A phrase coined by

Guillaume Apollinaire to describe the particular style of the cubists.

At the start of the

20th century there were two popular interpretations of the fourth

dimension. One was that it is

time. This is fairly easy to imagine, and remains a common meaning for the term today. The second is that the fourth dimension is another

spatial dimension. This is much harder, if not impossible, to visualize. Nevertheless, several illustrators attempted to. One of the most popular illustrations of this was the four-dimensional hypercube. Just as we can form a cube by folding a cross-shaped piece of paper consisting of six squares, so it was believed one could form a four-dimensional "hypercube" by folding a similar arrangement of seven

*cubes*. (see

*Crucifixion,* Dali, 1954) A mathematician,

E. Jouffret, attempted to depict four-dimensional objects by drawing their projections on a

two-dimensional plane.

These representations were fairly well established when

Picasso and

Braque invented

cubism. While it would be incorrect to claim that they formed the basis of

cubism, it would be equally incorrect to say they had no influence on it. In the years before he started

cubism,

Picasso met regularly with a group of friends who called themselves

*la bande à Picasso*. Guillaume

Apollinaire, a member of

la bande describes the influence of the fourth dimension on cubism in his book

*Les Peintres Cubistes. *
"Until now, the three dimensions of Euclid's geometry were sufficient to the restiveness felt by great artists yearning for the infinite...
The new painters do not propose, any more than did their predecessors, to be geometers. But it may be said that geometry is to the plastic arts what grammar is to the art of the writer. Today, scholars no longer limit themselves to the three dimensions of Euclid. The painters have been led quite naturally, one my say by intuition, to preoccupy themselves with the new possibilities of spatial measurement which, in the language of modern studios, are designated by the term fourth dimension."^{1}

In this statement, he recognizes the

graphical representation of the fourth dimension in

cubist art, and also the

mental influence. The fourth dimension is representative of the

infinite possibilities that the cubists sought.

Apollinaire reaffirms this in

*La Peinture nouvelle*: "The art of the new painters takes the infinite universe as its ideal, and it is to the fourth dimension alone that we owe this new norm of the

perfect..."

^{2} Another member,

Maurice Princet had extensively studied

Poincaré's writings and is widely recognized as having exposed the cubists to his work. Poincaré's book

*l'Science et l'Hyposthese*, written in

1902, popularized four-dimensional geometry and is often linked to the cubists work through Princet. First hand accounts tell of Princet discussing problems of perspective and simultaneously representing objects from multiple viewpoints. This would be a consequence of the fourth dimension being a spatial one which acts as an "

astral plane", from which an object of the usual

three dimensions can be viewed from all sides

simultaneously. (Just as in our three dimensional

world we can see the entirety of a two dimensional object at once).

We can see this effect quite clearly in Picasso's

paintings. In

*Les Demoiselles d'Avignon*,

1907, the crouching

woman's body is seen from behind while her head is seen from the front. Similarly, the two central standing figures are shown in a frontal view, but their

noses are painted in

profile. The painting also shows influences of Jouffret's projections of a four-dimensional

ikosatetrahedroid on a

plane. The rightmost woman's upper body fits into a

diamond grid that is extremely similar to Jouffret's projections of 1903. (This is more clearly seen in

*Standing Nude with Joined Hands (Study of Proportions)*, 1907.) The

faceting which has become synonymous with Picasso's name is also very similar to Jouffret's drawings, in which he superimposes his projections on top of each other in an attempt to display multiple sides of a polyhedron simultaneously. Though this may have been more a similarity in appearance than a direct depiction of the fourth dimension, it was certainly a rejection of three-dimensional

perspective.

Later cubists, particularly

Metzinger and

Gleize show an even greater influence from Princet's lectures. Unlike Jouffret, who conceded to project his four-dimensional figures into two dimensions, Metzinger believed that the mind was capable of perceiving all four at once. In a style similar to Picasso's he shows figures from various perspectives in the same painting, though often with less fragmentation than Picasso used. In

*Le Gouter* (1911), he varies perspectives as the viewer looks from one side of the woman's face to the other. The left side of her face is seen in a

frontal view, her nose in a three-quarters view, and her right eye in profile. Of even greater interest is the

bowl from which she eats. The left side of it is seen from the side, while the right side is seen from above. This is almost exactly the same problem posed by Princet in 1910:

"You represent by means of a trapezoid a table, just as you see it, distorted by perspective, but what would happen if you decided to express the table as a type? You would have to straighten it up onto the picture plane, and from the trapezoid return to a true rectangle. If that table is covered with objects equally distorted by perspective, the same straightening up process would have to take place with each of them. Thus the oval of a glass would become a perfect circle."^{3}

Princet, in turn, took this from Poincaré, who writes, "We can even take of the same

four-dimensional figure several perspectives from several different points of view. We can easily represent to ourselves these perspectives, since they are only three dimensions. Imagine that the various perspectives... succeed one another..."

^{4} In

1880, a mathematician,

W.I. Stringham, attempted to illustrate such

perspectives. In 1910, two painters also made use of Stringham's work. The vase in Gleizes's

*Woman with the Phlox* and the fruit in Le Fauconnier's

*Abundance* bear a striking resemblance to Stringham's figures.

The

futurists also used the term fourth dimension, but not in the same way the cubists did. While the cubist fourth dimension was spatial, the futurists' was temporal.

Boccioni describes this in his

*Plastic Dynamism* (1913).

"...Instead of the old-fashioned concept of sharp differentiation of bodies, instead of the modern concept of the Impressionists with their subdivision, their repetition, their rough indications of images, we would substitute a concept of dynamic continuity as unique form. And it is not by accident that I say form and not line, since dynamic form is a species of fourth dimension in painting and sculpture, which cannot exist perfectly without the complete affirmation of the three dimensions that determine volume: height, width, depth."^{5}

This "

dynamic form" he writes about is easily seen in his work. In

*The City Rises*, 1910, the elongated

brushstrokes create a blurred motion that is similar to the kind created when a

photograph is taken of a moving object. His sculpture

*Unique Forms of Continuity in Space*,

1913 perfectly captures the essence of motion, as the figure seems to

liquefy and move forward through itself. Later futurist works, most notably

Balla's "

dynamisms" show a scene over the course of a period of time with much less

distortion, and clearly demonstrate the idea of the fourth dimension being

time. The futurist method was so effective that even Picasso reconsidered the possibilities of the fourth dimension. Around the same time as Balla's paintings, it was reported by

Kahnweiler that Picasso "considered setting his pictures in motion using a clockwork mechanism or producing a series of works which could be shown in rapid succession."

^{6}
Marcel Duchamp also used the Fourth Dimension (and other topics from Poincaré's books). In 1911 he began meeting regularly with Princet, who, as mentioned above was a leading proponent of art of the "new geometries". A good example of the fourth dimension in his work is

*The Bride Stripped Bare by Her Bachelors, Even*,

1915-

1923 (often called

*Large Glass*) In this work, Duchamp's goal was to depict the bride as four-dimensional, and the bachelors in three dimensions. The shapes which make up the bachelors' machine are textbook examples of geometric solids seen in one-point perspective. The bride, however, is composed of parabolic and hyperbolic forms which Duchamp considered idealized and typical of a four dimensional object's projection in three dimensions. Duchamp arrived at the fairly logical conclusion that because a three-dimensional object has a two-dimensional shadow, a four-dimensional object must have a three-dimensional shadow. This is the same reasoning that Jouffret used in his projections, and these were certainly Duchamp's inspiration. Although it is not readily apparent in

*Large Glass*, Duchamp did extensive research into four-dimensional perspective. He likens the three-dimensional projection to "the method by which architects depict the plan of each story of a house"

^{7} and continues to discuss how the four-dimensional object is constructed: "A 4-dim'l figure is perceived (?) through an ∞ of 3-dim'l sides which are the sections of this 4-dim'l figure by the infinite number of spaces (3-dim'l) which envelope this figure."

^{8} Though these methods cannot be practically applied, they show the devotion which Duchamp gave to this idea. Some scholars have even gone as far as to claim that Duchamp's famous "

ready-mades" have their roots in Poincaré's work. In an essay on mathematical thought, Poincaré describes how the unconscious mind cannot supply "ready-mades", but that it does constantly sift through ideas which the conscious mind can then select from. A related suggestion is that the photos he took of these ready-made objects from various perspectives were the result of his fascination with projecting

higher-dimensional objects into

lower-dimensional spaces.

The spread of the fourth dimension continued, and the depiction of it became more

abstract as art did. The

de stijl artists in

Holland interpreted the fourth dimension as

negative space.

Van Doesburg used shades of gray to represent negative space, and the

primary colors as

positive space. This interpretation differs from earlier ones because it does not try to represent the fourth dimension a as a physical reality, but conceptually.

Mondrian's appreciation for mathematics led him to his unique style of representing the fourth dimension. He believed that his use of colored planes "by both their dimensions (line) and values (color), can express space without the use of visual perspective."

^{9} By eliminating perspective while maintaining the appearance of a three dimensional space, Mondrian has indirectly represented the fourth dimension. In a sense,

color was Mondrian's fourth dimension. Van Doesburg continued using the fourth dimension after Mondrian abandoned it. He did this in a sort of natural continuation of Mondrian's work, by combining colored planes in three dimensional compositions. In

*Color Construction in the Fourth Dimension of Space-Time*, 1924 he draws colored planes in perspective to create the four dimensional view which Mondrian denounced in 1918. Around this time, Van Doesburg also began experimenting with the fourth dimension as it was interpreted in

Einstein's relativity theory, which was confirmed in

1919 and quickly gained popularity. Van Doesburg also applied the fourth dimension to architecture. In a plan for a house which he drew in 1923, he combined his three-dimensional colored planes with the idea of a hypercube. In this way he combined Mondrian's abstract notion of the fourth-dimension with the original, concrete notion of it. Van Doesburg explains:

"The new architecture is anticubic, in other words, its different spaces are not contained within a close cube. On the contrary, the different cells of space (balcony volumes, etc., included) develop excentrically, from the center to the periphery of the cube, so that the dimensions of height, width, depth, and time receive a new plastic expression.
Thus, the modern house will give the impression of floating, suspended in air, in opposition to the natural force of gravity...
The new architecture takes account not only of space but also of time as an architectural value. The unity of space and time gives architectural vision a more complete aspect."^{10}

By eliminating gravity, Van Doesburg eliminated the idea of an

absolute coordinate system in architecture. No longer was one direction defined as "

down" and opposed by "

up", nor did the words "

left" or "

right" have meaning. All directions were equal, and only their relative orientation to each other mattered. The complex shape of his buildings also required motion in time to view. Van Doesburg was also the only major

artist of the time to embrace Einstein's relativity theory. While it may seem

natural the popularization of relativity theory would lead to a popularization of the fourth-dimension in art, the actual result was just the opposite.

Why? Well it's hard to say. The

driving goal of using the fourth dimension had always been to make an art form that was somehow more ideal, more perfected than previous works. With the realization of the fourth dimension, this idea had in some ways been confirmed, but the thrill of pursuing it was lost. Another thing that should be mentioned is the frequent interpretation of relativity as the basis for cubism. Picasso denied any scientific roots to cubism but his recollections often contradict history as well as each other. Einstein, however, flat out states, "This new artistic ‘

language' (of cubism) has nothing in common with the

Theory of Relativity."

^{11} He seems to reject any connection between

science and art, stating, "In science, the principle of order which creates units is achieved through logical connection while, in art, the

principle of order is anchored in the

unconscious."

^{12}
Works Cited:

1 Apollinaire, Les Peintres Cubistes, 1912, p. 15. Cited in Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art, 1983, p. 75

2 Apollinaire, La Peinture Nouvelle. Cited in Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art, 1983, p. 75.

3 Delaunay, 1957, p. 146. Cited in Miller, Einstein Picasso, 2001, p. 114.

4 Poincaré, La Science et l'Hypothese, 1902, p. 89

5 Boccioni, "Plastic Dynamism", 1913. in, Futurist Manifestos, ed. Apollonio, trans. Brain, Flint, Higgitt, Tisdall, p. 93

6 Wolter-Abele, "How Science and technology changed art", History Today vol.46 no.11 , November 1996, p. 64

7 Duchamp, A l'infinitif, 1966. Cited in Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art,1983, p. 139.

8 Ibid.

9 Mondrian, "The New Plastic Painting", 1917. In The New Art – The New Life: The Collected Writings of Piet Mondrian, 1986, ed., trans. Holtzman, James, p.38

10 Van Doesburg, "L'Evolution de l'architecture moderne". Cited in Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art,1983, p. 325

11 Einstein, letter to Paul Laporte, in Laporte, "Cubism and Relativity with a Letter of Albert Einstein", from Leonardo, vol.21 no. 3, 1988, p. 313.

12 Ibid.

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