association of chess, mirrors, and higher dimensions. At the beginning of a chess game, the opposing halves are arranged as mirror images of one another (“Ce Façonnement symetrique," p. 354). In Lewis Carroll's Through the Looking Glass, Alice's adventure occurred on a chessboard. On the higher dimensional aspects of this tale, see again Taylor, The White Knight, pp. 85-116, as well as Chapter 1, n. 50, above.