"Jetting endlessly from cause to cause and camera to camera, he was sprinting toward his own vanishing point." Michael Kelly
"We were doing cubism. ...nobody new what is was. Even Einstein did not know it either!" Picasso
We live in a three dimensional (3D) world, spanned by three directions, left-right, forward-backward and up-down. This seems natural, and obvious, we never think about the consequences of this particular attribute of space. Imagine however, what would happen if we had only one or two dimensions at our disposal. Imagine a one dimensional world where we would have only one line along which we could move about. It would be as if we were confined to a narrow corridor. Our neighbors would remain the same throughout our lives. We would move in unison with them. We would loose much of the freedom we enjoy in the higher dimensions. A similar life style has been described by A. Abbott in his novel Flatland, in which he refers to a two dimensional (2D) universe.
We are somewhat more familiar with issues of dimension related to images or representation as many of the images we use such as, drawings, paintings and movies are projected to a lower, two dimensional world. These issues are much in the focus of our current technical innovations. Holography and sophisticated multidimensional photography enable us to expand the dimensions of representation. 3D is now a common word in the present. However, a classic example illustrates a problem that arises when changing from one dimension to another. In Plato's Republic prisoners are kept throughout their life in a cave. Their only contact with the outside world is trough the images created on the walls by a small opening. One prisoner after escaping to the outside, cannot comprehend the higher dimension: our three dimensional 'real world' which is so different from his world which is based on his two dimensional images.
Although we can not change the world we live in we can create situations where the physical processes we study occur in different dimensions. Electrons, which move around in solid materials are typical examples. In a typical metal like copper or aluminum electrons can move in all three directions. They can avoid each other and thus their interaction is not important. We can however, design materials where electrons move only back and forward in one direction along chains which build up the material. Or, in two dimensions, such as at the surface of materials. As the electrons cannot avoid one other they strongly feel one others presence and line up in a periodic array, called density waves or electron crystals. They can be observed in a fashion similar to how crystals of atoms are seen. The consequence of such a state is the collective motion of all electrons, much in the way soldiers or a marching band proceeds. The motion of all electrons is step by step all at the same time. We can hear this by designing sensitive experiments.
Electrons also respond differently to the presence of a magnetic field in one or two dimensions than from electrons in a higher dimensional space. This observation (Which goes by the obscure name "Quantized Hall effect") is regarded as one of the most fundamental discoveries of the last decade in physics.
Let us return to three dimensions and to the task of representing this on a flat surface (2D). The method of achieving this is referred to as linear perspective. It is based, at least in an intuitive way on the notion that parallel lines never meet except at infinity. This is the famous fifth axiom of Euclid. It has to be postulated as one of the cornerstones of the (so-called Euclidean) geometry, as it cannot be proved starting from other known facts or observations. The concept of progressive diminution of objects which are further away from us is the consequence of this seemingly trivial geometry. This concept is governed by strict rules which relate the size of the object or the distance between objects to their distance from the viewer. Parallel lines, running away from us look as they would meet somewhere, we know however that this could happen only at infinity. The point where they apparently meet is called the vanishing point, the representation we call linear perspective. A simple sketch illustrates the concept. (Figure 6.1) The Greeks viewed stenography as:
the shading of the front and retreating sides, and the correspondence of all lines to the vanishing point, which is the center of the circle.
Vitruvius defines their approach as highly scientific, as explained in books written about perspective, the first perhaps by Agatharcus of Samos in 500 B.C..
According to Ervin Panofsky, a prominent art historian. "Each period in western civilization had its own "perspective", a particular symbolic form reflecting a particular Weltanschauung or world view"
Perspective the way we know it was not exercised by the Egyptians, the painting was not intended to be a faithful representation of our world, instead they represented the best, most desirable aspects of humans and their surroundings.
Strict geometric construction was applied by the Greeks and Romans, but - surprising in the light of the scientific concepts known at the time- the notion of vanishing point was not employed. Instead the reclining lines are either parallel, or converge towards a different point, all meeting in the central axis (Figure 6.3)
This had been lost in the middle ages. Size and proportion was used to emphasize not physical but spiritual relations. (Figure 6.4) Real space representing our everyday world was regarded as less important than the spiritual space which provided a background for the figures represented by undefined dimensions.
The linear perspective was rediscovered by Della Francesca, Brunelleschi and Alberti late in the 14th and early in the 15th century, at the start of the Renaissance; in a period where most of the artists were also well versed in geometry, and other aspects of mathematics. There are conflicting accounts of who can be regarded as the reinventor of the technique. The first examples of linear perspective emerge through a gradual progression of ideas throughout the 13th and 14th centuries. Then Pier Della Francesca, a painter, used the concept of the vanishing point in his art. In addition, Brunelleschi developed a method which best illustrated the concept of perspective. But most clearly, the book Della Pittura by Alberti summarizes in a systematic and scientific way the main elements of this new representation of three dimensional space on a two dimensional surface. In accordance with the Greeks depth is represented by converging lines pointing towards the vanishing point placed at infinity and the size of the objects corresponds to the distance at which they are shown. The first painting where this concept was tested to full scientific rigor is the Flagellation of Christ by Pier della Francesca. This work was used throughout the renaissance as the example of linear perspective. (Figure 6.5) This linear, single point perspective representation has been fully employed by many painters who also developed an array of tools to achieve the correct proportion and therefore perception. A page from Durer's instruction manual serves as the illustration of what tools were used to achieve perfect perspective. (Figure 6.6)
The focus provided by the vanishing point has also been used to emphasize meanings and allegories. On the painting The Last Supper by Leonardo da Vinci the vanishing point coincides with the imaginary halo above the head of Christ. (Figure 6.7)
What we did before is to transform, (or represent) the three dimensional universe in a two dimensional flat universe given by a canvas or a wall. We can also represent the world of different dimensions in another dimension and describe how such representation - or transformation - would work for simple objects and for cases where both dimensions are flat, and there are no curved surfaces.
Let us look at a cube, a three dimensional object (one of the perfect bodies) and view it from one side. The vanishing point is in the middle and we see the front and back faces. (Figure 6.8) Albers squares are an example of such a construction. The figure can be regarded as a representation of the cube in two dimensions. Another way to represent the cube is to spread out the surface of the cube until you arrive at six squares. (Figure 6.9) Of course in both cases we have to provide some additional information if we intend to reconstruct the cube. In the first case, the lengths of the lines connecting the two faces have to be specified. They have to be equal to the sides or we have to specify the distance we are from the cube. In the second case we have to provide instructions for folding the surface to fully enclose a space.
The same can be done if we want to represent a two dimensional object, such as a square, in one dimension. The one dimensional space is a line. The faces of the square now correspond to lines, and the spreading out of the boundary leads to four equivalent lines representing the 4 sides. (Figure 6.10)
We can then follow the same procedure for a hypothetical four dimensional (4D) cube, called the hypercube. In both cases we end up with a 3D construction, reducing the representation by one dimension going from 4 to 3. (Figure 6.11), (Figure 6.12) Instead of two squares we end up with two cubes for the face on view representation. The surface of a 4D cube is a collection of 3D cubes, and now we need eight (instead of the six squares) to have the "surface" of the hypercube.
All this can be summarized in a Table summarizing the surfaces of objects in various, zero 0, one 1, two, 2 dimensions and so on :
| 1D | line | represented in 0D by | 2 points |
|---|---|---|---|
| 2D | square | represented in 1D by | 4 lines |
| 3D | cube | represented in 2D by | 6 squares |
| 4D | hypercube | represented by in 3D by | 8 cubes |
A similar table can be assembled using the concept of face on representation discussed earlier.
Of course we can perform the same exercise for other perfect bodies, such as the tetrahedron. It takes five triangles to make a tetrahedron and therefore six tetrahedra to make a 4 dimensional hypertetrahedron. The procedure only works for objects defined by flat surfaces. Curved objects cannot be reduced to a lower dimension without being distorted.
The linear perspective, has been abandoned because of scientific discoveries in geometry and physics which revealed new dimensions and new perspectives. This influence is expressed by one of the great Hungarian abstract painters of this Century, Gyarmathy:
Art - as always- delt with logical, intellectual connotations only. It was the Euclidean space above all, and emphatically so. Streets were constructed in a way to make houses smaller, stairs converging, just to let the principles of perspective to assert themselves. All this has allowed Euclidean space to go to an extent where the whole thing became doubtfulThis statement was probably inspired by the scientific discoveries on space and dimension. As the notion of space and dimension underwent drastic transformation in geometry and physics a conceptual transformation also occurred in the arts of the same period. The idea emerged that there is a different to represent the third dimension: showing the same object from different angles on the same painting. This concept is an important aspect of Cubism, which was started by Braque and Picasso early this century. In the painting of a mans head by Picasso the nose, mouth, ears and the head itself are displayed from different perspective. (Figure 6.13) This style provides a much fuller representation than linear perspective can offer. We loose the geometric visual order we are accustomed to, the gain different sides of an object. This aspect was expressed by the Egyptians centuries earlier.
Cubism, and the scientific concept of higher dimensional space in our physical word appeared nearly simultaneously. Einstein's theory of relativity dealing with a four dimensional space was published in 1906, the first cubist paintings emerged around 1910. This is probably not a coincidence, but most likely an excellent example of the influence of scientific notions of the time on the artistic expression of the same period.