A Protocubist anecdote
Henri Matisse’s and Leo Stein’s reaction at first seeing Pablo Picasso’s “Demoiselles D’ Avignon” (1907) at the “Bateau Lavoir” was to half jokingly exclaim that the painter was trying to create a fourth dimension. The art of Painting may indeed be considered as a pathway across dimensions, as it has been for millennia the pursuit of convincingly squeezing the three dimensional world perceived by humans onto a two dimensional surface. Yet any discussion about a fourth dimension in Painting appears paradoxical: Painting is about reducing dimensions rather than expanding them.
Dali’s “Corpus Hypercubus”
Following the development of Mathematics, where spaces with more than three dimensions are routinely addressed, the unfathomable, metaphysical character of possibly unperceived dimensions attracted wider attention and, not surprisingly, some of these mathematical ideas found their way towards artistic expression. A notorious example is Salvador Dali’s “Crucifixion” or “Corpus Hypercubus” (1954), a painting where Jesus Christ is depicted crucified upon the cross – like three dimensional net of a hypercube, the four dimensional analog of a cube. Though some mental gymnastics have been created to assist, after considerable exercise, towards the understanding of the nature of objects inhabiting a world our mind is not tuned to, full perception of objects such as the hypercube may be even impossible. Yet some at least superficial understanding may be achieved by creating analogs in spaces of lower dimensions. A cube for example, the 3D analog of the hypercube, can be formed by properly folding a 2D net consisting of six squares. When rotated, a cube casts shadows of a variety of geometric shapes on a 2D wall, two of them being a hexagonal shape and a square. Similarly, a hypercube, inhabiting a 4D space, casts “shadows” of a variety of three dimensional shapes upon 3D space and it can be formed by properly folding a 3D net consisting of eight cubes (though this kind of folding is far from possible to imagine), such as the one depicted in “Corpus Hypercubus”. From this point of view, Dali’s painting represents a pathway from 4D space (hypercube) towards 3D space (hypercube net) and then towards 2D space (the canvas surface).
A view from a higher dimension
Elementary Mathematics provide the means of understanding any point in 2D space (the plane) as represented by two numbers (coordinates), one for each of the two dimensions of the 2D space, namely length and width. The first number (abscissa) measures the horizontal displacement while the second (ordinate) measures the vertical displacement with respect to a fixed point selected as the origin. A distance in 2D space can be measured by simply applying the Pythagorean Theorem and, as it easily turns out, it is the square root of the sum of squares of the two displacements (coordinates). Similarly, any point in 3D space can be specified using three coordinates, each representing the displacement corresponding to each of the three dimensions of 3D space, namely length, width and height. A distance in 3D space is thus the square root of the sum of squares of these three coordinates. Though it is impossible to visualize where a fourth dimension (beyond length, width and height) would extend to, mathematicians routinely manipulate points and objects in 4D space, represented by four coordinates, and accordingly measure distances as the square root of the squares of those.
Painting reduces dimensions of 3D objects by one, providing an image (projection) of the subject as viewed from a single viewpoint, a process that inevitably creates ambiguities. The painting of a cube for example may be a 2D hexagonal shape properly rendered in order to create the illusion of viewing a cube from a certain viewpoint. In fact, the hexagonal shape is ambiguous and may correspond to an infinite number of 3D objects. Recognizing the object as a cube is a mental process that only emanates from previous experience. It is exactly this previous experience the artist relies upon in order to create a convincing illusion. For example, Albrecht Duerer’s notorious solid depicted on his engraving “Melencolia I” (1514) is of unresolved nature as its 2D rendering does not clearly and undoubtedly correspond to a geometric solid recognizable from previous experience. The same ambiguity holds for any object viewed from a single viewpoint, whether this object inhabits the 3D space or any other space. To clarify the nature of an object one has to be provided with multiple views from different viewpoints. Suppose that a tiny bookworm lives within a single, isolated book page. One may imagine the page, and consequently the bookworm, extremely thin to a degree that both the page and the bookworm may be considered as virtually two dimensional. A square on the page is a 2D object inhabiting the bookworm’s flat world yet, viewed by the bookworm from a specific viewpoint, it is indiscernible from a line segment. The bookworm cannot comprehend the square’s nature unless it follows a path around it in order to catch multiple “views” of it. Remarkably, simultaneous multiple views of the square from a single viewpoint would be also provided were it possible for the bookworm to hover above the page and into 3D space – a journey to the third dimension, something unimaginable for a creature accustomed to the conditions of a 2D world. From there, could the bookworm understand this unnatural perspective, the exact nature of the square would become apparent. By analogy, ambiguity of the image of a 3D object, such as Duerer’s solid, would be eliminated either by going around the object and having multiple views of it from various viewpoints or by hovering somewhere beyond the three dimensions, into 4D space, where simultaneous multiple views of the object would be available from a single viewpoint – though these would require sufficient understanding of this new perspective.
Cubism and fourth dimension
The Cubist movement, initiated by Picasso with the Protocubist “Demoiselles D’ Avignon”, seemed to allude to similar concepts, as Cubist paintings consisted of amalgamated fragments of the subject’s simultaneous views from various different viewpoints. This is particularly evident in the “Demoiselles”, where a seemingly distorted and somehow disturbing perspective presents female faces displaying simultaneously a side and a frontal view. Similar multiple perspective views of the Eiffel tower can be seen in Robert Delaunay’s “The Tower Behind Curtains” (1910). And a strikingly clear example can be found in Jean Metzinger’s “Tea Time” (1911) where the detail of a teacup is fragmented in half by a frontal and an oblique view. Such unusual images, typical in Cubist paintings, may be interpreted as attempts to catch views of the subjects by hovering into an unimaginable 4D space and then projecting this new perception back onto the 2D canvas.
Time as the fourth dimension
Following the advance of Physics in the beginnings of the 20th century, time came to be accepted as a fourth dimension, though it is of quite different nature than the ordinary three spatial dimensions. For example, any point in spacetime, the 4D space of the ordinary three spatial dimensions together with time, is free to move along any spatial dimension forward or backwards but is obliged to move only forward in time and thus traces a unique path, called its world line. Each one of us similarly traces a unique four dimensional path across spacetime, beginning with birth and ending with death. To create some visual representation of such a path, we may use the usual trick of making an analogy by reducing dimensions. For the 2D bookworm described above, spacetime is a 3D space every point of which can be specified by two spatial and one time coordinate. A time conscious bookworm would understand its own path in spacetime as a 3D world volume, like a long soap bubble, created by the compulsive forward movement in time. Every slice of this volume represents an image of the bookworm at a specific moment in time.
Among the necessary coordinates to specify any point in spacetime, the time coordinate stands out as being measured in time units while all three spatial coordinates are measured in units of length. How could one then measure distances with the usual, Pythagorean – like square root rule when one of the displacements under the radical sign is measured in units incompatible to the others? This issue was addressed by observing that, once an absolute velocity is accepted, time may be used to express distance and vice versa. For example, it is ordinary to say that the distance between two cities is three hours by plane and thus express a distance using time units, where the accepted absolute velocity is that of an airliner. It would not be unusual to similarly express time intervals using units of length, when saying for example that reading a magazine article takes 10km of travelling by train, where the absolute velocity considered is that of a train. The absolute velocity in nature is the speed of light c (approximately 300000 km/s) which means that a spatial distance of 1m is equivalent to 1/300000000 seconds of time and a time interval of 1 second is equivalent to 300000 kilometers. To compensate for the different nature of time dimension, Albert Einstein conceived the idea of expressing distances in 4D spacetime using the ordinary plus signs for the squares of spatial displacements and the unusual minus sign for the square of the time displacement, a choice that permits distances to inhabit the realm of complex numbers yet proves to be particularly successful in conveying the physical properties of time.
Chronophotography and spacetime
The idea of a 4D spacetime is thus intimately connected to the notions of speed and movement, subjects that were also addressed by artists after the development of the technology of photography. Early attempts to capture motion, and thus introduce the time dimension into images, yielded exotic devices with names reminiscent of dinosaurs such as the electrotachyscope, the phenakistoscope, the praxinoscope and the zoopraxiscope, the device employed by Eadweard Muybridge (1830 – 1904) to answer the galloping question (i.e. whether all four hooves of a horse are off the ground during gallop). Muybridge, in collaboration with the French naturalist Etienne Jules Marey (1830 -1904) and the American painter Thomas Eakins (1844 – 1916), succeeded in capturing successive images of a galloping horse and invented the zoopraxiscope to reproduce them in succession, creating the illusion of motion. Marey was later able to capture a number of such images on a single photographic surface to produce sequences of different, successive phases of motion in one picture. The result is an array of overlaid slices of a world volume, similar to that of the 2D bookworm. This was a revolution in image making, either by manual or mechanical means. Up to that early era of chronophotography, painting and the new art and science of photography were exclusively dedicated to capturing fleeting moments. It was the first time that a systematic way of capturing the motion in a specific time interval was introduced and the implications in science and art became immediately obvious.
The Futurist “dynamic sensation”
The Futurist movement, an offshoot of Cubism initiated by the Italian writer Filippo Tommaso Marinetti (1876 – 1944), displayed an obsession with motion and speed to such an extent that led Marinetti once declare, in a fit of exaggeration, that a racing car was more beautiful than the Nike of Samothrace. Though the Futurists had parallels in several countries (most notable of which was the Russian offshoot, with artists of magnitude such as Kasimir Malevich, Nataliya Goncharova, Mikhail Larionov and even the poet Vladimir Mayakovsky), Futurism was mainly an Italian phenomenon that soon began producing images reminiscent of Marey’s sequential photographs, systematically introducing the time dimension in painting. In a “Futurist Technical Manifesto” signed by Giacomo Balla (1871 – 1958), Gino Severini (1883 – 1966), Umberto Boccioni (1882 – 1916), Carlo Carra (1881 – 1966) and Luigi Russolo (1885 – 1947) in April 1910, a direct reference to world volumes and time as a dimension is made: “The gesture which we would reproduce on canvas shall no longer be a fixed moment in universal dynamism. It shall simply be the dynamic sensation itself […] On account of the persistency of an image upon the retina, moving objects constantly multiply themselves; their form changes like rapid vibrations, in their mad career. Thus a running horse has not four legs, but twenty, and their movements are triangular”. Typically, such Futurist compositions were often followed by descriptions including the term “dynamism”, a half artistic and half scientific terminology referring to evolution in time. For example Giacomo Balla (1871 – 1958) with “Dynamism of a Dog on a Leash” (1912) provides first a conventional pathway from the ordinary 3D space to a 2D image and then a leap towards 3D spacetime by allowing the time dimension to enter the scene and thus breaking the dimension limits of the painting surface: his dog, formed by overlaid slices of the corresponding world volume, appears smudged and multi – legged. Similar attempts to extend painting over the time dimension can be observed in scores of Futurist paintings, such as Balla’s “Rhythm of a Violonist” (1912) and the wonderful Neo – Impressionist “Girl running on a Balcony” (1912). However, the most remarkable work combining both the Cubist simultaneous multiple views and the Futurist world volume slicing technique is Marcel Duchamp’s “Nude Descending a Staircase No. 2” (1912), a geometric depiction of a human form in motion. The exceptionally vibrant picture seems to allude at the same time to a possible new perspective acquired by hovering outside the ordinary 3D space, somewhere in the unimaginable space of four spatial dimensions, and to the necessity of considering time as an additional dimension.
Such considerations make Matisse’s and Stein’s intuitive exclamation over a “fourth dimension” in Picasso’s “Demoiselles” appear less puzzling and less paradoxical. Painting has traditionally been the pathway from the 3D world we perceive to the 2D painting surface and, at least during the last century or so, it has also raised the question of other possible dimensions and the corresponding new perspective. Probably from different points of view, either rigorously or intuitively, Painting and Mathematics are both concerned about spaces and dimensions and as the Futurist and Cubist examples indicate, artistic expression has found the way, just like Mathematics, to probe into dimensions far from possible to imagine.