Flatland, Curved Space: How M.C. Escher Illustrated the History of Geometry

Steven Gimbel
Bridges: Mathematical Connections in Art, Music, and Science (2001)
Pages 123–132


M. C. Escher's work Circle Limit III is a graphic representation of one of Henri Poincare's relative consistency proofs for hyperbolic geometry. Poincare, an opponent of the modern conception of mathematical truth, used this proof as the center point of an argument defending the competing view of mathematical truth put forward by Immanuel Kant against the charge that it is completely undermined by the existence of non-Euclidean geometries. This defense led Poincare to limit the scope of geometry to spaces of constant curvature. This limitation was challenged by Hans Reichenbach who generalizes Poincare's argument. This generalization, pictured in Escher's work Balcony, however, uses Poincare's argument to support the modern view of mathematical truth that it was initially designed to attack.