4. Transformation of a Pattern
The basic version of the computer program that performs the
transformation requires that the motif be contained
in one of the
p isosceles triangles formed by the radii of a
pgon.
Figure 3 below shows the isosceles
triangles within a 6gon of {6,4}
that is the basis of Escher's
Circle Limit I
(shown in gray).
A natural motif in
Circle Limit I
is composed of a black halffish and an adjoining white halffish,
however such a motif has part of a white fish fin protruding outside
its isosceles triangle. This motif can be modified to the required form
by clipping off the protruding part and "gluing" it back between the tail
and the back edge of the fin of the black fish.
The program has been extended slightly so that this modification is often not necessary. The extended program also seems to work reasonably well with a motif that overlaps two adjacent isosceles triangles (with roughly half the motif in each triangle)  as is the case with Circle Limit IV (Figure 1).
The basic transformation process makes use of the Klein model of
hyperbolic geometry. As with the
Poincaré
model, the points are interior points of a bounding circle, but the
hyperbolic lines are represented by chords. We let
I
denote the isomorphism that maps the
Poincaré
model to the Klein model. Then
I
maps a centered
pgon
with its isosceles triangles to a regular
psided
polygon which also contains corresponding isosceles triangles.
Different tessellations
{p,q}
produce different isosceles triangles in the Klein model,
but an isosceles triangle from
{p,q}
can be mapped onto an isosceles triangle from
{p',q'}
by a simple (Euclidean) differential scaling, since those isosceles
"Klein" triangles are represented by isosceles Euclidean triangles.
Thus the transformation from a
{p,q}
pattern to a
{p',q'}
pattern can be accomplished by
Using similar techniques, another program has been written to transform isosceles Euclidean triangles to isosceles hyperbolic triangles, and thus Euclidean Escher patterns (of which there are many) can be transformed to hyperbolic patterns. 

