2. Hyperbolic Geometry
Unlike the Euclidean plane and the sphere, the entire hyperbolic plane
cannot be isometrically embedded in 3dimensional Euclidean space.
Therefore, any model of hyperbolic geometry in Euclidean 3space
must distort distance.
The Poincaré circle model
of hyperbolic geometry has two
properties that are useful for artistic purposes:
(1) it is conformal
(i.e. the hyperbolic measure of an angle is equal to its Euclidean measure)

thus a transformed object has roughly the same shape as the original,
and
(2) it lies within a bounded region of the Euclidean plane
 allowing
an entire hyperbolic pattern to be displayed.
The "points" of this
model are the interior points of a bounding circle
in the Euclidean plane.
The (hyperbolic) "lines" are interior circular arcs
perpendicular to the bounding circle, including diameters.
The sides of the hexagons of the {6,4} tessellation shown in
Figure 2
lie along hyperbolic lines
as do the backbone lines of the fish in Figures
3
and
4 .
