Escher Tile Deformation via Closed-Form Solution

TL;DR

This paper introduces a real-time, closed-form deformation method for Escher tiles, enabling seamless, controllable tessellations with applications in art, fabrication, and animation.

Contribution

It provides the first analytical, closed-form solution for deforming Escher tiles while maintaining seamless tessellation and interior consistency.

Findings

Effective deformation of Escher tiles across 17 wallpaper groups

Supports interactive, artist-controlled local deformations

Applicable to photo editing, shape sculpting, and fabrication

Abstract

We present a real-time deformation method for Escher tiles -- interlocking organic forms that seamlessly tessellate the plane following symmetry rules. We formulate the problem as determining a periodic displacement field. The goal is to deform Escher tiles without introducing gaps or overlaps. The resulting displacement field is obtained in closed form by an analytical solution. Our method processes tiles of 17 wallpaper groups across various representations such as images and meshes. Rather than treating tiles as mere boundaries, we consider them as textured shapes, ensuring that both the boundary and interior deform simultaneously. To enable fine-grained artistic input, our interactive tool features a user-controllable adaptive fall-off parameter, allowing precise adjustment of locality and supporting deformations with meaningful semantic control. We demonstrate the effectiveness of our method through various examples, including photo editing and shape sculpting, showing its use in applications such as fabrication and animation.