In reality, hyperbolic
space is infinitely large. Like the Euclidean plane it goes on forever.
But in order for us to depict it within our Euclidean framework
we have to make some compromises. The Poincaré compromise
is to represent angles truly while distorting scale. In this diagram
all the sides of all the triangular shaped areas are, in fact, of
equal length.
In his book Science and Hypothesis (1901), Poincaré wrote
of his model as an imaginary universe. To us, as observers of this
bubble world, the inhabitants of the disc appear to shrink as they
approach the boundary of the disc. They, however, see no such effect.
As far as they are concerned, they live in a perfectly normal non-shrink
space, albeit one that is not Euclidean. It is only we, confined
to view them in a Euclidean framework, who see their dimensions
behaving strangely.
The Poincaré disc model has entered the artistic lexicon
through the work of the Dutch artist M.C. Escher, who was introduced
to the concept by the great geometer Donald Coxeter. With his “Circle
Limit” series of drawings, Escher explored the endless symmetries
inherent in hyperbolic space: in “Circle Limit III,”
red, green, blue and yellow fish tessellate their world in a symphony
of triangles and squares. In “Circle Limit IV” angels
and demons disport themselves in a hyperbolic trinity, fluttering
out from a central point to fill their space with hexagons and octagons. |