199-A A note on the Circle Limit – III

A note on the Circle Limit – III

The Circle Limit III pattern can be named as (4, 3, 3) i.e  4 No of Fish meeting at Right Fins,3 No of Fish meeting at Left Fins, 3 No of Fish meeting at nose tips.

Great artists like M.C. Escher often explored the concept of infinity in their work. The Poincaré Disk Model for Hyperbolic Geometry provided them with a means to depict infinitely repeating patterns within a finite, bounded area while maintaining recognizable shapes, thanks to conformality. 

In this model, we can liken Escher's artwork to standing on a hilltop on Earth. Distant objects appear small, while closer ones seem larger—a consistent phenomenon regardless of location on the planet.

Each fish in Escher's Circle Limit III patterns undergoes an infinite pattern, yet equal hyperbolic distances correspond to progressively smaller Euclidean distances toward the edge of the disk. Consequently, all fish in the patterns are of the same (hyperbolic) size. The appeal of the Poincaré disk model to artists, including Escher, lies in its ability to represent infinitely repeating patterns within a defined space.