It’s an Illusion, Part 10 – A Different Kind of Space

Hyperbolic Space

Hyperbolic Space

You could think of hyperbolic space as a kind of sphere.  It has several interesting characteristics – if you could draw a loop around it and make it tighter and tighter eventually that loop would become a point.  None of the lines on it are straight, and none meet at right angles. If an object or shape moved from the point closest to you to the visible edges it would quickly grow smaller and smaller.

Escher created his own imaginative images of hyperbolic space, including “Circle Limit III”.  When you look at it you see that the main lines defining the patterns are curved, and there is no straight line anywhere. The repeated pattern is of fish, the lines through the fishes’ bodies defining a series of four-sided spaces each bordered by a three-sided space..  The fins closest to the center are larger, the fins further away are smaller.  The pattern grows smaller and smaller as it repeats, until it becomes too small to see.

It takes skill to represent a three dimensional shape in two dimensions.  It takes imagination to create that shape in a way that allows us to glimpse how hyperbolic space works.  Escher combines both skill and imagination to create this illusion.

Images:

M. C. Escher,  Circle Limit III

Hyperbolic Space Image:  From Wikimedia Commons, by Salix alba, shared under Creative Commons Attribution-ShareAlike 3.0 License.

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