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The Fourth Dimension And Non Euclidian Geometry in Modern Art

Copyright Linda Dalrymple Henderson

The Fourth Dimension And Non Euclidian Geometry in Modern Art (Introduction PDF)

The Fourth Dimension and Non-Euclidean Geometry in Modern Art

HENDERSON, Linda Dalrymple (1983)


During the years 1902 to 1907, at the time Albert Einstein was working in the Swiss Patent Office, Charles Howard Hinton, a little-known Englishman crucial to this study, was employed in the United States Patent Office in Washington, D.C. Hinton had published his last major book, The Fourth Dimension, in 1904, three years before his death at age fifty-four. Einstein, on the other hand, was at the threshold of his own life’s work, in 1905 formulating the first of his many contributions to science, the Special Theory of Relativity. In the long run, Einstein’s influence was to be far greater than that of Hinton, revolutionizing scientific theory and, after about 1919, the world view of laymen as well. However, in the first two decades of the twentieth century, the idea promulgated by Hinton and many others that space might possess a higher, unseen fourth dimension was the dominant intellectual influence.

The complex spatial possibilities suggested by a fourth dimension, as well as by the curved space of non-Euclidean geometry, were the outgrowth of developments in early nineteenth-century geometry. Popularized during the later years of the century, these notions had begun to capture the public’s imagination by the turn of the century in much the same way Black Holes have done in recent years. Like a Black Hole, “the fourth dimension” possessed mysterious qualities that could not be completely understood, even by scientists themselves. Yet, the impact of “the fourth dimension” was far more comprehensive than that of Black Holes or any other more recent scientific hypothesis except Relativity Theory after 1919. Emerging in an era of dissatisfaction with materialism and positivism, “the fourth dimension” gave rise to entire idealist and even mystical philosophical systems, such as that of Hinton. Only the popularization of Einstein’s General Theory of Relativity, with its redefinition of the fourth dimension as time instead of space,’ brought an end to this era in which artists, writers, and musicians believed they could express higher spatial dimensions.

Besides the artists who are the subject of this study, the list of prominent figures interested in the fourth dimension is an impressive one. Between Dostoevsky’s references to higher dimensions and non-Euclidean geometry in The Brothers Karamazov of 1880 and P. G. Wodehouse’s offhanded use of the term in “The Amazing Hat Mystery” of 1922, the fourth dimension attracted the notice of such literary figures as H. G. Wells, Oscar Wilde, Joseph Conrad, Ford Madox Ford, Marcel Proust, and Gertrude Stein.’ Among musicians, Alexander Scriabin, Edgar Varese, and George Antheil were actively concerned with the fourth dimension, and were encouraged to make bold innovations in the name of a higher reality.

For early twentieth-century artists “the fourth dimension” and non-Euclidean geometry had an equally liberating effect, In the past, art historians have most often ignored or dismissed references to either of the “new geometries” In the writings of modern artists and critics.’ Without a knowledge of the widespread popular interest in these spatial concepts, historians tended to misinterpret the terms as purely mathematical or purely mystical, missing the variety of views between the two extremes, When understood in their original context, however, “the fourth dimension” and non-Euclidean geometry are far from being the “scourge of every history of modern painting,” as they have been termed. Instead, these concepts open the door to our understanding more fully the goals of many seminal artists of the early twentieth century.

That references in Cubist literature to higher dimensions and to non-Euclidean geometry had nothing to do with Einsteinian Relativity Theory was first suggested by John Adkins Richardson and myself in the early 1970s. My historical arguments against such a connection, presented in a 1971 Art Quarterly article,5 are in Appendix A below, which includes additional information on the subject as well. In his 1971 text Modern Art and Scientific Thought Richardson supported his case by citing the works of J.C.F. ZölIner and H. G. Wells as proof that the term the fourth dimension was used independently of Relativity Theory in this period.6

Although my Art Quarterly article presented the first extended discussion of the popular tradition of “the fourth dimension” and non-Euclidean geometry as an outgrowth of nineteenth-century geometry, several other art historians, in addition to Richardson, had earlier touched upon one or another of its manifestations. Christopher Gray in Cubist Aesthetic Theories of 1953 had noted the mystical writings of P. D. Ouspensky on the subject.’ Similarly, Apollinaire scholars LeRoy C. Breunig and Jean-Claude Chevalier in 1965 had mentioned the 1909 Scientific American essay contest on the fourth dimension, as well as the 1912 Comoedia serial Voyage au pays de la quatrième dimension.’ Nevertheless, only one historian of modem art, Meyer Schapiro, seems to have sensed the broader philosophical implications of the new geometries in the early twentieth century. As early as his 1937 essay on the “Nature of Abstract Art” Schapiro noted, “Just as the discovery of non-Euclidean geometry gave a powerful impetus to the view that mathematics was independent of existence, so abstract painting cut at the roots of the classic ideas of artistic imitation.”9

My reinterpretation of Cubist references to a fourth dimension and to non-Euclidean geometry was expanded in a 1975 Ph.D. dissertation to include Marcel Duchamp, as well as a number of artists working outside of France during the period 1900 to 1930.10 The present text is a more fully developed version of that work, with a new chapter added on American art. France, however, remains the central focus of this study as well, for it was among the Cubists that the first and most coherent art theory based on the new geometries was developed. From Cubism (and the speculation of Duchamp) successive artistic explorations of the subject occurred in Italy, America, Russia and Holland. The varying approaches of artists in each of these nations toward the new geometries and their frequent alterations of the original prewar Cubist ideas on the subject provide valuable new insights into the character of a number of modem movements.

As wide ranging as the present study is, the absence of two avant-garde centers, England and Germany, may initially raise questions. Yet this omission was dictated by my requirement that for each artist or movement to be examined there be a body of writings on the fourth dimension and non-Euclidean geometry by an artist and his contemporaries. Thus, while the fourth dimension was certainly well known in England and Germany, it does not figure prominently in Vorticist or German Expressionist literature.11“

Germany, and particularly the Russian-born leader of the Munich Blaue Reiter group, Wassily Kandinsky, proved the most surprising in this regard. As is discussed in Chapter 5, Kandinsky was clearly aware of the fourth dimension and shared an anti-materialist stance with his Russian colleagues that would have made the fourth dimension a logical part of his artistic theory. The term, however, does not appear in Kandinsky’s published writings, which instead bear the stamp of Rudolf Steiner’s Christian Theosophy. In fact, in Germany it may have been the strong influence of Steiner that overwhelmed “the fourth dimension” with a compatible but more mystical and elaborate philosophy.12

Among German Expressionist painters there may also have been a conscious rejection of a fourth dimension identified with cerebral French Cubism. Significantly, the Cubist painter who had the most widespread impact on Germany, Robert Delaunay, was not a major advocate of the fourth dimension.” Thus, Franz Marc, cofounder with Kandinsky of the Blaue Reiter group and an admirer of Delaunay, documented his own interest in the fourth dimension only in 1916. In a letter to his wife of 24 January 1916, Marc wrote enthusiastically of hearing from a physicist friend about the progress of science beyond three dimensions to four-dimensional space-time.14“

The question of the artistic influence of the fourth dimension in Germany, nevertheless, deserves further research. Germany, after all, had produced more scholarly and semipopular articles on this subject than any other country by 1910.’5 This literature needs to be surveyed carefully, and the writings of German artists, as well as figures such as Kandinsky’s friend Arnold Schoenberg, should he examined closely for evidence of the influence of the new geometries, even in their most subtle, non-mathematical forms, r’ For instance, Schoenberg’s advocacy of a new atonal language for music reflects one of the most persistent themes associated with “the fourth dimension”: the inadequacy of present language to deal with the new reality of higher dimensions.

The present study, however, concentrates on the early twentieth-century artists and movements that have left the most direct records of their interest in the spatial concepts associated with the new geometries. Even in the absence of advocates among the German Expressionists and English Vorticists, the fourth dimension and non-Euclidean geometry emerge as among the most important themes unifying much of modern art and theory.


1. See Appendix A for the principles of Relativity Theory in the various phases of its development and the timing of its popularization.

2. Except for Dostoevsky and Wodehouse, all of these authors are discussed in the pages that follow. Dostoevsky’s Ivan Karamazov refers to higher dimensions and non-Euclidean geometry in the course of his speculation on the existence of God (Fyodor Dostoyevsky, The Brothers Karamazov, trans. Constance Garnett [New York: New American Library, 1957, pp. 216-17). For Wodehouse’s tale, see P. G. Wodehouse, Young Men in Spats (Harmondsworth, England: Penguin Books, 1971), pp. 68-86. The magical properties of higher spatial dimensions, which fascinated H. G. Wells and, before him, Lewis Carroll, have continued to stimulate writers of science fiction. See, for example, Robert A. Heinlein, “‘—And He Built a Crooked House—,'” Astounding Science Fiction, February 1941; reprinted in Analog’s Golden Anniversary Anthology, ed. Stanley Schmidt (New York; Davis Publications, 1980), pp. 95-110. For an overview of this body of literature, which continued to grow even after the popularization of Relativity Theory, see Pierre Versins, Encyclopedie de l’utopie, des voyages extraordinaires, et de science-fiction (Lausanne: L’Age d’Homme, 1972).

3. “New geometry” is used throughout this study as a relative term, in that the n-dimensional and non-Euclidean geometries, which seemed so novel and modern at the turn of the century, had actually existed since the first half of the nineteenth century.

4. William Rubin, “Reflexions on Marcel Duchamp,” An International, iv (1 Dec. 1960), 52. A recent sign of how the new geometries are increasingly recognized for their role in the evolution of modem an is the essay by Lucy Adelman and Michael Comp-ton, “Mathematics in Early Abstract Art,” in Towards a New Art: Essays on the Background to Abstract Art, ed. Michael Compton (London: The Tate Gallery, 1980), pp. 64-89.

5. Henderson, “A New Facet of Cubism: ‘The Fourth Dimension’ and ‘Non-Euclidean Geometry’ Reinterpreted,” The Art Quarterly, xxxiv (Winter 1971), 410-33. 6. See Richardson, Modem Art and Scientific Thought (Urbana: University of Illinois Press, 1971), ch. 5, “Cubism and Logic.” His chapter had been published in much the same form in France in 1969 as “Un Mythe de la critique moderne: Le Cubisme et la quatrième dimension,” Diogène, no. 65 (Jan.—Mar. 1969), pp. 103-15.

7. See Gray, Cubist Aesthetic Theories (Baltimore: Johns Hopkins Press, 1953), p. 85 and n. 74 to p. 85.

8.  Breunig and Chevalier, Guillaume Apollinaire: Les Peintres Cubistes (Paris: Hermann, 1965), p. 105.

9. Schapiro, “Nature of Abstract Art,” Marxist Quarterly, t (Jan.—Mar. 1937), 78.

10. Henderson, “The Artist, ‘The Fourth Dimension,’ and Non-Euclidean Geometry 1900-1930: A Romance of Many Dimensions,” Ph.D. dissertation, Yale University, 1975.

11. As will be seen, the notion of a vortex itself had been connected to the fourth dimension by Hinton. However, although there was a strongly geometrical orientation in Vorticist art (advocated particularly by T. E. Hulme) and Ezra Pound was later vocal in his support of George Antheil’s musical interest in a fourth dimension, Wyndham Lewis’s desire to prove Vorticism independent of its Cubist and Futurist sources may have caused him to downplay the fourth dimension. Only later did Lewis write on the subject—after its redefinition by Einstein—in Time and Western Man (New York: Harcourt, Brace & Co., 1928).

12. Although the fourth dimension never became a vital part of his Theosophical system, Steiner had actually lectured on the subject in 1904-1905 (Robert C. Williams, Artists in Revolution: Portraits of the Russian Avant-Garde 1905-1925 [Bloomington: Indiana University Press, 1977], p. 109).

13. By the time Apollinaire traveled to Berlin with Delaunay in early 1913, he was no longer so interested in the fourth dimension. Nevertheless, the German avant-garde would have learned of his earlier views on Cubism and the fourth dimension in Les Peintres Cubistes. For example, Paul Fechter in his 1914 text Der Expressionismus cynically chided Apollinaire for neglecting in Les Peintres Cubistes to discuss Riemann and non-Euclidean geometry in connection with the fourth dimension (Paul Fechter, Der Expressionismus [Mu-nich: R. Piper & Co., 1914], p. 34).

14. Franz Marc letter to Maria Marc, 24 January 1916, in Marc, Briefe, Aufzeichnungen and Aphorismen (Berlin: Paul Cassirer, 1920), p. 104. Marc’s change in attitude toward science is discussed by Ida Katherine Rigby in “Franz Marc’s Wartime Letters from the Front,” in Franz Marc, 1880-1916, ex. cat. (University Art Museum, University of California, Berkeley, 5 Dec. 1979-3 Feb. 1980), p. 58.

15. See Duncan M. Y. Sommerville, Bibliography of Non-euclidean Geometry, including the Theory of Parallels, the Foundations of Geometry, and Space of n dimensions (London: Harrison & Sons, 1911), p, viii.

16. Another figure for investigation is Herwarth Walden, founder of the Berlin gallery and periodical Der Sturm. Walden must have had at least a passing interest in the fourth dimension, for he published in 1911 a review by S. Friedlaender-Halensee of Max Zerbst’s Die vierte Dimension (Der Sturm, it [Oct. 1911], 663-64).

Copyright Linda Dalrymple Henderson