Τετάρτη, 30 Ιουλίου 2008

THE LINE BETWEEN MATHEMATICS AND ART

(Article by Apostolis Papanikolaou – museum Herakleidon Athens)

M. C. Escher spent his artistic enterprise representing ideas “that overwhelmed him to such an extent he felt he had to share them with others”, “constantly crossing the line between Mathematics and Art”.

Given the limited amount of space, this text does not naturally aspire to expound on the mathematical and philosophical background of M. C. Escher’s artistic representations or pursuits but rather to serve as a reference guide for further and more in-depth search.

The bounds of Mathematics and Art were defined 2500 years ago by Plato in the tenth book of his dialogue Republic, where, having previously (in the fourth book ibid.) divided the psyche into two parts, a superior (logic) and an inferior (willful/ wishful) one, he went on to regard Mathematics as connected to the reasoning (logical) part while Art as connected to the inferior (emotional) part.

Having studied Plato’s treatment of Art, I stand assured that if Plato were to come to life and lay eyes on Escher’s works, he would recognize in the latter’s face one of the artists that he so passionately sought for his ideal State: «Seeking those creators that are intelligently able to search out the nature of the good and the decent», while he rejected the imitators who drew “merely holding a mirror up to nature”, thus ascribing it superficial, external characteristics. These are the words that Plato used some 2500 years ago to place artistic representation up on a high pedestal, refusing to consider Art a realistic-slavish depiction of reality. In another of his dialogues, “Philebus”, his description of the absolutely beautiful is reserved for the kind of drawing that uses geometrical shapes as its basic components. Many took this as a warning sign for the appearance of 20th century Modern Art, which, in the words of Paul Klee, “does not reproduce the visible; rather it makes visible”.

Observing Escher’s works, one perceives Plato’s very urge in “Philebus” come true. For the works’ basic constituents are geometrical shapes indeed and the existence of a mathematical substratum evident. But where does the purely mathematical background begin and where does Art come in?

The divide between Mathematics and Art, as it was conceived by Escher himself, becomes apparent in one of his 1958 passages Regular Division of the Plane. The text is about his renowned tessellations (repeated tilings), which by 1936 he had already devised, after the respective Arabian style, a technique he picked up while visiting Spain for the second time and through which he created his own tilings, using animal images (which the Koran prohibited Arab artists to use).

“The regular division of the plane has been theoretically examined by mathematicians ... Does this mean that it is a purely mathematical concern? Not
if you ask me. Mathematicians have opened the gate leading to an extensive domain but they themselves have not entered this domain. For, by nature, they are interested in the way the gate opens rather than in the garden that lies behind that gate ...”

Evidently, Escher could clearly make out the special aesthetic value that can result from the artistic representation of mathematical-philosophical concepts.

Mathematicians are not in search of the beautiful so much as the true. They seek out the structure of an appearance, and it is not necessarily a beautiful structure. Even symmetry is a structure conditional on another wider structure. Mathematics presents a mathematician with an intrinsically familiar aesthetic ring but which is inaccessible to the ordinary average mind. As well-known researcher-mathematician Thanasis Fokas puts it: “The existence of aesthetics in Mathematics is conditional, for Mathematics expresses truth, and beauty is the hallmark of truth.”

So, Escher determined to enter the “realm” of Mathematics and draw from it anything he could have artistically represented, thus enabling even the non-mathematician layman art-lover to perceive this kind of beauty.

As we have already mentioned, his first “visit” to the mathematical realm was the regular and semi-regular plane tessellation.

In October 1937, Escher showed some of his new works to his brother Berend, a geology professor with Leiden University at the time, while they were both visiting their parental home in the Hague. Berend, acknowledging a connection between those woodcuts and crystallography, sent his brother a list of articles that he felt would help him. That was Escher’s first encounter with Mathematics.

Escher paid special attention to an article by mathematician Polya, which referred to plane symmetry groups and from which he was able to get a very good grasp of the 17 plane symmetry groups that were described.

Between 1937 and 1941, Escher worked on periodic tilings, producing 43 coloured drawings that exhibited a wide range of symmetry types.

In 1941, Escher wrote his first article, entitled Regular Division of the Plane with Asymmetric Congruent Polygons, through which he practically researched the artistic representation of themes in crystallography.

In 1954, he met mathematician Coxeter, one of the 20th century’s (1907-2003) greatest geometricians, and they became close friends. Through their correspondence, as well as the study of articles and books suggested by Coxeter, Escher re-entered a mathematical realm, that of non-Euclidean Geometries. The hyperbolic Geometry model, already introduced by Poincare, as well as other similar models, led Escher to the creation of a series of woodcuts, entitled Circle Limit I-IV, a sample of which, Circle Limit III, you can see below.

Coxeter published a series of articles in which he admiringly commented on Escher’s works. One such article was The Non-Euclidean Symmetry of Escher’s Picture ‘Circle Limit III’ – Leonardo – 1979.

Escher’s contact with another great mathematician, Sir Roger Penrose, brought him into the field of topology and the impossible figures that Swedish artist Oscar Reutesvärd had already introduced. The “impossible triangle” and the “impossible scale” inspired him into creating his own impossible forms: Waterfall, Relativity, Ascending and Descending, and so on.

The big question that arises from this series of works, which held Escher’s fascination, is how the brain can “read” reality through pictures. To what extent is the knowledge of an object feasible through its mere image? The great surprise is that these representations, which at first sight seem impossible, can actually be images of existing objects!

Swiss crystallographer Necker’s ambiguous cube and the three-dimensional cube, an impossible form, inspire Escher into creating the “Belvedere”.

Still drawing on the field of Topology and in a uniquely artistic way he represents Moebius strip: a seemingly double-faced surface which is in reality single-sided and “non-orientable”. Ants do not live in separate compartments and are able to meet. In his works Print Gallery and Cube with Ribbons he explores the logic and topology of space.

Escher could not have stood indifferent to mathematicians’ probes into the concepts of truth and falsehood, the foundation of Logic and the production of thinking machines, i.e. Artificial Intelligence. The woodcut entitled “Drawing Hands” depicts the problem-concept of self-reference, a sentence by which a person referring to themselves attaches an attribute, or predicate, to them. A historical self-referencing proposition is the one put down to Cretan Epimenides: “All Cretans lie”. The search for the truth or falsehood of this assertion leads to continuous contradictions. A ‘logical’ machine, a computer, cannot decide on the truth or falsehood of such a proposition. But let us have a look at the proposition underlying Escher’s woodcut: A painter’s hand can draw anything. If this holds true, can the drawing hand draw itself in the act of drawing?

But beyond the matters of typical Logic and Artificial Intelligence that this woodcut raises, there is also the great philosophical matter of self-awareness to consider. Only through cognizance can a cognizant being represent, and thus come to know, itself. A painter’s hand is, in Escher’s case, the hand of cognizance that can draw its very image.

Escher’s frequent and favourite concern was to offer his idea of the creation of the Universe and how it evolved. His way was invariably geometrical and profoundly inspired by diachronic philosophy, a thought which pervades his painting “Verbum” (logos):

To ancient Greeks, Apollo, who is shown in the middle, was the god of the sun, but also that of Logic. The Universe was created through mathematical ratio – specific arithmetic and geometric analogies – this is why to come to know it is “true opinion (belief) with an account (logos)” (Plato – Theaetetus).

The concept of self-similarity, that is a property according to which certain shapes exhibit the same structure in any scale change and are thus similar to one or more of their parts, is clearly shown in the painting entitled Exploring the Infinite 116.

What is shown in the paintings entitled Depth and Cubispace is the concept of discreet infinity, while in Limitsquare we can see the concepts of the infinitesimal and limit. But, of course, there is a long list of similar samples.

Plane symmetry groups, impossible figures-topology, space tessellation, self-similarity, hyperbolic geometries, reasoning paradox and the philosophy of creation all found their way into Escher’s works, through unparalleled artistic representation.

Can the deservedness of the gratitude and respect to which Escher has been – and still is – held by mathematicians and other scientists alike be questioned?

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