Ohrimenko+ / Barnsley. Superfractals

T : R L ∞ → D+ . See also Figures 2.61 and 2.63. Where is the fish’s eye?

then we have

c − a b − d cross-ratio(A, B, C, D) = b − c d − a .

When one of the points A, B, C, or D, lies on L∞ the cross-ratio is given by the same formula in the limit as a, b, c, or d respectively tends to infinity.

For example, if the four points lie on the x-axis with x-coordinates a = 1, b = 2.3, c = −1, d = 10 then the cross-ratio is

T h e o r e m 2.7.30 Let P : R2 L∞ → R2 L∞ be a projective transformation and let A, B, C, D R2 L∞ be a sequence of four distinct collinear points. Then

a, b, c, d lie on the x-axis and P is an interesting projective transformation, which you choose.

Does the cross-ratio correspond to some property of pictures you can somehow ‘see’, say in a picture that contains a line of equally spaced fence posts or four windows in a row on the front of a house? If the cross-ratio of four distinct collinear points, belonging to a picture of four copies of the same, is 43 or one of the numbers 4, −3, 14 , − 13 , 34 , depending upon the order in which the points are taken, then the answer, suitably qualified, may be positive, for these crossratios correspond to sets of points that can be transformed into a row of equally spaced points on the x-axis. The ‘unique fourth-point theorem’, [25], p. 141, tells us that if we know cross-ratio(A, B, C, D) and the locations of A, B and C then the location of D is uniquely determined. So, by looking at a picture containing three points A, B, C R2, can you locate by eye the point D such that the four points A, B, C, D could be approximately the result of applying a projective transformation to the points a = 0, b = 1, c = 2, d = 3 on the x-axis? Try it, in Figure 2.64, then calculate cross-ratio(A, B, C, D). Draw your own conclusions.

The theorems of geometry

We have introduced projective transformations and Mobius¨ transformations. We have mentioned some basic deep geometry results, such as Theorems 2.7.4, 2.7.10, and 2.7.30, the cross-ratio theorem, and illustrated to some extent what they mean for sets, measures and pictures. But there are many more results that we have not mentioned, such as Steiner’s porism, Pappus’ theorem, La Hire’s theorem, the three tangents and three chords theorem and so on.

Figure 2.63 See also Figures 2.61 and 2.62. Here a picture of flowers and birds, represented both on R2 and on D+ , is transformed by a projective transformation. Here and in Figures 2.61 and 2.62 some picture information has been lost owing to numerical effects arising because points have been mapped too close to L ∞ or to the boundary of D+ . Can you find some examples of such information loss?

This introduction of ours may serve as an invitation to you to consider afresh the existing body of theorems. Begin by reading or rereading a good work on geometry, such as the very practical book [25], the more abstract books by Coxeter [26] and by Berger [23] or the good brief historical review of geometry in Encylopaedia Brittanica [94]. Take theorems from such sources as appeal to you and think about what they may say explicitly and specifically about transformations of pictures, over and above what they say about transformations of points, lines, planes and conics. What do they not tell you? What things that you might think are obvious from a visual colourful point of view, what visual intuitions that have not yet been captured by mathematics do these theorems suggest but not prove?

Figure 2.64 Using your intuition alone, can you sketch the locations of the missing windows? Do so, then calculate some cross-ratios. Are your answers close to 1.33?

2.8 Transformations on code spaces

Here we describe some simple transformations, on code spaces, that are relevant to fractals. These transformations are continuous and they can be interpreted geometrically in terms of affine transformations acting on trees in R2. Moreover, remarkably, they preserve a quantity – quite unlike angle, length, or cross-ratio – that is related to information-carrying capacity. This fact enriches our theme of the connections between code spaces and meristems.

Recall that we introduced the code spaces A and A in Chapter 1. Also, recall that we can think of {0,1} {0,1}as a tree-like structure, which in this section we call simply a tree, embedded in R2; finite strings of zeros and ones, points of{0,1}, are represented by the nodes of the tree while the points of {0,1}, infinite strings of zeros and ones, correspond to the tips of the twigs, the canopy of the tree, as illustrated in Figure 2.65.

Let ω A. Then we define the branch transformation fω : A A →

A A by

fω (σ ) = ωσ for all σ A A.

and whose canopy represents {0,1} . The points with addresses 0, 1 and 010 are examples of periodic points for the shift transformation.

For example,

f1(0100101 · · · ) = 10100101 · · ·

It is easy to see that fω is one-to-one and continuous with respect to the natural topology. Also fω ( A) A (and fω ( A) A), so we can restrict fω to A. We denote this restricted transformation by fω : A → A. In terms of the tree representation, fω maps the whole tree onto the branch at the node ω, as illustrated in Figure 2.66(iii). In this representation the transformation is a similitude acting on R2.

E x e r c i s e 2.8.1 Let ω, ν A. Prove that fω ◦ fν = fων .

E x e r c i s e 2.8.2 Let I denote [0, 1] R2 minus all the points that possess two binary addresses (cf. Exercise 1.4.3). What does fω look like geometrically when interpreted as acting on I ? What happens if you try to include points with two binary addresses?

for all σ A A { }. We might say that f0 and f1 are branches of the inverse of S.

The shift transformation is illustrated in Figure 2.66(iv).

E x e r c i s e 2.8.3 Show that, when A = {0, 1},

(S ◦ S)−1(σ ) = { f00(σ ), f01(σ ), f10(σ ), f11(σ )}

for all σ A.

E x e r c i s e 2.8.4 Describe the shift transformation S : {0,1} → {0,1} in terms of arithmetical operations on the point 0.σ1σ2σ3 · · · [0, 1] R2 corresponding to σ {0,1}.

A subset A is called a shift-invariant subspace of A if

S( ) = .

Note that a shift-invariant subspace of A need not be, according to our terminology, an invariant set for S : A → A because we have not required that S−1( ) = . However, it is an invariant set for S| : → .

There are various types of shift-invariant subspaces. They relate to geometrical aspects of fractals as well as to information theory. They include: (i) subsets ofA defined in terms of the periodic points of S, see below; (ii) sets of points defined via topological entropy, see below; (iii) sets of points defined via fractal tops, discussed in Chapter 4; (iv) sets of points defined via stationary Markov processes; see for example [88].

We need the concept of periodic points elsewhere, so we define it generally here.

D e f i n i t i o n 2.8.5 Let X be a space and let T : X → X be a transformation.

A point p X is called a periodic point of T of period n ≥ 1, n N, iff T ◦n ( p) = p and T ◦k ( p) =p for all k = 1, 2, . . . , n − 1, where

T ◦k ( p) := T (T (· · · (T ( p)) · · · )).

,-. /

k times

A periodic point of T of period 1 is simply a fixed point of T.

Examples of periodic points of the shift transformation S : {0,1} → {0,1} are 0 = 000 · · · , 010 = 010010010 · · · and 01 · · · 11 = 01 · · · 1101 · · · 1101 · · · .

Here 0 is of period 1, 010 is of period 3 and 01 · · · 11 is of the same period as the number of symbols under the bar. Examples of shift-invariant subspaces

of S : {0,1} → {0,1} are the set of all periodic points of S and the set of points { p, S( p), S◦2( p), . . . , S◦n−1( p)}, where p {0,1} is a periodic point of S of period n ≥ 1. Figure 2.65 includes illustrations of some periodic points.

Other types of shift-invariant subspace can be defined using a quantity that is related to information theory.

where Uk (σ ) is the set of all distinct substrings of σ of length k.

The following theorem tells us that this definition works.

T h e o r e m 2.8.8 The limit h(σ ) exists for all σ A.

P r o o f This follows from the observation

|Umk (σ )| ≤ |Um (σ )| |Uk (σ )|.

The topological entropy of a point σ A is a measure of the informationcarrying capacity of the string σ . It takes account of the diversity of substrings of σ but not of the relative frequencies of occurrence of the different substrings.

Let σ A be such that all the elements of A are substrings of σ . Then clearly, on the one hand,

h(s ) = 0

because s contains only one substring of length k for each k Z+. The following theorem tells us that the topological entropy is invariant both under shift transformations and branch transformations.

T h e o r e m 2.8.9 Let S : A → A be a shift transformation. Then

h(S(σ )) = h(σ ) for all σ A.

Let ω A and let fω : A → A be a branch transformation. Then

h( fω (σ )) = h(σ ) for all σ A.

P r o o f Let σ = σ1σ2σ3 · · · A and k {2, 3, . . . }. Then

|Uk (σ2σ3 · · · )| ≤ |Uk (σ1σ2σ3 · · · )| ≤ 1 + |Uk (σ2σ3 · · · )|

invariant pictures for a transformation of this kind, in the case where the gaps in the Cantor set are infinitesimal.

The group of transformations generated by S : 2A → 2A conserves the topological entropy of both components of each point on which it acts, since

(h(σ ), h(ω)) = (h(Sσ ), h(σ1ω)) for all (σ, ω) 2A.

Finally we define nodal flip transformations, for which the alphabet is A = {0, 1}. Let ω A. Define flipω : A A → A A by

where 0 = 1 and 1 = 0; see Figure 2.66(i), (ii). Nodal flip transformations are continuous, one-to-one and onto, and pairs of such transformations commute. Furthermore we can compose infinite sequences of such transformations to obtain new continuous invertible transformations on code space; see Exercise 2.8.12 below.

In Chapter 4, in connection with fractal tops and colour-stealing, we show how to use continuous invertible transformations on code space to define transformations on subsets of R2 that are continuous almost everywhere. Also, we show how you can apply such transformations to some beautiful pictures to obtain other beautiful pictures.

E x e r c i s e 2.8.12 Let {ω(n) A}∞n=1 be such that |ω(n)| ≤ |ω(n+1)| for all n. Define Fn : A A → A A by

Fn = flipω(1) ◦ flipω(2) ◦ · · · ◦ flipω(n)

for each n {1, 2, 3, . . . }. Show that {Fn }∞n=1 converges uniformly with respect to the metric d to a continuous invertible function F : A A → A A. Show that F−1 = F.

In this section we have played with the fact that code space can be embedded in a tree in R2. This allows us to handle some transformations on code space by using classical geometrical transformations on R2. We can also embed code space in diverse other geometrical structures in R2, such as products of Cantor sets and not-quite-touching Sierpinski triangles. Then we may define transformations on R2 that map these new structures into themselves. By such means we may define and think about transformations on code space in terms of transformations of a more classical type.