Ohrimenko+ / Barnsley. Superfractals

In turn this simplifies to

F◦(k+1)(P0) = P0 f1(P0) f2(P0)

f11(P0) f12(P0) f21(P0) f22(P0)

f111(P0) f112(P0) f121(P0) f122(P0)

f211(P0) f212(P0) f221(P0) f222(P0)

f1111(P0) f1112(P0) · · · f2221(P0) f2222(P0)

· · · f 1···11 (P0) f 1···12 (P0)

· · · f 2···21 (P0) f 2···22 (P0)

This almost completes the proof.

We need also to show that the result is remarkable! Equation (3.5.7) implies that

Theorem 3.5.5 tells us that we can compute approximations to P = O(P0) by recursion. For example, we can compute an approximation to, say, the sequence of functions in the mapping (PN )◦4, use it to apply this mapping to P0 to obtain P N (N 4−1) (P0), then apply it again to yield (PN )◦4 applied to P N (N 4−1) (P0), to obtain

N −1 N −1

P N (N 8−1) (P0), and so on. This type of recursion may be used quite efficiently, as

N −1

only a few iterates are needed to produce a ‘high-order’ approximation to the orbital picture. In some cases, for example when all the transformations are strict contractions, this allows us to minimise the growth rate of the cumulative error due to successive rounding errors by keeping low the required number of iterates both of functions and of pictures.

In the top row of Figure 3.26 we illustrate four approximants to an orbital picture. The approximants are

P0, P340(P0), P87380(P0) and P22369620(P0).

They were computed in three steps, according to

P340(P0) = (P4)◦4(P0), P87380(P0) = (P4)◦4(P340(P0))

Figure 3.26 The top row shows four approximants, from left to right, to the orbital picture of the buttercup P0. The bottom row shows four underneath pictures. In this case the sequence does not converge to some final picture; instead, a restless sequence of textures is produced. See the main text.

and

P22369620(P0) = (P4)◦4(P87380(P0)).

In this case the IFS semigroup was generated by the four projective transformations

where the coefficients are given in Table 3.1.

The set attractor of this IFS is the domain of the textured green and yellow leafshaped segment that is the bottom right element of Figure 3.26; this was discussed briefly in the Introduction. What you can see from the top row in Figure 3.26 is that the sequence of approximants converges efficiently to an approximation to the orbital picture, which ceases to change, at viewing resolution, if further iterations are effected.

We may refer to algorithms for the computation of approximants to orbital pictures based on Theorem 3.5.5, as above, as being deterministic. This is in contrast to random iteration algorithms, such as the chaos game algorithm, which are discussed in Chapter 4.

We note that Theorem 3.5.5 implies that the orbital picture of P0 is the same as the orbital picture of P0 f1(P0) f2(P0) · · · fN (P0). A little algebra then provides us with the following result.

C o r o l l a r y 3.5.6 Let P(P0) = O(P0) denote the orbital picture of P0 under the IFS semigroup S{ f1, f2,..., fN }(X). Let

P0 = f1(P0) f2(P0) · · · fN (P0)'P0.

Then

P(P0) = P0 P(P0).

E x e r c i s e 3.5.7 Prove Corollary 3.5.6. Look at some orbital pictures and identify P0 and P(P0).

The self-referential equation obeyed by some orbital pictures

The definition of an orbital picture may be expressed as

P= O(P0)

=P0 f1(P0) f2(P0) · · · fN (P0)

f11(P0) f12(P0) · · · f1N (P)

f21(P0) f22(P0) · · · f2N (P0) · · · .

Thus we can always write an orbital picture as a union of disjoint segments, which we call global segments, of the form fn (Rn ) P for n = 1, 2, . . . , N ,

where Rn P for n = 1, 2, . . . , N . Typically each global segment contains multiple ‘tiles’.

We refer to Equation (3.5.10) as a self-referential equation because it says that the orbital picture P is the disjoint union of P0 with at most N transformations of segments of itself. It is this self-referencing property that makes many orbital pictures, including wallpaper patterns, beautiful and mysterious. The orbital pictures illustrated in Figures 3.20, 3.21, 3.23 and 3.24 involve overlapping ‘tiles’. Look at each of these pictures, to visualize how it obeys a self-referential equation like (3.5.10).

Under the condition (*) in the following theorem, the segments R1, R2, . . . , RN can be chosen to be the whole orbital picture. These conditions might at first sight look difficult to check. But they apply in quite simple situations, for example if the fn (P)\P0 for n = 1, 2, . . . , N are disjoint, or if the fn (P) are disjoint, or if the sets fn (X) are disjoint or if N = 1.

T h e o r e m 3.5.8 Let P = O(P0) denote the orbital picture of P0 under the IFS semigroup S{ f1, f2,..., fN }(X), and suppose that (*) for each n = 1, 2, . . . , N − 1 the following set of pictures is disjoint:

fn (P)'P0 f1(P0) f2(P0) · · · fn (P0)

and

fm (P0)'P0 f1(P0) f2(P0) · · · fm−1(P0)

−

Then we start by proving that under the condition (*) we have, for all k =

0, 1, 2, . . . ,

F◦(k+1)(P0) = P0 f1(F◦k (P0)) f2(F◦k (P0)) · · · fN (F◦k (P0)), (3.5.12)

for all P0 . We will demonstrate this result for the case N = 2. The general case is a straightforward generalization of the same ideas. We proceed by induction. When k = 0 and N = 2, Equation (3.5.12) reads

F◦1(P0) = P0 f1(P0) f2(P0),

which is true. Suppose that Equation (3.5.12) is true for k = 0, 1, . . . , K . Then, choosing k = K , N = 2 and P0 to be F◦1(P0) = P0 f1(P0) f2(P0) in

f2 F◦ (F◦ (P0))

for all P0 . By Theorem 3.5.5 it follows that

F◦(K +2)(P0) = P0 f1(P0) f2(P0) f1 F◦(K +1)(P0) f2 F◦(K +1)(P0)

f2(F◦(K +1)(P0)).

Hence Equation (3.5.12) is true when k = K + 1, which implies completion

It is tempting to think that P = O(P0) is the unique solution of the selfreferential equation (3.5.11). This is not the case, as the following example shows. Let P0 have domain {(x, y) R2 : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}, let f1(x, y) =

( 12 x + 2, 12 y) and f2(x, y) = ( 12 x, 12 y + 2). Let A denote the closed line segment that joins the pair of points (0, 4) and (4, 0). Then A is the attractor of the IFS

{R2; f1, f2} and obeys A = f1(A) f2(A), and it is disjoint from the domain of P. Let PA denote a picture of constant colour, with domain A. Then

PA = f1(PA) f2(PA) = f2(PA) f1(PA).

Figure 3.27 An example of a picture P which obeys the self-referential equation P = P0 f1(P) f2(P) but which is not the orbital picture P of the buttercup P0. The difference between P and P is the red segment, whose domain is the fractal set, the attractor of the IFS.

Now let P = P PA = PA P. Then it is readily verified that

P = P, P = P0 f1(P) f2(P) and P = P0 f1(P) f2(P).

See for example Figure 3.27.

A commonly used technique in the fractal compression of a given picture P involves seeking a set of segments S of P each of which can be transformed, under one of a given family of transformations T , into a segment belonging to a given set of segments S of P; see for example [12], [53] or [38]. Typically the given segments S are obtained by chopping the domain of P into square blocks, with little regard for the geometry of P. Figure 3.28 illustrates that domains of the segments f1(R1), f2(R2), . . . , fN (RN ) occurring in Equation (3.5.10) may be very complicated even when the domain of P0 is rectangular. This suggests that, in the future development of fractal image compression technology, more

Figure 3.29 The photographs on the left and right show two quite distinct leaves taken off the same plant, like the one in the middle, which was growing near Lake Padden in northern Washington State, U.S.A. in June 2003. It seems as though the branching veins crowd together, in the leaf on the right, and either stop growing, or go ‘underneath’. Can orbits of pictures be used to model the geometry of leaf veins? Can an underlying code space be identified, yielding biologically meaningful topological invariants?

attention should be given to the geometry of the segments into which pictures are partitioned. Without such attention, the compression would be inefficient for many orbital pictures; given the ‘fractal’ and self-referential character of the latter, it would seem to be a minimum requirement for fractal compression to work well, at least for orbital pictures where N is small.

The IFS used in Figure 3.28 is

The code space of the orbital picture

In this subsection we define and investigate code spaces of orbital pictures. This relates to our meristem theme, as we discuss below; see also the caption of Figure 3.29.

Code spaces of orbital pictures, with a few side conditions, enable us to:

(i)establish the existence of invariant quantities associated with orbital pictures, including the growth rate of periodic cycles and the topological entropy;

(ii)establish a dynamical system on panels, i.e. certain segments of orbital pic-

tures, see below; (iii) construct a certain ‘space of limiting pictures’, LP0 , from the set of panels; (iv) relate some of the limiting pictures, elements of LP0 , to the periodic cycles of the dynamical system. These constructions (i)–(iv) provide us