The Fourth Dimension in Painting: Cubism and Futurism …

Posted: March 19, 2011 | Author: Theodor Pavlopoulos | Filed under: Visual Arts | Tags: Art, Cubism, Futurism, Geometry, Mathematics, Visual Arts |

A Protocubist anecdote

Henri Matisses and Leo Steins reaction at first seeing Pablo Picassos 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.

Giacomo Balla, "Girl Running on a Balcony", 1912

Dalis Corpus Hypercubus

Salvador Dali, "Corpus Hypercubus" (1954)

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 Dalis 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, Dalis painting represents a pathway from 4D space (hypercube) towards 3D space (hypercube net) and then towards 2D space (the canvas surface).

A two dimensional projection of the three dimensional "shadow" cast by a rotating hypercube

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.

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The Fourth Dimension in Painting: Cubism and Futurism ...