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Introduction: A Musico-Logical Offering

12

remarkable. Bach and Escher are playing one single theme in two different "keys": music

and art.

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 (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. 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

FIGURE 7. Hand with Reflecting Globe. Self-portrait In, M. C. Escher (lithograph,

1935).

Introduction: A Musico-Logical Offering

13

Introduction: A Musico-Logical Offering

14

FIGURE 8. Metamorphosis 11, b\ M. C. F.seller (woodrut. 19.5 cm. x 400 cm., 1959- 40).

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. 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? (Incidentally, the

illusion underlying Ascending and Descending and Waterfall was not invented by Escher,

but by Roger Penrose, a British mathematician, in 1958. However, the theme of the

Strange Loop was already present in Escher's work in 1948, the year he drew Drawing

Hands. Print Gallery dates from 1956.)

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 n 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. 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,