The approach to infinity

In 1959 Escher would write that: "as long as there have been men [...] upon this globe [...] we have held firmly to the notion of [...] all of which must continue to be everlasting in time and infinite in space". He was very interested in finding new ways of depicting the idea of infinite space, and the idea that he found and later developed is depicted in "Development II" (1939). Here "the figures used to construct this picture are subjected to a constant radial reduction in size, working from the edges toward the center, the point at which the limit is reached of the infinitely many and the infinitely small" (Escher)

Another example in the same direction is the wood engraving from 1956 "Smaller and smaller I". The figures no only get small in the center, but they are also metamorphosing, up to the point where they become unidentifiable.

His works in this direction can be grouped into three parts:

They are the starting point and simple in construction. A few examples: "Smaller and Smaller I" (1956), " Plane Filling" (1957),"Square Limit" (1964)

The purpose of these is not so much to represent inifinty as to depict an expansion from infinitely small to infinitely large, and back to small again (birth, growth, decline in a cyclic way). Some examples: "Development II", "Path of Life I and II" (1958),"Path of Life III" (1966), "Butterflies"(1950), "Whirlpools"

After discovering a diagram in a book by the Canadian mathematician H.S.M. Coxeter, Escher built his famous "Circle Limit" series (I-IV) and "Snakes"(1969). These are the works that interests mathematician the most, because they are related to representing the hyperbolic plane using the Poincare model.

• Square-Division Prints

• Spiral Prints

• The Coxeter Prints

 

Geometry behind "Square Limit"

"Square Limit" is a woodcut from 1964.

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