Left: Act 1, Scene 4 of Victory Over the Sun, 1913. Right: Kasimir Malevich, sketch for backdrop in photograph

Stephen Luecking

Popular ideas about the fourth dimension wielded significant influence from two directions on the earliest decades of modern abstraction. The first and most popular direction emerged from burgeoning esoteric beliefs where the fourth dimension served as the habitat for spiritual beings. The second direction resulted from extending spatial relationships in two-and three-dimensions of Euclidean geometry into analogous relationships in four dimensions. Depending on the nature of the artists, i.e. whether spiritually inclined or analytically inclined, the effects of both directions impinged on the forms of early abstraction.
In 1913 both directions shaped the Russian Futurist opera, Victory over the Sun, created and produced in Saint Petersburg by three friends and leaders of the Russian avant-garde: Mikhail Matyushin, who wrote the music, Aleksei Kruchenykh, who penned the libretto, and Kazimir Malevich, who designed the sets. A fourth friend, the poet Velimir Khlebnikov, wrote the introduction.
Matyushin admired Pyotr Ouspenskii, a popular proponent of the esoteric sciences who resided in St Petersburg at the time. These ‘sciences’ studied spiritual phenomena as subject to physical verification and not to supernatural justification. Key to the science was the existence of a physical fourth dimension containing our own three dimensions.
An actual fourth dimension would account for many of a spirit’s abilities. Take the case of the penchant for suddenly appearing within a closed space. The esoteric scientists explained this phenomenon by relying on a geometric induction termed ‘dimensional analogy’. This induction held that relationships of 3D entities to 2D objects shed light on parallel relationships of 4D entities to 3D objects. The analogy favored by the spiritualist follows from the ability of a 3D being to view and enter from above a closed 2D space such as a square or circle. This was equivalent, then, to a 4D being seeing inside a closed room, entering it from ‘above’ and seen by 3D beings as materializing out of empty space. To this ‘spirit’ the room appears to be wide open and easily accessed.
A corollary to the fourth dimension’s housing of spirits was that it also housed the spiritual component of a human being, that is, one’s consciousness. The physical body, this corollary implies, is simply the visible component of oneself residing in three dimensions with the invisible component extending into four dimensions. To theosophists such as Ouspenskii the supernatural was, by way of the fourth dimension, perfectly natural.
The aspect of the esoteric sciences most integral to Victory over the Sun, however, lay in Ouspenskii’s view of time, which he believed to be an illusion. He propounded a variation on eternalism. Eternalism was a philosophy of time which regarded all future and past events as existing simultaneously by virtue of being bound up in the fabric of four dimensions. Time, as experienced in three dimensions, emerges from the progressive encounters by a reality of limited scope as it moves along its spatial extensions in a higher reality. In effect, we humans age by subsequently inhabiting the pre-existing extensions of our future selves. The plotline of Victory had humans of a future race entering the three-dimensional space of the stage.
To provide these denizens of eternity with imagined access to the stage, Malevich depicted portals from a fourth dimension onto his set designs. Like the spiritualists Malevich drew on ideas of dimensional analogy to design his imagery.
Malevich’s exposure to 4D visualizing most likely came from contact with a publication by American architect Claude Bragdon. Bragdon had won a contest held by Scientific American in 1910 to write the best rational exposition of the fourth dimension, which he later expanded into a book published in 1913. As a Russian speaker and noted theosophist, Bragdon shared the book, A Primer of Higher Spaces, with Ouspenskii and other Russian theosophists, some of Malevich’s friends among them. Bragdon, who was nationally renowned as an architectural draftsman, had filled every page of the book with precise illustrations of various dimensional analogies.
Malevich’s design drawings borrowed something of Bragdon’s visual style, but curiously he used none of the dimensional analogies presented by Bragdon but developed his own. The result of Malevich’s analogy invented for the opera might be labelled ‘the window in the choron’.
Chorons are the three-dimensional forms that work as boundaries for four-dimensional objects. The reasoning behind the chorons goes something like this: lines of one dimension bound two-dimensional shapes while two-dimensional planes enclose three-dimensional volumes, leaving three-dimensional spaces, termed chorons, to border four-dimensional entities. This is perhaps the most basic of the dimensional analogies.
Malevich extended this analogy to include windows and doors by which future peoples can enter the stage from the higher space beyond the walls of the set. As seen in the first diagram, extracting a line segment from the middle of the side of a square creates the analog of a window in two dimensions. As shown in the second diagram, similarly extracting a planar segment from the middle of the face of a cube yields a window in three-dimensions. The third diagram denotes how, by extension, extracting a cube from the center of a cubic choron opens the analog of a window in four dimensions
The third diagram is clearly counter-intuitive. How does a hollow inside a volume function as an aperture through that volume? First of all, ‘through’ in the fourth dimension is not the same ‘through’ in the third dimension. Second, the diagram is not the actual image of a choron but the image of its 3D projection. In 3D projections of 4D space the center of the depicted volume is actually the most distant portion of the drawing. When the 4D object rotates, this projection appears to be turning inside out, an illusion forming as the center of the image moves to the fore.
Malevich’s ‘windows’ comprised the entire backdrop of the set opening the stage to the fourth dimension. In contrast to their large size on the stage, the artist executed his original designs as modest pencil drawings. He began each drawing by framing out the interior of a cube with a few lines defining its interior faces. He then treated each face as separate views into the fourth dimension.
Below: Template for Malevich’s studies of the backdrops for Victory over the Sun; followed by Study published on the cover of the playbill for the opera

Despite the innovation and sophistication of this design in connecting dimensions, Matyushin regarded Malevich to be naïve regarding the fourth dimension. This was likely due to antipathy Malevich held toward Ouspenskii and which he held in common with the Russian constructivist artists as a whole. Ouspenskii did not share these artists’ fervor for the revolution and he hung out in the St Petersburg café scene with established figures like Tolstoy, who shared his views. Ouspenskii returned this antipathy by dismissing Malevich’s later Suprematist works that the artist had titled ‘Colored Masses in the Fourth Dimension’. Ouspenskii declared that, “Painting the fourth dimension was like making a sculpture of a sunset.”
Malevich viewed geometry as coming in many different guises as hypothesized by Nikolai Lobachevsky, the Russian mathematician who developed the first system of Non-Euclidean geometry. In his book Pangeometry, Lobachevsky asserts that there could exist many other forms of non-Euclidean geometry other than his own. (There are at least 200 variations of geometry used today by scientists and engineers.) Although Lobachevsky’s discovery occurred early in the 19th century, his star didn’t rise until near the end of the century after his death. Improvement in astronomical instruments proved that, as Lobachevsky theorized, space at astronomical scales curved, inducing parallel lines to meet. In addition other mathematicians sanctioned his theories by developing their own geometries of curved space.
Malevich extended Lobachevsky’s pangeometry to art, which he believed could entertain its own forms of geometry. By contrast Ouspenskii discounted Lobachevsky’s geometry because it did not account for an actual fourth dimension.Although Malevich never stated any belief in a physical fourth dimension, the geometric ideas it suggested were to influence his work on at least two more instances. In 1913 he devised Futurist lithographs appearing in the poetry book Troe, published by his friends Kruchenykh, Khlebnikov and Elena Guro, Matyushin’s wife. Two years later he included allusions to the fourth dimension in the titling five of his Suprematist paintings.
Typically mum about his working methods and his influences, Malevich only wrote once on the fourth dimension and then only after Albert Einstein’s theories had reached the public stage. By 1918 he believed that the fourth dimension existed as time and not space.

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