Escherian Images Using Hyperbolic Tilings

A guided research project in Geometry & Dynamics.

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Experience Research Geometry & Dynamics

Question

How can one motif repeat across a hyperbolic tiling without gaps or overlaps?

Hyperbolic tiles appear smaller near the disk boundary even though their true sizes are equal.
Hyperbolic tiles appear smaller near the disk boundary even though their true sizes are equal.
A dense symmetric tiling in a Poincaré disk.
A computed hyperbolic tiling from the source research repository.

Goals

  • Construct a symmetric tiling in the Poincaré disk.
  • Transform a chosen motif into an Escher-like repeating image.

Method

Use reflections and hyperbolic isometries to copy one decorated tile.

  • Choose a regular hyperbolic tiling and draw its fundamental polygon.
  • Design a motif whose boundary matches neighbouring copies.
  • Apply the tiling symmetries, clip the copies, and render the full disk.