Basic concepts

The major objective of this article is to demonstrate through art the concept of dimension beginning with the point and ending up with a multidimensional space. These concepts are usually perceived as abstract notions and it is believed that through art the brain can grasp and visualize them more easily.

The concept of dimension appears in many fields in science such as mathematics, physics, science fiction, topology, etc. The following definitions are based on Wikipedia, the free encyclopedia. For common usage, dimension gives an indication about the shape and size of an object. On the other hand space is defined as that part of the boundless four-dimensional continuum in which matter is physically extended. In a more daily language, it is that part of the universe that lies beyond the earth’s atmosphere.

The building up of the concept of dimension becomes easier if we display first the definition of point, line and plane. A point is a zero-dimensional object having no extension in space. If we think of the point as a geometric vector, it would look like an arrow but without length. This vector is called the zero vector. A line is a one-dimensional object, namely, a nonzero vector with some definite length. Even though a line has no thickness, if we are to draw a line, it has some small thickness, so we can see it. A plane is a two-dimensional (2D) object. It has length and breadth but no thickness, somewhat like a sheet of paper (but paper too has some thickness). Space, as we perceive it, is three-dimensional (3D) and is composed of planes. These planes are "stuck together" like a sandwich. And, finally, a multidimensional space, having three or more dimensions, for example, the space-time in which we live, is four-dimensional (4D), namely, three spatial dimensions and one of time.

Following the above definitions the concept of dimension is elaborated. For one dimension, we think of stretching only one nonzero vector. For two dimensions, we need to stretch and add together two nonzero vectors. For three dimensions, we need to stretch and add together three nonzero vectors. Notice that the dimension of the object we think about matches the number of vectors needed to form that object. The vectors we need are called a basis, and this idea of dimension is called the Hamel dimension.

In reality, human beings are used to spaces ranging from one to three dimensions, which are also perceived by the brain. However, when looking at a two-dimensional picture, the retina of the eye is a two-dimensional array of receptors but can perceive the nature of three-dimensional objects using some clues. For example, artists use perspective to give three-dimensional depth to two-dimensional pictures as well as combinations of light and shadow to achieve depth. When these tricks are absent, the 3D view is reduced to 2D. These effects will be demonstrated in the following where the picture in Fig.1, slightly modified, painted by Alex Grey (1953) an American artist of the anatomy of the body, demonstrates perception of perspective in the brain.

Fig.1: Perception of perspective in the brain

Demonstration of the basic concepts by artworks

The fundamental quantity of point is demonstrated in Fig.2 by two artworks by the French Pointillist artist Georges Seurat (1859-1891). Here the artist applied pure colors in small dots in order to achieve more subtle areas in the artwork.

Fig.2: The point by artworks of Pointillism stream

The straight line is demonstrated in Fig.3 by different artworks. Although a line in principle has no thickness, if we are to draw it, it has some small thickness, so we can see it. Fig.3a, painted by Piet Mondrian (1872-1944) a Dutch Geometric constructivist, demonstrates parallel and perpendicular lines. Fig.3b demonstrates lines that look non-parallel. However they are totally parallel once you look at the picture from the side. The illusion of non-parallelism arises from the fact that each line is composed of slightly inclined black and white lines that give the mistaken impression. Fig.3c are lines of growths of different colors in fields in Holland.

Fig.3: Lines by artworks

A plane is demonstrated in Fig.4 by different artworks. The American artist Jasper Johns (1930) painted Fig.4a while the German artist Ludger Tom Ring the Younger (1522-1584) painted Fig.4b, in which the page demonstrates a plane. Ludger belonged to the Trompel’oil art stream that painted artworks with absolute similarity to reality. Fig.4c was painted by the Dutch artist Cornelis Norbertus Gijbrechts (1926-1987) and Fig.4d by the American artist John Haberle (1856-1933); both artists belonged to the above art stream. Fig.5 demonstrates the transition from a plane to a spatial construction where the basic elements are Azrieli Towers in Tel-Aviv. The left-hand side picture demonstrates the 2D shape of the towers where on the right-hand side the real towers are demonstrated. The two other pictures are the transition steps.

An important effect in generating 3D patterns is shadow. This is demonstrated on the right-hand side of Fig.6. Once the shadow is removed, as on the left-hand side, a two-dimension sensation is achieved. The artwork in Fig.7 by Ron Miller [1], a space artist, demonstrates the solar system. Different effects are used by Miller to create the feeling of a 3D space on the bottom picture. Once these effects are removed, a 2D space is generated on the top painting, which is a modification of the bottom one. It should be emphasized at this point that from Fig.8 and forth, different techniques are presented for creating a 3D space and higher dimensions on a 2D space, namely on a paper page.

Fig.4: Planes by artworks

Fig.5: Azrieli Towers: from 2D on the left to 3D on the right

Fig.6: The effect of shadow on the transition from 2D to 3D spaces

Fig.7: 2D and 3D spaces

Fig.8 is the lithograph “Reptiles” (1943) by M.C.Escher (1898-1972), a Dutch Illustrator. Here we see how Escher has been putting together some ideas for periodic drawings. At the lower left-hand edge the small, flat 2D, sketchy figures begin to develop a fantastic three-dimensionality and thereby the ability to creep right out of the sketch.

Fig.8: The transition from 2D to 3D according to Escher

Figs.9-1 to 9-4 demonstrates stereograms in which a transition from 2D to 3D takes place. A stereogram is a picture comprised of a repeating pattern with very small differences between them. The combined image of the 2D patterns is more than the sum of its patterns. It is a 3D stereo picture created in the brain. The word “stereo” comes from the Greek word “stereos” which means firm or solid. With stereo vision you see an object as solid in three spatial dimensions – depth, height and width. It is the added perception of the depth dimension that makes the stereovision so incredible. It should be noted that according to Bekenstein [4] the universe could be like a gigantic stereogram.

There are a few techniques for looking at a stereogram. For example:

• Place your nose on the center of the picture.
• Imagine you are looking through the picture towards a point far in the distance.
• Move the picture (or your head) a way extremely slowly and continue to look through the picture. Keep also your peripheral awareness of the four corners of the picture.
• At a certain distance a 3D image will appear in your brain.

Fig.9-1: In the 3D stereogram you see two animals in the center

Fig.9-2: The 3D space demonstrates Uranus, Saturn and Jupiter at different depths

Fig.9-3: In the 3D stereogram you see a butterfly (Gene Levine)

Fig.9-4: In the 3D stereogram you see Rodin’s “The thinker”

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