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The Mathematical Art of M.C. Escher
“For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art.” — M.C. Escher
— BBC-4, 2005
Douglas R. Hofstadter on M. C. Escher’s drawings
“To my mind, the most beautiful and powerful visual realizations of this notion of Strange Loops exist in the work of the Dutch graphic artist M. C. Escher, who lived from 1902 to 1972. Escher was the creator of some of the most intellectually stimulating drawings of all time. Many of them have their origin in paradox, illusion, or double-meaning.
Mathematicians were among the first admirers of Escher’s drawings, and this is understandable because they often are based on mathematical principles of symmetry or pattern… But there is much more to a typical Escher drawing than just symmetry or pattern; there is often an underlying idea, realized in artistic form. And in particular, the Strange Loop is one of the most recurrent themes in Escher’s work. Look, for example, at the lithograph Waterfall, 1961 (Fig. 5), and compare its six-step endlessly falling loop with the six-step endlessly rising loop of the J.S. Bach’s “Canon per Tonos”. The similarity of vision is remarkable. Bach and Escher are playing one single theme in two different “keys”: music and art.
Figure 5, M. C. Escher, Waterwall, 1961
Escher realized Strange Loops in several different ways, and they can be arranged according to the tightness of the loop. The lithograph Ascending and Descending, 1960 (Fig. 6), in which monks trudge forever in loops, is the loosest version, since it involves so many steps before the starting point is regained.
Figure 6, M. C. Escher, Ascending and Descending, 1960
A tighter loop is contained in Waterfall, which, as we already observed, involves only six discrete steps. You may be thinking that there is some ambiguity in the notion of a single “step”-for instance, couldn’t Ascending and Descending be seen just as easily as having four levels (staircases) as forty-five levels (stairs)% It is indeed true that there is an inherent haziness in level-counting, not only in Escher pictures, but in hierarchical, many-level systems. We will sharpen our understanding of this haziness later on.
But let us not get too distracted now’ As we tighten our loop, we come to the remarkable Drawing Hands (Fig. 135), in which each of two hands draws the other: a two-step Strange Loop.
Figure 135, M. C. Escher, Drawing Hands
Figure 136, Abstract diagram of M.C. Escher’s Drawing Hands. On to, a seeming paradox. Below, it’s relsolution.
“And yet when I say “strange loop”, I have something else in mind — a less concrete, more elusive notion. What I mean by “strange loop” is — here goes a first stab, anyway — not a physical circuit but an abstract loop in which, in the series of stages that constitute the cycling-around, there is a shift from one level of abstraction (or structure) to another, which feels like an upwards movement in a hierarchy, and yet somehow the successive “upward” shifts turn out to give rise to a closed cycle. That is, despite one’s sense of departing ever further from one’s origin, one winds up, to one’s shock, exactly where one had started out. In short, a strange loop is a paradoxical level-crossing feedback loop.” — (D. Hofstadter, I Am a Strange Loop, p. 101-102)
And finally, the tightest of all Strange Loops is realized in Print Gallery (Fig. 142): a picture of a picture which contains itself. Or is it a picture of a gallery which contains itself? Or of a town which contains itself? Or a young man who contains himself’? (…)
Figure 142, M. C. Escher, Print Gallery
Implicit in the concept of Strange Loops is the concept of infinity, since what else is a loop but a way of representing an endless process in a finite way? And infinity plays a large role in many of Escher’s drawings. Copies of one single theme often fit into each’ other, forming visual analogues to the canons of Bach. Several such patterns can be seen in Escher’s famous print Metamorphosis (Fig. 8). It is a little like the “Endlessly Rising Canon”: wandering further and further from its starting point, it suddenly is back. In the tiled planes of Metamorphosis and other pictures, there are already suggestions of infinity.
Figure 8, M. C. Escher, Metamorphosis II
But wilder visions of infinity appear in other drawings by Escher. In some of his drawings, one single theme can appear on different levels of reality. For instance, one level in a drawing might clearly be recognizable as representing fantasy or imagination; another level would be recognizable as reality. These two levels might be the only explicitly portrayed levels. But the mere presence of these two levels invites the viewer to look upon himself as part of yet another level; and by taking that step, the viewer cannot help getting caught up in Escher’s implied chain of levels, in which, for any one level, there is always another level above it of greater “reality”, and likewise, there is always a level below, “more imaginary” than it is.
This can be mind-boggling in itself. However, what happens if the chain of levels is not linear, but forms a loop? What is real, then, and what is fantasy? The genius of Escher was that he could not only concoct, but actually portray, dozens of half-real, half-mythical worlds, worlds filled with Strange Loops, which he seems to be inviting his viewers to enter.”
Inspirations: A Short Film Celebrating the Mathematical Art of M.C. Escher
☞ The Official M.C. Escher Website
☞ The Strange Worlds of M C Escher, Escape Into Life
☞ Mathematical Art of M.C. Escher, Platonic Realms MiniText
☞ Nature by Numbers - a short movie inspired on numbers, geometry and nature by Cristóbal Vila