5. Examples Based on Circle Limits I and IIM. C. Escher created four patterns using hyperbolic geometry: Circle Limit I, Circle Limit III, Circle Limit III, and Circle Limit IV. THese are shown as Catalog Numbers 429, 432, 434 (and p. 97), and 436 (and p. 98) of (Locher (1982)). We have seen renditions of Circle Limit I (Figure 3) and Circle Limit IV (Figure 1). Later we will also show renditions of Circle Limit II (Figure 7) and Circle Limit III (Figure 9). Escher's Circle Limit prints
First we consider the pattern of
Figure 5 below,
which shows another transformed
version of the Circle Limit I pattern that is
based on the {6,6} tessellation.
Escher was unsatisfied with his Circle Limit I pattern
(Figure 3) for three reasons:
In examining
Figure 5, we notice that three
backbone lines pass through the meeting points of the fish noses.
This leads to the conjecture that the pattern could be colored
with three colors, one color per line of fish. Indeed this can
be done as is shown by Figure 6 below.
This is an answer to Escher's second criticism. Of course Escher
himself successfully addressed all three criticisms with his
Circle Limit III pattern
(Figure 9).
Circle Limit II is the least well known of Escher's
hyperbolic patterns. A rendition of Circle Limit II is
shown below in
Figure 7.
This pattern is based on the tessellation {8,3}.
There are 3fold rotation points to the left and right at the end
of each of the four cross arms. These eight 3fold points form
the vertices of a regular octagon. As with Circle Limit VI,
the number of cross arms must be doubled to obtain the number of
vertices of the underlying octagon, due to the bilateral symmetry
of the cross arms.
This pattern was undoubtedly the first repeating hyperbolic pattern
with perfect 3color symmetry.
It seems natural that this pattern could be transformed to a
pattern of crosses with any number of arms and linked together
at rotation points of any valence. Indeed such a pattern can
be constructed from the tessellation {p,q} as long as
p is even (due to the bilateral symmetry of the crosses).
The pattern will be hyperbolic if (p2)(q2) > 4.
Figure 8 below shows a sample
pattern of 5armed crosses based on the tessellation {10,3}.


