Ohrimenko+ / Barnsley. Superfractals

D

B

A E

C

Figure 1.7 Nested bisections of the line segment L [ A , B ] constructed with straight-edge and compass. Identify here some of the shields shown in Figure 1.9.

L01, and we bisect L1 to produce two intervals, L10 lying to the left of L11. This successive bisection process is done in such a way that the geometrical ordering of the intervals, from say left to right, starting at A and going to B, as L00, L01, L10, L11 corresponds to the lexicographic ordering of the strings 00, 01, 10, 11 {0,1}. We now have

L0 = L00 L01 and L1 = L10 L11

as well as

L = L00 L01 L10 L11.

We can repeat this bisection process inductively. At the nth generation we obtain 2n intervals, denoted by {Lσ : σ {0,1}, |σ | = n}. These intervals form a partition of L[A, B], that is,

Lσ is of length 1/2|σ | times the length of L[A, B].

Figure 1.8 L is partitioned into smaller and smaller subintervals L σ , where σ belongs to the code space {0,1}. Any point x L belongs to an infinite sequence of such intervals, and the sequence of addresses of these intervals determines an address in {0,1} for the point x . (In the case illustrated the address begins 010 · · · .) Conversely, given any address in {0,1} we can define uniquely a corresponding sequence of nested intervals and a point x L .

We call the elements of the space C := {Lσ : σ {0,1}} the cylinder sets of

L[A, B].

By construction, we have

That is, Lω1 contains Lω2 and so on. We say that the sequence of subsegments

{Lωn : n = 1, 2, . . . } forms a decreasing sequence of sets.

As we will explain in Section 1.11 each set in this sequence is compact and, since the length of Lωn shrinks towards zero as n increases towards infinity, this

J

A

Figure 1.9 Space made of a ‘tree’ of shield-shaped tiles. The shields are defined by regions produced in the repeated bisection construction illustrated in Figure 1.6. The limit set of the tiles – reached after the construction is repeated infinitely many times – is the set of points in the line segment L [ A , B ]. Finite binary strings provide addresses for the tiles, while infinite binary strings address the points in the line segment.

sequence defines a single point x L[A, B]. It is this procedure that defines the mapping ϕ : {0,1} → L[A, B].

To show that ϕ : {0,1} → L[A, B] is indeed an address function, we need to show that it is an onto mapping. But, given any point x L[A, B] and any n = 1, 2, 3, . . . , Equation (1.4.1) tells us that we can find at least one string ωn{0,1} of length n such that x Lωn , and clearly we can do this in such a way that Lω1 Lω2 · · · Lωn · · · . Equation (1.4.2) implies that

ωn+1 = ωn σn+1,

where σn+1 {0, 1}. As above, this sequence of subsegments defines a unique point, and that point must be x. So ϕ : {0,1} → L[A, B] is onto, and hence provides an address function for L[A, B]. This completes our excursion into how an addressing scheme may depend upon geometrical properties of the space.

Here is another example of a space and an addressing function. In Figure 1.9 we show a branching tree of shields, tiles defined by four circular arcs. The circular arcs are produced during the iterative bisection construction described above. DFEG is the single zeroth-generation shield, the ‘base’ of the tree, to which we assign the address . We denote this zeroth-generation shield S . It is formed by arcs from circles used to construct the zeroth and first generations of bisection points. The two first-generation shields, PHJK and QLMN, are denoted S0 and S1

respectively and are formed by arcs of the circles used to construct the first and second generations of bisection points. The four second-generation shields, of one quarter the linear dimensions of DFEG, are denoted S00, S01, S10, S11. Similarly there are eight third-generation shields, S000, S001, S010, . . . , S111, sixteen fourthgeneration shields and so on.

In this way an unique shield Sσ is defined corresponding to each σ {0,1}. Then

S := Sσ : σ {0,1}

is a space. We might call it shield space. A convenient addressing function is ϕ : {0,1} → S, defined by ϕ(σ ) = Sσ . In this example the elements of the space are sets and the addressing function maps codes onto sets.

E x e r c i s e 1.4.3 Let ϕ : {0,1} → L[A, B] be the address function defined above. Which points in L[A, B] have more than one address? Show that the point C L[A, B], which is one third of the way from A to B, has only one address. What is the address of C?

E x e r c i s e 1.4.4 Show that the two circles used above to bisect the line segment L[A, B] in Figure 1.7 intersect at 120◦.

may be used to model the set of pictures of S in which some shields are coloured red and the others green. Devise an address function and code space for X. Using this address function, give a possible address of the point x X represented by the picture in Figure 1.9 with S coloured green.

1.5 Metric spaces

In this section we introduce a second property which a space may possess and through which we may consider its points to be organized. It is the property of possessing a metric.

D e f i n i t i o n 1.5.1 A metric space (X, d) consists of a space X together with a metric or distance function d : X × X → R that measures the distance d(x, y) between pairs of points x, y X and has the following properties:

(i)d(x, y) = d(y, x) for all x, y X (i.e. the distance from x to y is the same as the distance from y to x);

(ii)0 < d(x, y) < +∞ whenever x and y are distinct points of X (i.e. distance is always greater than zero when x =y);

(iii)d(x, x) = 0 for all x X;

Figure 1.10 A metric space (X, d ) consists of a space X together with a function d : X × X → R having certain properties that make it behave like ‘distance’. Here X is a leafy-looking subspace of R2 and distance is measured using a conveniently positioned binary ruler of length unity. Determine the approximate binary distance between the tips of the second and fourth fronds, counting up from the bottom. (The figure depicts two fractal sets with colours given by IFS colouring; see Section 4.6.)

(iv)d(x, y) obeys the triangle inequality, namely d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z X.

When it is clear from the context what the metric is, or the particular metric does not matter, we may write X in place of (X, d).

Metric spaces of diverse types play a fundamental role in fractal geometry. They include familiar spaces like R and R2, code spaces and many other examples; see Figure 1.10. One example of a metric space is (R, d(x, y) = |x − y|), where |x − y| denotes the absolute value or norm of the real number x − y. Suppose that x, y [0, 1] are both represented in base N , that is,

0.1010 and y = 0.101 using Equation (1.5.1). What does this distance become if you interpret these expansions of x and y as being in base 3? Explain what is going on here.

Another example of a metric space is (R2, deuclidean), where

deuclidean(x, y) := (x1 − y1)2 + (x2 − y2)2 for all x, y R2.

Quite generally, (Rn , dp) is a metric space for all n = 1, 2, 3, . . . and p > 0, where

In Rn we define |x − y| := deuclidean(x, y) = d2(x, y).

E x e r c i s e 1.5.3 Show that the following are both metrics in R2:

(i)dmax (x, y) := max{|x1 − y1| , |x2 − y2|} for all x, y R2,

(ii)dmanhattan(x, y) := |x1 − y1| + |x2 − y2| for all x, y R2.

E x e r c i s e 1.5.4 Check whether you agree that if (X0, d) is a metric space and X X0 then (X, d|X×X ) is a metric space. We say that (X, d|X×X ) is a subspace of (X0, d).

We now draw attention to the following wonderful method for constructing metrics. We will use it to make ‘geometrical’ metrics on code spaces in Section 1.6.

T h e o r e m 1.5.5 Suppose that X is a space, that (Y, dY ) is a metric space and that ξ : X →(Y, dY ) is an embedding function. Then (X, dX ) is a metric space, where

dX (x, y) := dY (ξ (x), ξ (y)) for all x, y X.

P r o o f This is straightforward. (i) dX (x, y) = dY (ξ (x), ξ (y)) = dY (ξ (y), ξ (x)) = dX (y, x) for all x, y X. (ii) Suppose that x and y are distinct points of X. Then ξ (x) and ξ (y) are distinct points of Y because ξ , being an embedding function,isone-to-one.Hence0 < dY (ξ (x), ξ (y)) < ∞,andso 0 < dY (x, y) < ∞. (iii) dX (x, x) = dY (ξ (x), ξ (x)) = 0 for all x X. (iv) dX (x, y) = dY (ξ (x), ξ (y)) ≤ dY (ξ (x), ξ (z)) + dY (ξ (z), ξ (y)) = dX (x, z) + dX (z, y) for all x, y, z X.

Figure 1.11 An inchworm tries to work out the shortest distance to a delicious morsel that she has spotted.

E x e r c i s e 1.5.6 Suppose that X is a subset of R2 that ‘looks like’ a ragged leaf; see Figure 1.11. Argue that the following is a metric:

dcaterpillar(x, y) = length of shortest path, on the leaf, from x to y.

E x e r c i s e 1.5.7 Let (X, d) be a metric space. Define d : X × X → R by

We will sometimes write

f : (X, dX )→ (Y, dY )

to denote a transformation between two metric spaces (X, dX ) and (Y, dY ).

and (Y, dY ) are called equivalent metric spaces, iff f is one-to-one and onto and

In Section 1.14 we will discover that each bounded subset of Rn has associated with it a number, called its fractal dimension, whose value depends upon the underlying metric. This number is unchanged when the metric is altered to another equivalent metric, and hence fractal dimension is invariant under any metric transformation.

A metric transformation for which C = 1 in Equation (1.5.2) is called an isometry or isometric transformation. Distance is invariant under an isometric transformation.

Throughout this book we will be mentioning properties of mathematical objects – points in appropriate spaces – that are invariant under transformations of one type or another. This is a recurring theme. Quite generally, geometry studies the properties of sets that are invariant under groups of transformations; see Chapter 3. Geometrical properties are properties that are invariant under a group. Here we are getting our first taste of this idea: the set of metric transformations forms a group and so does the set of isometries. Fractal dimension is a geometrical property of metric transformations just as distance is a geometrical property of

denote the shortest path in S from x to y. Show that (S, dshortest) and (S, deuclidean) are not equivalent.

Figure 1.12 In Rn , it is possible to find n + 1 points {x1, x2, . . . , xn } such that deuclidean(xi , x j ) = 1 for all i =j , for n = 1, 2, 3, . . . In (i) and (ii) we illustrate a way to do this when n = 1 and n = 2 respectively.

Use the hint provided by (iii) to find the coordinates of such a set of points in the case n = 3.

{x1, x2, . . . , xn+2} such that deuclidean(xi , x j ) = 1 for all i =j, where i, j {1, 2,

. . . , n + 2} for all n = 1, 2, 3, . . . See also Figure 1.12.

1.6 Metrics on code space

In this section we show how any code space A A can be embedded in R2 in diverse ways and consequently can be endowed with numerous different metrics. A simple metric on A is defined by d (σ, σ ) = 0 for all σ A, and

for σ = σ1σ2σ3 · · · and ω = ω1ω2ω3 · · · A, where m is the smallest positive integer such that σm = ωm .

E x e r c i s e 1.6.1 Show that ( A, d ) is indeed a metric space.

E x e r c i s e 1.6.2 Evaluate d (1010, 101) when A = {0, 1} and when A = {0, 1, 2}.

Figure 1.13 This shows the code space {0,1} represented as a subset of the real interval 0 ≤ x ≤ 1. To obtain this figure we represented the points of [0, 1] in base 3 and then plotted all those points whose representation does not include the symbol 2.

We readily extend d to A A by adding a symbol, which we will call Z , to the alphabet A to make a new alphabet A = A {Z }. Then we embed A A

It is readily verified that ε is one-to-one and hence that d does indeed furnish a metric on A A. This metric is a very simple one to work with.

But there is another metric, of a different type and with a more geometrical character, that we can define on A A. It is constructed with the aid of the embedding technique of Theorem 1.5.5. It depends explicitly on the number of elements |A| in the alphabet A, so we denote it by d|A|.

Assume, without loss of generality, that A = {0, 1, . . . , N − 1}. Then we treat the addresses in A as representing points in the real interval [0, 1] = {x : 0 ≤ x ≤ 1} in base N + 1; and we take the distance between two addresses to be the euclidean distance between their representations. Note that the base number is one more than |A|, the number of elements in the alphabet. See Figure 1.13.