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Unformatted text preview: ed by the values of
r and θ = ϕ. There are 8 possibilities
(r, θ) = (+, 0), (+, π/2), (+, π ), (+, 3π/2), (−, 0), (−, π/2), (−, π ), (−, 3π/2),
so each of these IFS tables is determined by a point in an 8 × 8 × 8 lattice. A map
of those lattice points that give each topological type might reveal interesting
patterns, but so far this approach has not been pursued. So far as we know,
the ﬁrst general attempt at determining the topological type of gasket relatives
from their IFS parameters was made by Taylor .
Sketch of the proof of Thm 2.9.1: Though it is tedious to ﬁnd a precise relation between small changes in a parameter and changes in the image of the
transformation with modiﬁed parameter, the basic idea is illustrated in Fig.
2.59. On the left is the image of the unit square S under a transformation T .
Continuing to the right, in turn we increase r, s, θ, and ϕ and compare T (S )
with the corresponding T ′ (S ), in bold. Certainly, for any ǫ >...
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