Unformatted text preview: fractals. Of course, the changes presented in
Fig. 2.55 aren’t exactly small.
On the right of Fig. 2.55 we see the eﬀect of θ1 = ϕ1 = 0.01. Holding the
page at arm’s length, or squinting slightly, we see that the changes between
this shape and the gasket are imperceptible. On the other hand, topological
characteristisc, such as being connected, are discrete: a set is connected or
not connected, there is no middle ground. Even this tiny change in θ1 = ϕ1
disconnects the attractor. Topological changes are abrupt, as they must be.
Back to Fig. 2.55, we see the attractor appears to be connected again at
θ1 = ϕ1 = π . Are there other angles giving connected attractors? Look at θ1 =
ϕ1 = π/4. These attractors are not connected, but suppose θ1 = ϕ1 were slightly
larger than π/4. Say enough that the line joining T1 (1, 0) and T1 (1/2, 3/2)
passes through the point T3 (0, 0). Would thiat attractor be connected?
For now, the last topological question we ask about IFS involves the topological types of the attractors of IFS consisting of t...
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