About Hyperbolic Tessellations

In July 2009 I visited the MC Escher exhibit at the Portland Art Museum (Oregon)…twice! Seeing MC Escher’s Circle Limit IV (Angels & Demons) blew me away.  The display of the graph system he used to help him create it and his subsequent sketches, intrigued me even more.  After several months of research, I created my own graph to use as a guide for creating my own Hyperbolic Tessellation mandalas.

The Process

I free-hand draw my ideas on a piece of tracing paper over my hyperbolic graph.  When I finish the sketch, I then scan it.  I import the scanned image into a vector graphics app where I free-hand trace over one section of the image. I take that section and copy/paste it around the circle. Because the design is free-hand drawn, it becomes a bit of a challenge to line everything up properly since some of the edges can be off just a bit and requires manual adjustments. The whole process takes about 3-4 hours per design.

I know that software applications exist to make this process easier and more precise however I prefer the natural/organic results.

About Hyperbolic Tessellation Mandalas

• A mandala, in Sanskrit, means circle, center, or sometimes sacred circle. Webster’s Dictionary defines it as “a design symbolizing the universe.” Carl Jung calls them “archetypes of wholeness” and used them for his own and his patients therapy.
• A tessellation is a pattern made of identical shapes, where the shapes fit together without any gaps and without overlapping each other.
• Simply put, hyperbolic plane is a flat, 2-dimensional representation of a 3-dimensional object.
• A hyperbolic tessellation uses a tessellation pattern, repeating it over a hyperbolic plane.
• A hyperbolic tessellation mandala uses a tessellation pattern, repeating it around a hyperbolic circle.

Disclaimer

First and foremost, I am an artist who is fascinated with and dabbles in geometry. I leave the science of it up to the amazing mathematicians of the world.

To Learn More…

• Sculptural Forms from Hyperbolic Tessellations
• David Joyce, Department of Mathematics and Computer Science at Clark University
• Math and Art of MC Escher, St. Louis University
• Don Hatch, Hyperbolic Planar Tessellations
• Dmitry Brant, Hyperbolic Tessellations
• Douglas Dunham, Transformation of Hyperbolic Escher Patterns
• Jos Leys, Mathematical Imagery Blog: Hyperbolic Chamber
• David Eppstein, The Geometry Junkard

The order of this list reflects how I found them via Google versus any judgment regarding their order of expertise.  – Maureen

About Tessellations