Tessellation Symmetry:
Resizing / Inflation / Dilation
in Circle Limits, Spiraling, Concentric Rings, & so on

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Tessellation: fish spiral by M C Escher
Escher paints a resizing spiral tessellation

We've already covered the types of symmetry that all tessellation experts agree upon: Translation, Reflection, Glide-Reflection, and Rotation. These are called "isometric", which is a fancy way of saying that the tiles don't change size.

But, what about patterns like "circle limits" that use gradually smaller and smaller tiles as they expand outward, and their opposites, the spirals and concentric circles that use larger and larger tiles as the patterns expand outward?

Many math experts say these are not tessellations. They say that the tiles must all be the same size, and the tessellations must entirely fill a plane.

It's true, these types of patterns might have trouble filling in the centermost point. However, the spirals and circles virtually finish the centerpoint. Such a pattern can so nearly fill the center as barely matters, in the way that a single atom is so small that it barely matters. As Spock of the original Star Trek TV series said, "A difference that makes no difference is no difference."

M. C. Escher did many spiral and circle-limit patterns. He even tried to make "square limit" patterns.

Tessellation: circle limit 3 by M C Escher
A Circle Limit Tessellation by M. C. Escher
Another Circle Limit Tessellation by M. C. Escher
Tessellation: circle limit 3 by M C Escher
An unusual resizing Tessellation by M. C. Escher
unusual resizing tessellation by Hop David Another unusual resizing tessellation. This one was created by Mr. Hop David

example of an inflation (resize, spiral) in tessellation
"Fish Vignette" (1956): This tessellation of concentric rings, done by Escher in 1956, uses rotational symmetry (turning) and inflation (resizing).

Spiraling Dogs: an example of a turn (rotation) and inflation (resizing) in tessellation
A spiral tessellation of dogs by Mr. Hop David of Arizona.
Like the Escher picture above it, this one uses both rotation and resizing.