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Projective Ornament Claude Bragdon

Bragdon Projective Ornament (color)
Bragdon Projective Ornament (full)

full text from ©Linda Dalrymple Henderson

Claude Bragdon and the fourth dimension in America

Born in 1866, Claude Fayette Bragdon had spent the 1890s through 1923 primarily in Rochester, New York, where he worked first as an apprentice draftsman and then as an architect.74 After the death of his second wife, Bragdon moved in 1923 to New York to continue the work in theater design he had begun with the design for Walter Hampden’s 1919 production of Hamlet. Bragdon’s autobiography is aptly titled More Lives than One, for during his life Bragdon was also the author of over twenty books and two hundred articles as well as a publisher, whose Manas Press was active in Rochester from 1909 to 1921. 75
Yet another side to Bragdon was his “occult life,” as he termed it. The son of a Theosophist, Bragdon helped to found the Genesee Lodge of the Theosophical Society in Rochester and always maintained an ardent belief in esoteric philosophy and occult phenomena.
76 Even spiritualism figured in Bragdon’s life through his second wife, Eugenie, whose posthumous communications from her spirit voice he published as Oracle in 1921. Given his background and personal philosophy, Bragdon would certainly have discovered “the fourth dimension” on his own had he not been introduced to the concept in 1907 by his friend Burgess and the hyperspace philosopher Hinton.77 For the next nine years, Bragdon devoted himself to the idea, producing four major books and at least two articles. After that, a chapter on the fourth dimension regularly figured in Bragdon’s subsequent publications on architecture, philosophy, and Theosophy.

Bragdon’s first readings on the fourth dimension were the books of Hinton. Although Bragdon worked closely in 1911 and 1912 with the mathematician Philip Henry Wynne to learn the laws of four-dimensional space,78 it was the philosophical meaning of the fourth dimension that was most important for him, as it had been for Hinton. If his geometrical drawings in the 1915 book Projective Ornament belong to the mathematical tradition of n-dimensional geometry, Bragdon’s hyperspace philosophy also grants time a limited role in his explanation of the fourth dimension, as Hinton had done. Yet, like the Russian mystic Ouspensky, whose philosophy was developing at the same time, Bragdon the Theosophist presented a far more overtly mystical view of the notion than Hinton. When Bragdon discovered Ouspensky’s 1911 Tertium Organum in 1918, he enthusiastically collaborated on a translation with a young Russian, Nicholas Bessaraboff, and his Manas Press published the text in 1920. Similarly, Ouspensky later recorded how moved he had been upon first reading Bragdon’s Man the Square, “which carried the message of a common thought, a common understanding. “79 

Bragdon’s Publications on the Fourth Dimension

Under the pseudonym “Tesseract” Bragdon submitted his first text on the fourth dimension to the 1909 Scientific American essay contest. Much like the other contributions to that contest, Bragdon’s entry, “Space and Hyperspace,” was nevertheless unusual in its reliance upon the ideas of J.C. F. Zöllner, the author of Transcendental Physics.80 Bragdon was obviously impressed by Zöllner’s argument that the seemingly “contradictory facts” evidenced by the feats of the American medium Slade proved that an additional dimension of space must exist. After this initial essay, however, Bragdon, the architect and designer, began to bring his artistic talents to bear on the subject, and the books that resulted are a unique contribution to the literature on the fourth dimension.

Man the Square: A Higher Space Parable was published as a pamphlet by Bragdon’s Manas Press in 1912 and was included as well at the back of his 1913 A Primer of Higher Space (The Fourth Dimension). Man the Square is a religious parable, employing the analogy of the world of two dimensions to convey a message of love and harmony. According to Bragdon, the disharmony in this world of two dimensions results because cubic men, who exist as cross sections of its plane, are at odd angles with one another and produce irregular geometric figures as their sections (Fig. 58). Their higher existences as cubes are in the end revealed to them when a “Christos” cube unfolds his six sides down into the plane in a cruciform shape. Once aware of their true three- dimensional nature, the men, too, can be “square with the world.” None of Bragdon’s other works exhibits a similar overtly Christian stance, but even Man the Square contains certain striking illustrations such as Figure 58, which transcend the book’s immediate religious purpose.81

Bragdon’s most important work on the subject of the fourth dimension is A Primer of Higher Space (The Fourth Dimension) of 1913. The Primer contains thirty striking plates drawn and lettered by Bragdon and illustrating almost every popular idea about the fourth dimension that had emerged from the nineteenth century.82 Figure 59 reproduces one of the Primer plates, which reveals the variety of sections that can result when a cube passes through a plane at different angles, the underlying notion of the Man the Square story. Bragdon was the first author to illustrate the two-dimensional analogy by which writers since the nineteenth century, including Hinton, had explained the fourth dimension. Bragdon’s philosophy of the fourth dimension is set forth in the introductory portion of A Primer of Higher Space. In the tradition of  Hinton’s hyperspace philosophy, Bragdon presents time as the first means by which we may sinse higher dimensions. Bragdon also elaborates the related notion, implicit in Man the Square, that “the whole evolutionary process consists in the conquest, dimension by dimension, of our successive space worlds. “83 Here Bragdon, like Ouspensky, was blending hyperspace philosophy with the interpretation of evolution by Theosophists and others, such as R. M. Bucke, as a process of expanding consciousness. 84

For his 1913 edition of the Primer Bragdon, as designer, employed W. I. Stringham’s famous figure of the projection of a four-dimensional hypercube (Fig. 3) as a decorative motif for the book’s endpapers. Bragdon saw not only philosophical significance in the concept of a fourth dimension, but also a source for a new style of ornament. His 1915 Projective Ornament is a treatise on this subject, which explores the decorative possibilities of the figures of n-dimensional geometry.85 An admirer and supporter of Louis Sullivan, Bragdon was convinced that a new ornamental mode was needed for the future. Arguing that “art must attune itself to this new (four-dimensional) keyhole of the modern world,” Bragdon writes, “Consciousness is moving towards the conquest of a new space; ornament must indicate this movement of consciousness; geometry is the field in which we have staked out our particular claim. It follows, therefore, that in the soil of the geometry of four dimensions we should plant our metaphysical spade.”86

In the foreword to Projective Ornament Bragdon acknowledges as sources for his new style Hinton’s The Fourth Dimension, H. P. Manning’s Geometry of Four Dimensions, William T. Campbell’s Observational Geometry, Hermann Schubert’s Mathematical Essays and Recreations, and Stringham’s “Regular Figures in n-Dimensional Space,” as well as an article in the Encyclopaedia Britannica on “Magic Squares.” To create the new ornament Bragdon used both the method of projecting four-dimensional figures onto a two-dimensional surface and of “cutting” them apart and folding them down into two dimensions. Once flattened, the proportions of figures could be adjusted to produce several different motifs from a single figure. Figure 60 illustrates Bragdon’s work with the four-dimensional hypercube or tesseract. The figure at bottom left is recognizable as Stringham’s hypercube projection. At the right is a star-shaped derivative of this basic form, which Bragdon employs in a design for a carpet shown in Figure 61. As the book progresses, Bragdon’s designs become increasingly complex and elaborate, such as the handsome, proto-Art Deco image in Figure 62 based on the four-dimensional hexadekahedroid.

In August 1914 Bragdon had published an article, “Learning to Think in Terms of Spaces,” in The Forum, which presented a simplified statement of his view that consciousness is evolving toward perception of higher spatial dimensions, and that time is merely a temporary guise for the fourth dimension of space.87 In 1916 Bragdon incorporated his Forum article, slightly altered, into his last major book on the fourth dimension, Four-Dimensional Vistas. In Four-Dimensional Vistas Bragdon appeals to mathematicians (including Poincaré) and the new developments in science, as well as to esoteric literature and spiritualism, to support his case for the fourth dimension. Chapters on “The Dimensional Ladder” and “Physical Phenomena” are followed by “Curved Time,” in which Bragdon refers (without naming Einstein) to a Theory of Relativity” to suggest that the “time-plane” may be warped because of the variations in time measurement which can be produced by extremely high velocities, 88 The remaining chapters, however, including “The Night Side of Consciousness,” “The Eastern Teaching,” and “The Mystics,” owe more to Theosophy and other such sources than to traditional literature on the fourth dimension.

If an unspecified “Theory of Relativity” had figured in one chapter of Four-Dimensional Vistas, aspects of both the Special and General Relativity Theories of Einstein were the subject of Bragdon’s last article devoted solely to the fourth dimension. “New Concepts of Time and Space” was published in the avant-garde periodical The Dial in 1920. Once again, however, Bragdon managed to relate the latest developments in science to his own mystical hyperspace philosophy. “Are we on the point of discovering that the only reality is thought–consciousness?” Bragdon asks. ” lf so, by a long detour, Western science arrives at the same conclusion as Eastern mysticism: that materiality is only maya, illusion–the mirror of consciousness,”89 he concludes.

Four-Dimensional Vistas was Bragdon’s last specialized book on the fourth dimension, but many of his successive publications would include a chapter on the notion. Acknowledging Relativity Theory in these later discussions, Bragdon nevertheless maintained his prerelativistic hyperspace philosophy largely intact and continued to advocate his system of projective ornament. Thus, the 1918 Architecture and Democracy includes, in addition to sections on contemporary architecture, Louis Sullivan, and “color music” (Bragdon’s new passion), two chapters on mathematical ornament based “The World Order” and “The Fourth Dimension.” The first of these develops the ornamental possibilities of “magic paths” within numerical magic squares, which Bragdon had also incorporated in Projective Ornament. 90 In his later publications, Bragdon would also utilize aspects of Jay Hambidge’s “Dynamic Symmetry” in his search for harmonic decorative modes. 91

In addition to Architecture and Democracy, chapters on the fourth dimension appear in Old Lamps for New: The Ancient Wisdom in the Modern World (1925), The New Image (1928), The Frozen Fountain (1932), and The Arch Lectures (1942). 92 Of these later books The New Image contains the most extensive philosophical discussion of the idea, and it is here that Bragdon first mentions contemporary painting and sculpture in relation to the fourth dimension. Bragdon’s chapter entitled “The Fourth Dimension” presents the standard theme of his hyperspace philosophy, that space is a reflection of consciousness, which itself is evolving toward an apprehension of the fourth dimension. However, another chapter, “The Archetype,” proposes a definition of four-dimensional forms as the ultimate archetypes or essential forms, whose shadows make up the ephemeral world of appearances. Bragdon then recasts the rise of modern art in terms of his “Delphic Brother-Sisterhood” whose members are developing “the fourth form of consciousness”:

This minority, with its incipient fourth form of consciousness, react to everything differently from their elders, because, without themselves realizing it, they are penetrating into the archetypal world …

This is amusingly indicated in their aesthetic preferences and reactions. In music it takes the form of a preoccupation with tone, timbre and rhythm, and with harmony rather than with melody, because these elements are more potent in arousing those universal, primal, orgiastic emotions than the mere tickling of the ear with sugary tunes. In painting and sculpture they prefer indication and suggestion to a literal and realistic rendering, because more emancipating to the imagination and stimulating to the intuition. They have dethroned the kings crowned by their elders, and set up in their places such men as Cézanne, Brancusi and Gordon Craig, for example. Now quite aside from his extraordinary qualities as a painter Cézanne was clearly preoccupied subjectively, at least with the idea of archetypes, the significant and “eternal” aspect of things, in contradistinction to their accidental and ephemeral aspects.

The success or failure of all such efforts in these various fields need not be discussed here, their direction alone is important, for they indicate that the animating spirit of modern art is increasingly abstract, subjective, concerned with the elimination of established barriers, the breaking down of existing conventions. Painters, practitioners of an art which, being of space, is naturally static, would make it dynamic–would paint movement, by means of such devices as the indication of an object in successively assumed positions, as Duchamps [sic] did, in his famous Nude Descending a Staircase.” Musicians, on the other hand, whose art is of time, and therefore dynamic, now endeavor to poach on the preserves of the painter, sculptor and architect, using new and surprising concatenations of sounds for chisel and brush. 93

By 1928 Bragdon was finally aware of a possible four-dimensional quality of modern art, although ironically and in line with much American criticism rom 1913 onward, he connected it with the formal qualities of Cézanne’s art. At this date Bragdon still seems to have been unaware that the Cubists and even Duchamp had consciously pursued higher dimensional space. Cubism is not once mentioned by Bragdon, who suggests that modern artists have entered the archetypal world “without themselves realizing it.” However, as in the case of Pawlowski, it is quite probable that Bragdon’s works were being read by avant-garde artists, whether Bragdon himself was interested in contemporary art or not.

Bragdon and the New York Art Word

Although his friend Burgess had been associated temporarily with the group around Stieglitz at “291,” Bragdon does not seem to have been known personally by the Stieglitz circle in the 1910s. The Rochester architect made only occasional trips to New York before he relocated there in 1923, apart from a period of relatively freg commuting during 1912 and 1913. At that time Bragdon was serving as architect for the New York Central Railroad Station in Rochester and simultaneously courting a New York widow, Eugenie Macaulay, who would become his second wife in July 1913.

Nevertheless, Bragdon’s diaries and correspondence indicate that he narrowly missed being in New York for both his friend Burgess’s “291” exhibition and the Armory Show, 4

Before his first stage-design commission by Walter Hampden in 1919 and his 1923 move to New York, however, Bragdon had begun to develop friendships within the progressive theater world. Through Walter Kirkpatrick Brice, whom Bragdon met in 1916 during a project involving the New York Community Chorus, Bragdon gradually came to know such theater figures as Robert Edmond Jones and, later, Norman Bel Geddes,95 In New York Bragdon also became a friend of the editor of Theatre Arts

Magazine, Sheldon Cheney, with whom he had corresponded as early as 1907.96 At this time Bragdon was also close to Van Deering Perrine and Thomas Wilfred, who shared his dream of creating a “‘color organ,” a project realized in one form by Wilfred’s “Clavilux“ of 1921.97

There had also been an earlier manifestation of Bragdon’s interest in light and music. that might have brought him to the attention of artists living in New York. In September 1916 he had collaborated with the director of the New York Community Chorus to present a “Festival of Song and Light” in Central Park. Bragdon had created his first “cathedral without walls” in Rochester the previous year, and in 1916 he brought his multicolored lanterns and screens, designed according to the principles of four-dimensional projective ornament, to New York (Fig. 63). For a second New York performance in 1917, the lanterns were carried around the lake on foot and by boat, adding a dynamic element of motion.98 If Bragdon’s projective ornament designs were lost on the general public at these patriotic songfests, an artist aware of his work on the fourth dimension would surely have appreciated their four-dimensional implications.

The reviews of Bragdon’s publications on the fourth dimension, the articles that he himself published, along with the records he kept of the sales of his books, all suggest that the New York art world did indeed have some knowledge of Bragdon’s ideas on the fourth dimension during the 1910s. The “Art Notes and Comments” column of the New York Sun quoted briefly from Man the Square on 13 April 1913, noting that this text might be “of great assistance to those who do not yet understand Cubism.”99 In July 1914 Bragdon’s Primer of Higher Space received an enthusiastic review in the New York Times, where it was pronounced “a splendid contribution to an ímportant problem.” Finally, in March 1917 the critic James Huneker praised Bragdon’s “fascinating speculations” in Four-Dimensional Vistas in an article on the book in the New York Sun.100

Before his essay “Learning to Think in Terms of Spaces” had appeared in The Forum in August 1914, Bragdon had been introduced to Forum readers in a March 1913 article, “The Gift of Asia.” Although Bragdon’s text made no specific mention of a fourth dimension, apart from allusions to a Higher Self and “ampler spaces,” it presented the “ancient wisdom” on subjects such as the soul and consciousness, ideas of particular interest to members of the Stieglitz circle at that moment. Readers of The Architectural Review encountered Bragdon’s theories of projective ornament in his “Art and Geometry” articles of March and April 1916. And, following the 1916 “Song and Light” performance in New York, Bragdon had discussed projective ornament and “color music” in articles in The Architectural Review, House Beautiful, and Theatre Arts Magazine.

Stimulated by these articles, the reviews of Bragdon’s books, and contemporary popular interest in the fourth dimension, the patrons of New York book dealers actively purchased the publications of Bragdon’s Manas Press. Bragdon’scarefully kept records show him repeatedly filling quantity orders for A Primer of Higher Space, Projective Ornament, and Four-Dimensional Vistas. The majority of the requests came from Brentano’s, Mitchell Kennerley, and Alfred Knopf, who, after several copublishing ventures in the later 1910s, would ultimately take over the publication of Bragdon’s books.101 

If Bragdon’s ideas appealed to the artistic avant-garde, his own taste in artistic matters was considerably less advanced than that of his artist readers. Bragdon’s artistic thinking was at its boldest on the subject of color music. His views on architecture rank on a scale somewhere between the extremes of experimental color music and a more conservative attitude toward painting and sculpture. As an architect and theorist, Bragdon was an early advocate of the “organic” architecture of Louis Sullivan. In his own practice, he had been most adventurous in the early years of the century.102 However, Bragdon’s entry for the Chicago Tribune Tower competition of 1921 was scarcely less “pseudo Gothic” than the winning design by Hood and Howells, which he chided in a 1931 article on “Skyscrapers.”103 Moreover, the “modern” system of four-dimensional ornament Bragdon proposed was connected in his mind with an idyllic vision of the classical era. Apart from a few plates in The Frozen Fountain, his illustrations are most often peopled by figures in togas or tunics (Fig. 62) who are far removed from the realities of contemporary American architecture.104

Bragdon’s conservatism in the area of painting is in part a reflection of his age. Born in 1866, the art with which he had identified as a young man was the “poster” style of the 1890s. Unlike Stieglitz, who was actually two years his senior, Bragdon did not respond enthusiastically to European modernism. By 1928, however, Bragdon was aware of Cézanne, Brancusi, and even Duchamp, including them, with some reservations about their value, in The New Image. Bragdon was probably educated about modern art during the 1920s by his friend Sheldon Cheney, whose 1924 Primer of Modem Art and succesive books mention Bragdon’s fourth dimension.

By the later 1920s Bragdon had a second tutor on the subject of modernism, none other than Stieglitz himself. This later friendship between Bragdon and Stieglitz, which is documented by their correspondence from 1932 to 1943, is further evidence that there was no contact between the two men in the 1910s.105 Bragdon and Stieglitz apparently met only sometime after 1924, when they both moved into the newly built Shelton Hotel. Bragdon lived in the Shelton until his death in 1946; Stieglitz and Georgia O’Keeffe, until 1936. Before 1932, when Stieglitz and O’Keefe stopped eating meals there, the three often dined together, so that the letters which commence in 1932 take the place of the more frequent personal interchanges which had occurred earlier, 106

In general, the letters between Bragdon and Stieglitz chronicle a warm friendship and admiration between the two men, now in their sixties. However, the picture drawn from the Bragdon-Stieglitz corresporndenceis enhanced by the letters of Bragdon to Dorothy Brett and Mabel Dodge Luhan, both of whom he met around 1930, presumably through Stieglitz and O’Keeffe.l07 For example, in a 1932 letter to “Brett,” in whom he found a willing pupil for his new ideas on dynamic symmetry and four-dimensional ornament, Bragdon wrote of Stieglitz in a moment of pique, “I think he is a wonderful man, but I have never been able to take seriously some of the people and some of the things he takes seriously. …I like them (Stieglitz and O’Keeffe), I even love them, but they are not, after all, in my dimension, or I in theirs.”108

Despite this remark, Stieglitz seems to have been very receptive to Bragdon’s ideas, praising each newly published text Bragdon sent him. He even recorded his enthusiasm publicly on the advertising circular for Bragdon’s 1938 autobiography, More Lives Than One: “A beautiful book in every way, a masterpiece of its kind.“101 Given this later attitude on Stieglitz’s part, it is all the more likely that Bragdon’s publications would have appealed to him in the prewar years, in the context of active artistic debate about the fourth dimension. Certainly, much in Bragdon’s hyperspace philosophy would also have recommended it to the Symbolist-oriented critics in the Stieglitz circle, Hartmann, de Casseres, and Caffin. Furthermore, devotees of Bucke’s Cosmic Consciousness, such as Hartley, would have responded favorably to Bragdon’s concept of a moving threshold of consciousness, which can progress to include the fourth dimension of space.110

Although Weber had broken with Stieglitz in early 1911, before any of Bragdon’s books appeared, Weber may have been a source for the first introduction of the books at “291,” by means of his connections with Mabel Dodge and Agnes Ernst Meyer. Weber himself must have read Bragdon’s books eagerly as they were published. As suggested earlier, his own reinterpretation of the fourth dimension in increasingly spiritual terms by 1914 may well owe a debt to Bragdon’s hyperspace philosophy, as presented in the 1912 Man the Square and even the 1913 Primer of Higher Space. 

Evidence for familiarity with Bragdon’s work is more direct for Duchamp and the group who gathered at the Arensberg apartment during the war years in New York. A number of Bragdon’s books and pamphlets, including A Primer of Higher Space, Projective Ornament, Four-Dimensional Vistas, and Architecture and Democracy are known to have been in the library of New York attorney John Quinn, an avid book collector and the important patron of numerous painters including Duchamp. 111 In addition to Quinn’s library, another source for Duchamp to have encountered Bragdon’s books was his friend Katherine Dreier. A proponent of Theosophy and number symbolism, Dreier may even have been familiar with Bragdon’s earliest Theosophical publications of 1909 and 1910. Dreier’s admiration for Bragdon is documented in an exchange of letters with Bragdon early in 1923.112

Judging from Bragdon’s answer to Dreier’s first letter, now lost, she had written to him on behalf of the Société Anonyme, asking him to lecture at their upcoming John Storrs exhibition. Bragdon replied to the Société Anonyme on March 1:

Dear Sir,

Answering your letter of yesterday I would say that while I remember Mr. Storrs very well, and pleasantly, and admire his work very much, I could not undertake to give a lecture, for I don’t do any of that sort of thing any more. I needn’t go into my reasons they are personal, I can only quote, in extenuation, the saying of Lao Tzú: Those who know do not speak, those who speak do not know.” Art and spirituality must be self-evident or else they are not art and spirituality.

Thanking you for the honor of considering me in this connection, I am,Yours very sincerely, 

Claude Bragdon

Dreier then answered Bragdon on March 9:

My dear Mr. Bragdon,

We were all enchanted with your letter to the Société Anonyme, in which you quoted Lao Tze (sic] who, of course, is an intimate friend of ours.

Mr. McBride feels, that since you refuse to speak, you at least ought to let our library have the benefit of your writings, and I hope, therefore, that you will be so generous as to send us copies of your various pamphlets and books. 

Our Reference Library means a great deal to us, and we not only carry works on modem art, but we try to have a certain amount of literature done in the modern spirit.

Hoping that all the members of the Board of Directors, of the Société Anonyme, of which there are twelve, may someday have the pleasure of meeting you, believe me

Very sincerely yours, 

Katherine Dreier

A curious aspect of this exchange is Bragdon’s recollection of Storrs. It is difficult to imagine where and when Bragdon might have encountered the modernist sculptor Storrs, who spent much of his time in France. However, since Storrs’s father was a Chicago architect and Storrs himself had studied architecture at the Art Institute between 1908 and 1910, Bragdon might have met Storrs on one of his professional visits to Chicago. 113

Whatever the history of a possible meeting with Storrs, Dreier’s letter confirms that none of the Société Anonyme’s directors (including Duchamp) had met Bragdon personally. Nevertheless, both she and critic Henry McBride show themselves familiar and sympathetic with his writings. 114 Bragdon evidently sent the Société Anonyme’s library only the two books he felt were most art-oriented, Projective Ornament and Architecture and Democracy, for these are the only two titles on a 1928 list of the library’s contents. 115 Dreier herself may well have owned copies of other of Bragdon’s books, however fitting contents for the library for which she had commissioned Duchamp’s multidimensional Tu m’ in 1918.

Even if he never met Bragdon, Duchamp, like Weber, was too interested in the fourth dimension not to have noted this primary American source. Although he had already completed most of the notes for the Large Glass by the time he arrived in New York, Duchamp would have read Bragdon with an eye to the actual execution of his project. In Bragdon’s Primer of Higher Space, Duchamp would have recognized many of the facts and speculations about the fourth dimension that he had encountered in Paris. Just as Jouffret had been his initial source for thinking about dimensions in terms of shadows, Bragdon reminded Duchamp in the Primer that “the representation of the form of an object is conditioned and restricted by the space in which such representation occurs. The higher the space, the more complete the representation.

…Lower-dimensional representations may be conceived as the shadows cast by higher-space forms on lower space worlds. “116

In A Primer of Higher Space Duchamp would also have found Bragdon’s illustration of “Evolution Interpreted in Terms of Higher Space” as a spiral, the form that so fascinated him in the “Handler of Gravity” for the Large Glass (Fig. 44), Tu m’, and his “precision optics.” Bragdon goes further than had Hinton, however, and suggests that a four-dimensional spiral passing through three-dimensional space may be the source of the paths of “the atom, the molecule, the cell-the earth itself.“117

Finally, Bragdon’s Four-Dimensional Vistas discusses mirrors, which were so important to Duchamp’s theorizing about the fourth dimension. Bragdon considers rotation in the fourth dimension in terms of a mirror, just as Duchamp had done: “In the mirror-image of a solid we have a representation of what would result from a four-dimensional revolution, the surface of the mirror being the plane about which movement takes place. If such a change in position were effected in the constituent parts of a body as a mirror image of it represents, the body would have undergone a revolution in the fourth dimension.” In his chapter on “Curved Time” Bragdon also uses the analogy of a spoon-man, based on Helmholtz’s discussion of a convex mirror as a model of non-Euclidean space. By means of an imagined conversation with the spoon-man, Bragdon explains the distortions in measurements that may occur and be undetectable under the new “Principle of Relativity. „118 This section of Bragdon’s text would have especially interested Duchamp after 1919, in the context of his friend John Covert’s Einsteinian collage/painting Time.

The one possible formal influence of Bragdon on Duchamp occurred only later, in the 1920s, in Duchamp’s poster for the French Chess Championship held in Nice in 1925 (Fig. 64). It has been suggested that Duchamp’s image is the three-dimensional analogue of a chessboard, with strong die-like overtones, 119 but his tumbling cubes are also very reminiscent of Bragdon’s falling cubes from A Primer of Higher Space (Fig. 59). Now, however, Bragdon’s two-dimensional black sections of the plane cling to the cubes themselves. In 1911 Duchamp had been stimulated by his reading of Jouffret to create a four-dimensional Portrait of Chess Players (Fig. 34). Perhaps in his 1925 poster Duchamp was now relying on another author on the fourth dimension to create a private symbol for the game of chess.

Claude Bragdon’s publications were the major vehicle by which his ideas and images reached the New York art world in the 1910s and 1920s. Largely unaware of the effect his theories had had on avant-garde artists in that period, Bragdon during the 1930s produced a series of paintings based on projective ornament. “It seems to me that Projective Ornament is important enough to warrant the Museum of Modern Art giving me an exhibition, “120 he wrote to Stieglitz in August 1940. Not surprisingly, Alfred Barr, the director of the Museum of Modern Art, took no interest in Bragdon’s geometry at a time when New York was focusing on the organic styles of French Surrealism.

The 1910s, and not the 1930s, were the years in which Bragdon’s “fourth dimension” offered exciting new possibilities to young artists. Indeed, “the fourth dimension” acted as a liberating influence in America much like the famous Armory Show of 1913. And it was in the context of the activities surrounding the Armory Show that Bragdon’s books came to the attention of additional members of the avant-garde.