Escher and the Infinite

In Escher on Escher Escher writes about the diffculty of comprehending the infinite. Just like the passage of time must be punctuated by the ticking of a clock, the infinite extend of space can be perceived by dividing the "universe in distances of a specific length, in compartments that repeat themselves in endless series."
Relatively straightforward approaches, including Sky and Water, were not altogether successful at capturing the infinite, because "... if only the plane on which they follow one another were infinitely large, an infintiely large number could be represented on it. However, we aren't playing a game of imaginings."
One possible solution, hinted in Encounter, is that "We can make a paper tube ... In this way, infinity is achieved in one direction." This approach is not completely satisfying, because of course it does not address the problem of representing the infinite along the axis of the cylinder.
"Curved Beechwood Ball with Fish gives a more satisfying solution: a wooden ball whose surface is completely filled with twelve congruent shapes. When we turn the ball around in our hands, we see fish after fish appear, continuing into infinity." But of course this is not quite the whole story: "twelve similar fishes are something different from infinitely many."
Graphically more interesting is to change the sizes of the pieces. The first attempt, Smaller and Smaller, places larger fishes around the periphery of a square, with ever-smaller fishes converging to the center of the square. While this gives the appearance of infinitely many pieces within a bounded region of the plane, Escher was not satisfied: "this composition also remains a fragment because we can expand it as far as we want by adding ever larger figures."
Inspired by Coxeter's picture of a covering of the Poincare disc by hyperbolic triangles, Escher produced his Limit Circle III pictures. Here the largest animals are in the center of the disc; approaching the boundary the animals appear to shrink till they become vanishingly small.

Return to fractal tilings.