Gödel, Escher, Bach and Hofstadter

Hofstadter in Bologna, Italy, in March 2002 by Maurizio Codogno

Douglas Hofstadter wrote a popular book in 1979 called Gödel, Escher, Bach to celebrate the work and ideas of Gödel, along with those of artist M. C. Escher and composer Johann Sebastian Bach. The book partly explores the ramifications of the fact that Gödel's incompleteness theorem can be applied to any Turing-complete computational system, which may include the human brain.

"Escher, Metamorphosis II" - Source: Official M.C. Escher website..

Waterfall - M.C. Escher

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, 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.

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. 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.