In order to demonstrate hyperbolic geometry, the French mathematician Jules Henri Poincare used a model in which the whole of an infinite flat plane was shown as being within a large finite circle.
From the hyperbolic point of view, no points exist on or outside the circle. All the characteristics of this type of geometry can be deduced from this model. Escher discovered it illustrated in a book by Professor H. S. M. Coxeter and he immediately recognised in it new possibilities for his approaches to infinity.
– Excerpted from The Magic Mirror of M. C. Escher
This was how the Circle Limit Series emerged and below is Circle Limit IV the last in the series.