Ever so plane and beautiful

M. C. Escher

January 14, 2005

Mathematics was the muse for a singular artist, says Roger Penrose

In his early work, before about 1937, the Dutch artist Maurits Cornelius Escher had followed a reasonably conventional route for a graphic artist, depicting such things as wild southern Italian scenery and striking architecture. But he had, in addition, experimented a little, as a student in the early 1920s, with repeating patterns of human figures. He had also, after his first visit in 1922 to the magnificent 14th-century Alhambra Palace in Granada, with its superb display of Moorish decoration, tried his hand at repeating and interlocking patterns, replacing inanimate Moorish designs with the shapes of lion-like creatures or other animals.

This early work did not attract significant attention, and Escher himself did not seem to find it fully satisfying. Nor did it clearly indicate the unique qualities we have come to recognise as characterising Escher's work.

In 1936, he again visited the Alhambra, and both he and his wife, Jetta, made careful copies of many of the Moorish designs that Escher had previously found particularly intriguing. After much experimentation with "animate" versions of such Moorish patterns, he developed one design into a cleverly interlocking pattern consisting of a Chinese boy that, in May 1937, he incorporated into his Metamorphosis . This extraordinary woodcut starts off, on the left, as a depiction of the Italian coastal town of Atrani, but as your eyes scan across to the right of the picture, the conglomeration of buildings gradually transforms into a geometrical configuration of cubes, which then seems to become a lattice of geometrically embellished hexagons, subsequently to be revealed as interlocking Chinese boys.

In a way, the entire display seems to represent Escher's artistic development, from his realistic depiction of architectural scenery, through repetitive geometrical patterns, to his illustration of this pure geometry in terms of human or animal patterns.

Moreover, the picture as a whole illustrates, self-referentially, the next "dynamic" stage of Escher's art, in which static repetition is brought to life through seemingly gradual changes in the repetition that eventually lead to dramatic reinterpretations of what his patterns actually represent.

Such things are familiar in much of Escher's art where, for example, starting from a repeating pattern of fish, we see the spaces between the fish become birds that then fly off as the fish disappear.

In October 1937, an important development took place in Escher's life.

Through his half-brother Beer Escher, who was a professor of geology at the University of Leiden, Maurits Escher became acquainted with various mathematical writings on the subject of such repeating plane patterns, but his limited mathematical expertise made it difficult for him to understand most of them. An important exception was an article written by the Hungarian mathematician George P"lya, which clearly explained, with the help of diagrams, the mathematical ideas underlying the Alhambra's Moorish designs. This paper was an eye-opener for Escher, and it sent him on a route of mathematical exploration that changed his outlook entirely.

Escher wrote some 20 years later: "I chanced upon this domain (of regular division of the plane) in one of my wanderings; I saw a high wall and as I had a premonition... something that might be hidden behind the wall, I climbed over with some difficulty. However, on the other side I landed in a wilderness and had to cut my way through with a great effort until - by a circuitous route - I came to the open gate of mathematics. From there, well-trodden paths led in every direction, and since then I often spent time there. Sometimes I think I have covered the whole area, I think I have trodden all the paths and admired all the views, and then I suddenly discover a new path and experience fresh delights."

Thus, it appears that Escher's new-found acquaintance with the mathematical principles underlying the Alhambra patterns led him on a route that further distinguished his art from that of all his contemporaries. It is curious that Escher, whose notebooks from this time demonstrate that he had the qualities of a research mathematician, considered himself to be lacking in mathematical ability. At school, his performance in the subject had been indifferent, and his expressed opinion was that he had no talent in this direction.

Perhaps the explanation for this apparent anomaly lies in the fact that mathematical ability consists of at least two main strands that may be considered as largely independent of one another. On the one hand, there is the ability to calculate, whether it be with ordinary numbers, as in performing arithmetical operations, or manipulating abstract algebraic symbols in algebraic calculations; and, on the other, there is the ability to work visually with geometrical configurations. When it comes to examinations, the ability to calculate is frequently favoured, and geometrical talents may go unrecognised if they are unaccompanied by a significant calculational skill. It is clear that Escher had considerable mathematical skills and insights, but these may well have been very much on the geometrical side and perhaps went unrecognised during his schooling.

In her superbly illustrated and presented book M. C. Escher: Visions of Symmetry - first published in 1990 and now in its second edition with the addition of some valuable new material - Doris Schattschneider makes a thorough and illuminating study of Escher's notebooks, explaining a good deal of the theory that Escher developed independently to explain the different kinds of patterns, with their various symmetries, that appear in the Alhambra designs and in a great deal of Escher's art. Indeed, Escher produced his own "layman's theory" for this, as he was not in possession of much of the terminology and technical notation used by the professional mathematician.

It seems that Escher's requirements, being very different from those of professional mathematicians, crystallographers and other scientists with interests in this area, caused his theory to develop in somewhat different directions from those of the professionals. For example, the necessity of using colour to distinguish the different elements of his designs from each other led Escher to consider things that are now called "colour groups", years before professional mathematicians did. Also, his need to be able to "metamorphose" one repeating pattern into another led him to consider the question of deciding which patterns can be so shifted into which others.

This seems to have been something new from the mathematical standpoint.

Accordingly, Escher indeed qualifies as a research mathematician.

For much of his later life, Escher was disturbed by the fact that other artists did not seem to want to pursue the same kind of interests as he did. Indeed, it seems that the artistic community as a whole was for a long time reluctant to accept that Escher's work was really "art" in their sense. Escher was more widely appreciated by mathematicians and scientists than he was by artists. He found this interest gratifying and was most pleased when scientists could appreciate some of the subtleties that would leave many of his artist colleagues cold.

It must be said that many aspects of his work are of a character that is more often associated with scientists than with artists. This applies, perhaps, not only to the underlying mathematical themes, but also to the meticulous precision to be found in Escher's renderings. Yet one also sees mathematical themes and "Escherian" logical paradox in Dal!, in Magritte, in Victor Vasarely and in the works of a good many other artists. Even the exploitation of "animate" repeating patterns had been carried out before Escher, such as in the work of Koloman Moser in the late 19th century.

Nevertheless, there is something unique about the work of Escher, in that his art gains much from the genuine aesthetic qualities inherent in the mathematical themes themselves. The aesthetic qualities of mathematics are not readily appreciated by non-mathematical people, and Escher is able to bring something of those qualities to a more readily accessible level.

Schattschneider's book does an excellent job of conveying not only the wonder of Escher's work but also much of its underlying mathematics. It concentrates almost exclusively on the material arising from Escher's "regular divisions of the plane", but there is some discussion of other topics such as Escher's magnificent depictions of hyperbolic (non-Euclidean) geometry and of three-dimensional geometry.

Schattschneider also provides fascinating historical insights into what compelled Escher to pursue specific directions. We find, for instance, that the creative impetus that informed his early work seemed to be satisfied largely by realistic depiction of Italian scenery. But he left Italy with the rise of the Fascists, and the flat countryside of his Dutch homeland did not offer him the same visual resources. Thus, he began to turn inwards for inspiration, and he found that mathematics, particularly the symmetry of plane designs, supplied him with what he needed.

Finally, this volume provides a great source of Escher reproductions, many of which - so far as I am aware - are not to be found in other books. The reproductions are of fine quality. It is a wonderful experience simply to leaf through the many beautiful designs, to marvel at their cleverness and their artistry, even to appreciate that so many of the repetitions are fundamentally different despite a frequent superficial appearance of similarity - and not worry at all about any of the underlying mathematics.

Sir Roger Penrose is emeritus professor of mathematics, Oxford University, and the creator of the "Penrose tile". He knew M. C. Escher personally.

M. C. Escher: Visions of Symmetry

Author - Doris Schattschneider
Publisher - Thames and Hudson
Pages - 354
Price - £19.95
ISBN - 0 500 511691

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