Fractals in Physics

By L. Pietronero and E. Tosatti

Fractals in Physics

• Language: English
• Publisher: Elsevier Science
• Release date: Dec 2, 2012
• ISBN: 9780444598417

Book preview

fractals

SELF-AFFINE FRACTAL SETS, I: THE BASIC FRACTAL DIMENSIONS

Benoit B. MANDELBROT,     Physics Department, IBM Research Center; Mathematics Department, Harvard University, Cambridge, MA 02138 USA*

The notion of -fractal dimension is explored -for various -fractal curves or dusts that are not self -similar, but are diagonally self - affine. A diagonal self -affinity stretches the coordinates in different ratios. It is showed that, in contrast to the unique -fractal dimension of strictly self-similar sets, one needs in general several distinct notions. Most important are the concepts of dimension obtained via the mass in a sphere and via covering by uniform boxes. One -finds it does not matter which definition is taken, but it matters greatly whether one interpolates or extrapolates. Thus, one obtains two sharply distinct dimensions: a local one, valid on scales well below, and a global one, valid on scales well above, a certain crossover scale.

1 INTRODUCTION

This paper examines, on three levels corresponding to three parts, what happens to diverse alternative definitions of fractal dimension when they are generalized from self-similar fractals to certain self-affine fractals. The substance of my Scripta paper¹ is incorporated. Self-similar fractals were the original objects on which diverse fractal dimensions had been tested in detail, and their values were found to coincide.² When a method works well in one case, it is tempting to apply it under increasingly wide conditions. The more general context of self-affine fractals now deserves systematic attention.

I have coined self-affine and self-similar in 1964 (the latter is so accepted now, that its age has become hard to believe), but affine goes back to Euler. In this paper, no specific knowledge of affine geometry will be required, but it is amusing to quote a characterization of that field by E. Snapper and R.J. Troyer: Roughly speaking, affine geometry is what remains after practically all ability to measure length, area, angles, etc… has been removed from Euclidean geometry. One might think that affine geometry is a poverty-stricken subject. On the contrary, it is quite rich. I hope to convince the reader that self-affine fractals also prove to be a very surprisingly rich topic.

One well-known but very special example of self-affine fractal is the record of Wiener’s scalar Brownian motion, which is the random process with independent and stationary Gaussian increments. This record has a well-known invariance property: setting B(O) = O, the processes B(t) and b−1/2B(bt) are identical in distribution for every ratio b > O. One observes that the rescaling ratios of t and of B are different, hence the transformation from B(t) to b−1/2 B(bt) is not a similitude but a more general affinity. This is why B(t) was called statistically self-affine on page 350 of my book².

While a similarity is a linear transformation that shrinks or expands all the vectors implicit in a geometric figure in the same ratio, an affinity is a linear transformation that shrinks different vectors differently according to their directions. More precisely, B(t) is unchanged statistically under diagonal affinity, a notion explained in Section 2.

In this paper, diagonally self-affine fractal curves, dusts, and other sets, are examined in detail on several successive levels of generality, first in the plane: records of functions (random or not) analogous to B(t), and two levels of more general sets, and then in space. In each case, the fractal-dimensional properties are shown to exhibit new and surprising complications. The different roles of the single all-purpose fractal dimension of a fully self-similar set are now shared by a multitude of different special purpose fractal dimensions.

.

Part II covers the dimensions obtained by walking a divider along a curve or triangulating a surface. The values one obtains are doubly anomalous, namely: distinct from those obtained in Part I.

Part III starts with recent mathematical findings³,⁴,⁵ concerning certain cases of the Hausdorff-Besicovitch dimension DHB, extends them, and discusses the meaning of the double anomaly found in certain cases.

The richness and complexity of this study are not purely mathematical, but reflect the richness and complexity of nature. Again, increasingly complex fractals are considered, the continuing refinement of our description of their structure demands a continuingly increasing number of fractal dimensions.

2 THE NOTIONS OF AFFINITY, DIAGONAL AFFINITY, AND SELF-AFFINITY

2.1 Background to diagonal affinity

Let us think of the Brownian record B(t) again. In Wiener’s original interpretation, t is time and B is a physical particle’s location on a spatial axis. The two coordinates play sharply different roles, and the units of B and t (cm. and second?) can be chosen independently. Rotation is not allowable, because it leads to sets that are no longer records of functions.

Forming the expression B(t)-δt (another application of affinity) introduces a function called Brownian motion with a drift, which is a very different process -from B(t). In an even earlier interpretation by Louis Bachelier (in 1900), t is time and B is a price in francs. The same remarks apply. However, an interpretation that I advanced later is substantially different: B(t) describes the vertical section of one of my Brown landscapes (my book², chapter 28); the coordinates still play different roles, since gravity defines the vertical direction, makes overhangs an exception and makes it is useful to represent the relief by a (single-valued) function. However, both B and t are lengths in this example, and their units can no longer be chosen independently. For reasons that will transpire later, the best is to choose as unit of both B and t to be the tc such that |B(t+tc) - B(t)| ∼ |tc|. This tc will be called the crossover scale. This unit’s counterpart in the original Brownian motion interpretation depends upon the units that happen to be selected for t and B; therefore, the crossover is in general not intrinsic.

The local and global dimensions that are introduced examined below are separated by this crossover. Without it, the local versus global distinction could not be clearcut.

2.2 Diagonal affinities

and an array of reduction ratios rm (0< m< E − 1), and considering the map

The ratios rm need not be positive. And they must not all be equal, because otherwise the transformation would fall be a similitude. The inverses 1/|rm| = bm, called bases, are integers in the simplest examples that are constructed recursively.

Most of the examples will be sets in the affine plane A² (E=2). We shall write b′=max bm, b′′=min bm, and H=log b′′/log b′. This H, called affinity exponent, will satisfy 02, there are E(E-1)/2 affinity exponents and crossover scales.

Formally, a linear transformation is the sequence of a translation and a multiplication by a matrix; we only tackle the cases where the matrix is diagonal and its diagonal terms are not identical. The product of two diagonal affinities is a diagonal affinity. Thus, a collection of diagonal affini ties can be used as the basis for a group.

The issues to be addressed involve the meaning of square, distance, and circle in affine geometry. (See second paragraph of Section 1). These notions remain meaningful for relief cross-sections, but for records of noise or of price, the units along the t axis and along the B axis are set up independently of each other. There being no meaning to equal height and width, a square cannot be defined. Similarly, a circle cannot be defined, because its square radius R² = Δt² + ΔB² would have to combine the units along both axes. Furthermore, one cannot walk a divider along a self-affine noise record, to measure its approximate length, because the distance covered by each step combines a Δt and a ΔB. On the other hand, a noise record is always represented on the same graph paper as a relief section or an isotropic set. This does not cause the distinction between the affine AE and the Euclidean RE to disappear, but sometimes it is elusive, and one is tempted to evaluate various prohibited dimensions mechanically. One should not.

2.3 The recursive constructions in a grid of many standard self-similar fractals extend easily to the self-affine case

For an example of how to generalize the Sierpinski carpet, take the semi-open unit square as initiator. (Semi-open means that the top and the right sides are open and the bottom and left sides are closed. The rectangles to be considered will also be semi-open). As generator, take the array on Figure 1.

FIGURE 1

Thus, we divide initiator into 3 × 4 = 12 subrectangular parts, and erase the middle two parts, shown in black. Then we erase the middle two of the 12 sub-subrectangular parts, etc… The resulting self-affine carpet is the union of N=10 tenths. Each tenth is obtained from the whole by a diagonal affinity with rn′ = 1/3, rn″ = 1/4 for n=l to n=N=10. I propose that those signs be summarized by representing the generator in the form of a stick generator:using arrows along the diagonals of the ten rectangles. In the present example arrows must be placed as marked to insure that the tenths of this carpet do not overlap. The fixed points are the four vertices, the midpoints of the left and right sides, and the points 1/3 and 2/3 along the top and bottom sides. Indeed, an affinity’s fixed point is the point of intersection of the four straight lines that join the vertices of the whole to these vertices’ transforms, each of which is the vertex of the part.

The choice of a unit square as initiator deserves comment. It fixes the units of the coordinates t and B in analogy with the condition |B(t+tc) - B(t)| ∼ tc. The extrapolation that yields scales above this tc is discussed in section 2.6.

A general fractal generator in a self-affine lattice is obtained by drawing b′ × b′′ subrectangles, and keeping NFigures 2 and 3 play especially important roles.

FIGURE 2

FIGURE 3

(Surprisingly, Part III will show that DHB depends on which variant is chosen!)

When one keeps both the rectangles and their diagonals, the resulting -fractal is obtained as a limit of nested collections of rectangles, nested meaning that each is contained in the preceding one. When one only keeps the diagonals, and these diagonals -form a curve, a self-affine -fractal curve is obtained as a limit of broken lines. Connectedness of the stick generator imposes a constraint on the retained subrectangles.

However, as in my book,² chapter 13, the stick generator may split into several curves, creating islands and/or lakes.

Important special case: When N=b′, a stick-generated -fractal curve is the record of a (one-valued) continuous function.

2.4 Definition of self-affinity

The preceding examples adequately motivate the following definition.

A set S is self-affine with respect to a list of N diagonal affinities, αn, if one can write S = UαnS, where αnS n αmS is empty unless n=m. That is, S decomposes into N parts (no two of which overlap) each of which results from the whole by an affinity in the list.

2.5 Fractally homogeneous measure

To be able to define a fractal’s mass dimension, one must define the mass within a cube. A uniform measure on the fractal is easy to define here.

For records of functions, one takes measure as proportional to time elapsed.

For recursive fractals within lattices, the k-th stage is made up of rectangles; they are given equal measures.

For records of functions constructed recursively, the two methods yield the same result, because the X-projection of the measure according to the second definition is uniform.

2.6 Extrapolation

To understand extrapolation is especially important in the case of self-affine fractals, because (as has been announced and will be proven), the extrapolate is ruled by its own global fractal dimensions. Recall that a Cantor dust’s extrapolation is not unique, in fact there is an extrapolate for every infinite sequence of base N=2. Since every point in the interpolated dust is determined by a sequence written towards the right, the extrapolative sequence is best written towards the left, as … a-k… a−3, a−2, a−1, a0. If a0=0 (resp., a0=1), our dust is viewed as the left (resp. the right) portion of a dust that has been upsized in a ratio b=3. And so on. The same applies to all multi-dimensional recursive constructions, extrapolative sequences being written in the base N. In our self-affine case, the values of n0 identifies the square initiator with one of the N parts of a super-initiator b’ wide and b′′ high; therefore, n0 identifies the super-initiator. Recursively, when the super−k-initiator is known, the values of n-k identifies it within a superk+1-initiator. And so on.

3 THE MULTIPLE FRACTAL DIMENSIONS OF A SELF-AFFINE SET

From either the purely mathematical or the fractal literature, the only very widely known fact on this topic is that the Hausdorff-Besicovitch dimension DHB of the Brown record is 3/2. This result becomes less murky if extended to the more general fractional Brownian motion BH(t), where 0 < H < 1. If BH(0) = 0, the random processes BH(t) and b−HBH(bt) are identical in distribution. The -fractional Brown record satisfies DHB = 2-H. The value H = 1/2 brings B(t) as a special case of BH(t).

But what about DHB for self-affine sets that are records of other functions, or are not the records of functions? And what about definitions of fractal dimension other than DHB?

Two words suffice to deal with the (self-) similarity dimension, DS. This notion applies most directly to self-similar fractal sets, which are made of N parts, each obtained from the whole by a similitude whose reduction ratio satisfies |r| = 1/b. For the self-similar fractal sets, DS = log N/log b. For self-affine fractal sets, DS simply cannot be evaluated. (However, a naturally generalized DS will enter in Section 4).

As a substitute for DS, several authors have made guesses, which by and large prove unjustifiable. For example, take B(t). If DHB is written as 1 + 1/2, its value happens numerically to be 1 + logb,b′′, where (again) b’ is the larger, and b′′ the smaller base. But in fact, DHB=2-logb,b′′, as shown by the study of the fractional Brownian motion BH(t). Other guesses are less obviously wild.

The basic fact established in this paper (and in part already established earlier¹) is that different roles that in the self-similar case are taken up by one number called fractal dimension must in the self-affine case be shared among different quantities. Some are local -like DHB - but the newest ones are global. Of particular significance are the global mass dimension DMG and the local box dimension DBL.

Recall that I once defined a fractal, as a set for which DHB>DT (=topological dimension). This tentative definition had looked less and less attractive, and I have long abandoned it (my book’s second and later printings, p. 458). An alternative was to view as fractal those sets for which the dimensions in a certain list coincide. This alternative is no longer promising.

4 THE GAP DIMENSION

We begin with a fractal dimension that is simple, but of narrow validity and interest. Since the similarity dimension log N/log b is meaningless under self-affinity, it is tempting to save it formally, by replacing b by some suitable effective base b*, and then to try and interpret the outcome. Taking for b* the geometric mean of the bm, namely b* = (b1, b2, … bE)¹/E, let us show that DG = log N/log b* is indeed a dimension in case one can define either gaps or islands. The formula for DG is symmetric with respect to the bases, which is why DG figures among guesses of what the D should be. We shall see that the other and more important D’s are not symmetric.

The notion of gap dimension applies to the self-similar fractals in RE exemplified by the Cantor dust on the line and by the Sierpinski gasket and carpet in the plane. These shapes have the following properties. Their measure in vanishes (fat fractals - my book², chapter 15 - are excluded). Their complement splits into an infinity of gaps (maximal open sets) which are domains in RE, similar to each other and differing solely by their linear scale. In all these cases, it is known that the following relation holds for all L:

number of gaps of linear scale > L ∝ L-DG with DG < E.

The exponent DG is now called gap fractal dimension, and all other definitions of the fractal dimension of a self-similar fractal give the same value. Now consider self-affine fractals that have gaps. There is good news and there is bad news.

The good news is that it is still true that the number of gaps scales like Nk and volume scales like b1 b2 … bE. Define linear size as the (1/E)-th power of volume the geometric mean of the sides. If so, the number-size relation for gaps or islands continues to be a power law valid for all sizes L. The exponent independent of L can continue to be called gap dimension; it is the DG defined earlier.

The bad news is that DG bears no direct relation to DHB. For example, consider the generator in Figure 4.

FIGURE 4

, while Part III will show that DHB=1.34, and we shall soon see that the basic fractal dimensions are DBL =1.38 and DMG=1.20.

For thin fractal dusts (dusts of Lebesgue measure 0) in the special case E=1 (e.g., Cantor dust, my book, p.78), the gap dimension is a stronger form (valid for all scales) of an exponent that Besicovitch and Taylor introduced for small scales, and showed to be identical to DHB (my book², p.359). The generalization of the identity DBT=DHB to E >1 is known to be correct in some self-similar cases (Sierpinski carpet; my book, p. 134), but now we see that in the self-affine cases, DG stands alone.

5 SELF-AFFINE PLANAR FRACTAL CURVES DEFINED AS RECORDS OF FUNCTIONS

Sections 5.1. and 5.2. are summary excerpts from my Scripta paper, which includes additional material and more detailed wording. Section 5.3. is new and important.

(One word about the use of the letter H for logb,b′′. It is appropriate in this section, because H is the Holder exponent of Bh(t) and other functions we study. However, I originally picked H in homage to H. E. Hurst; my book, Chapter 27).

5.1 The mass dimension’s local value for small R is 2-H. For large R the mass dimension is 1

When a set S is a self-similar fractal, the mass M(R) contained in the intersection of a S with a disc or ball of radius R behaves like M(R) ∝ RDM. One can also replace the disc or ball by a square or cube whose sides are parallel to the axes and of length 2R. For some physicists, this property has become almost a definition of the notion of fractal. But this step is not justified. In fact, we must be careful about the meaning of the symbol ∝.

tC, any one of these records is effectively a horizontal interval. Within a square of side 2R, it occupies a very thin horizontal slice. Therefore, if we follow Section 2.5 and weigh our record proportionately to time elapsed, we find that M(R) ∝ R. That is, we obtain the striking result that DMG = 1. The subscript G stands for global.

tC. There, our record is effectively a collection of vertical intervals, one for each zero. Some algebra yields for the local mass dimension the value DML = 2-H, which is familiar as the DHB of records of BH(t). The subscript L stands for local.

Conclusion: We discover that two limits that are identical for self-similar fractals can differ. Besides, 1<2-H for all H in 0

The rest of the paper extends this discovery.

5.2 The box dimension’s local value for large b is 2-H. The global value is 1

After a lattice made of boxes of side r = 1/b is made to cover a set, let N(b) denote the number of lattice boxes that intersect the set. Box dimension is my present term for an exponent that characterizes sets for which N(b) behaves like N(b) ∝ bDB. Box dimension is short for box counting dimension. Its virues are to be short and to have no other meaning, while the alternatives do (metric dimension applies to all fractal dimensions, and capacity dimension has long been reserved for the Frostman dimension).

Again, what does this α mean?

Most important is its meaning to the mathematician, who thinks of local behavior of the form limb → ∞ log N(b)/log b = DBL. In the case of recursively defined records, try to cover a piece b′−k=b−1 wide and b′′−k high, using boxes of side b−1. Clearly, these boxes must be piled into vertical stacks. The number of boxes in each stack is (b′′/b′)−k=b¹-H, and the number of stacks is b. Hence, DBL=2-H. Similarly, in the case of BH(t), the heuristic box argument given in my book², bottom left of page 237 yields DBL =2-H.

1 is covered by a single box. Hence, limb → 0log N(b)/log b = DBG = 1.

5.3 Behavior of the dimensions of BH(t) under section. Bounded locally self-similar fractals

As a rule (my book, p. 135), when a self-similar fractal in the plane is cut by a line, the fractal dimension goes down by one. The rule is fundamental but has numerous exceptions, To apply it to the record of BH(t), which is not self-similar, care is both needed and well rewarded. Assume that BH(t) has an intrinsic scale equal to one.

Horizontal cuts. They are self-similar and their local dimension also applies globally. It is 1-H=(2-H)-1, thus, it is obtained -from DBL, and the fact that DBG=1 does not matter.

Vertical cuts. They reduce to 1 point, whose dimension used to count as exception to the rule.

Skew sections by Y=σt, with 0< σ<∞. When a cell of size b′ −kxb′′−k is upsized to a unit square, Y=σt is replaced by Y=σ(b′′/b′) kt. Thus, Y=σt is locally indistinguishable -from Y=0. The cut is locally self-similar, indeed locally identical to the horizontal cut.

BUT! Self - affinity has sensitized us to also check the global properties and the intrinsic scale.

Globally, a skew cut is not self-similar. As a matter of -fact, it is bounded, therefore DMG=DBG=0. This is confirmed by observing that, globally, Y=σt scales up to a vertical line. Indeed, DMG=DBG=0 are the values obtained -from the basic rule, if it is applied to the original curve’s DMG=DBG=1.

I confess having been insensitive until now to the special status of bounded locally self-similar fractals. They are extremely common in nature, the best examples being individual island coastlines and DLA.

Now to the cut’s intrinsic scale. For a bounded dust, the scale is the length of the smallest interval that contains it. As σ → ∞, intrinsic scale → 0, which is why a vertical cut reduces to one point and the only thing that matters in the limit is the global dimension, which is O. In other words, vertical cuts have now ceased to count as exceptions. As σ →0, intrinsic scale → ∞, which is why the horizontal cut is unbounded self-similar. This explains why horizontal cuts are not at all affected by global quantities.

5.4 Behavior of the dimensions of the recursive self-affine fractals under section

This topic, surprisingly complex and interesting, must be postponed to Part III.

6 SELF-AFFINE RECURSIVE PLANAR FRACTALS WHOSE PROJECTIONS FILL THE AXES

6.1 Projections of a self-affine recursive fractal on the axes

When a projection of the generator fills the corresponding side of the initiator, the limit fractal’s projection fills the axis. When the projection of the generator fails to fill the side of the initiator, the limit fractal’s projection is a dust that leaves uncovered gaps. In particular, when rn′>0 and rn″>0 for all the basic affinities, the limit fractal’s projection is a Cantor dust. The above two cases are examined in Sections 6 and 7. Cases when rn′<0 and/or rn″<0 for some of the basic affinities cannot be discussed here.

6.2 The global mass dimension is 1-1/H+logb″N=logb″(Nb′/b′′). The local mass dimension is 1-H+logb, N = logb, (Nb′′/b′)

Again, we begin with the notion of greatest interest to physics: the extrapolative global mass dimension. On the advice of Section 2, we use a unit square as initiator, and attach a unit mass to it. The extrapolation (Section 2.6) is uniquely specified in a sequence of increasing boxes of mass Nk and area (b′b′′)k. This seems to point towards the gap dimension DG = log N/log b* as mass dimension. But things are more complex, because the mass-radius relation requires the mass to be evaluated within square boxes, not specially adapted rectangular boxes. A square box of side b′′k, if chosen at random within a rectangular box of sides b′k and b′′k, contains on the average the mass Nk(b′′/b′)k. Hence, the surprising new result

In the case of function records, Section 5, N = b′, hence this DMG duly yields the already known value DMG = 1.

A similar argument applied to local behavior yields

Again, in the case of function records, Section 5, N = b′, hence this DMG duly yields the already known value 2-H.

The above formulas would be hard to guess, and their most striking feature lies in their being asymmetric in b’ and b′′, and symmetric of each other.

6.3 The global and local box dimensions take the same values as the corresponding mass dimensions

The formula for DBL was obtained (implicitly) long ago by Kline⁶

6.4 one has DMG=DBGDG DBL=DML. Proof: DMGb′′

Hence,

.

7 SELF-AFFINE PLANAR RECURSIVE FRACTALS AT LEAST ONE OF WHOSE PROJECTIONS IS A SIMPLE CANTOR DUST

7.1 Definitions and example

What is meant in this section’s title is that, a) rn′>0 and rn″ >0 for all n, and b), the X- and/or Y-projections of the limit fractal are Cantor dusts, made (respectively) of N′ parts with r′ = 1/b’ and of N″ parts with r′′= 1/b′′.

The very simplest example uses the generator in figure 5. It is clear that, globally, the resulting fractal is the original dust obtained by midthirds removal. The global dimensions are, therefore, log3 2. Locally, on the other hand, the resulting fractal is the Devil’s staircase, minus its flat steps. The local dimensions are known to be 1.

FIGURE 5

7.2 The global mass dimension is DMG= (1-1/H) logb, N′+logb″N=logb″ (NN′H-1. The local mass dimension is

The argument concerning DMG runs exactly like in Section 6.2., up to the point where the average mass in a box of side b′′k is evaluated. The new feature is that this mass is now allowed to vanish, but massless squares cannot be part of our hierarchy of boxes. Therefore, it is necessary to exclude the massless boxes, and take a conditional average mass, which is larger than the average mass. Observe that b’kH = b′′k. The strip of width b’k and height b′′k decomposes into (b′/b′′) k boxes of side b′′k, of which N′k(1-H) are not empty. Hence, the conditional average mass is NkN¹-k (1-H), and

A similar argument applied to local behaviour yields

The first (second) expression for DML is the exact symmetric of the second (first) expression for DMG.

7.3 Except when N=N′N″, one has DML > DMG

Writing 1/log b′′ − 1/log b′ = F,

Since F>O and N< N′ N″, DMG < DML, with equality only if N=N′N″, in which case our planar Cantor dust is the Cartesian product of two linear Cantor dusts. A linear Cantor dust is self-similar, thus the simplicity of the self-similar situation carries on to the self-affine case, when it is obtained as Cartesian product.

7.4 The global and local box dimensions take the same values as the mass dimensions

8 SELF-AFFINE SURFACES

This section comments briefly on two functions Z(x, y), where the (x, y) plane is isotropic and there is one scaling parameter H, and on a function T(x, y, z), when the (x, y, z) space is itself affine and there are two scaling parameters G and H.

8.1 Fractal functions of a variable in an isotropic plane. Relief

My simplest model of the Earth’s relief (my book, chapter 28) is a fractional Brown surface BH (x, y), the point of coordinates x and y being in an isotropic plane. Everything concerning this surface depends on the single parameter H. It is easily seen that DBG.= DMG =2, while DBL = DML = 3-H. Also, DHB = 3-H.

Dimensions’ behavior under vertical or horizontal plane sections. Come back to the rule that, when a fractal is intersected by a plane, the dimension goes down by 1. This rule, again, is fundamental but with the reputation of having many exceptions; let us show how some of these can be irnoned away.

The vertical sections of BH(x, y) have both local and global properties, and the rule applies both to DBL and to DBG, with no fuss.

The horizontal sections are the coastlines of all the islands taken together. They are self-similar and have only one dimension, which is the local dimension of vertical sections. Horizontal sections’ intrinsic scale is infinite. Thus, a dimension that tells a lot about the horizontal sections, tells only half of the story about the vertical sections.

Dimensions’ behavior under skew plane sections. As for the skew lines in Section 5.3., a skew plane Z=σx downsizes locally to a horizontal plane, and upsizes globally to a vertical plane. Both the local and the global dimensions are decreased by 1.

8.2 Fractal functions of a variable in an isotropic plane. Clouds/rain

My fractal model of coastlines has been empirically shown by S. Lovejoy to extend to cloud boundaries’ vertical projection on the Earth’s surface. This has in turn led Lovejoy and Mandelbrot⁷ to a two-dimensional model of rain areas or clouds. It is based on fractal sums of pulses, a self-explanatory new term for a family of self-affine surfaces that I had introduced for other purposes. In the FSP model, some quantity (like temperature, opaqueness or rain intensity) is ruled by a self-affine function ZH(x ⋅ y), where the plane of the (x, y) is isotropic. The main mathematical contrast, compared to BH(x, y) as applied to relief, lies in the contrasting common experiences that a mountain’s altitude is by and large a continuous function, while rainfall intensities are sharply discontinuous in time and space. In the simplest case⁷, there is no parameter other than H.

8.3 Fractal functions of a variable in an affine plane. Clouds

As everyone is aware, large clouds tend to be like pancakes parallel to the Earth’s surface, and the conventional argument of the meteorologists is that the atmosphere is three-dimensional on small scales and two-dimensional on large scales, with a crossover scale in between. In a counter-argument, Schertzer and Lovejoy⁸ argue from available empirical evidence that the atmosphere itself is self-similar in x and y, but self-affine in x (or y) and z. I view this suggestion as excellent, and I like the way op. cit. adapts various of my models to make them self-affine, or more fully completely self-affine.

The dimensional properties of the corresponding fractals are therefore worthy of exploration. Unfortunately, op. cit.⁸ quotes numbers with scant motivation, or none, and shows no awareness of the interesting complications the topic presents.

With little cost, one can immediately consider self-affine functions T(x,y,z), where the horizontal variables (x,y, and z) are isotropic. The basic self-affinity property is invariance under a map whose diagonal terms can be written as r, r, rG, and rGH with G<1. In addition, using the awkward but self-explanatory notation of op. cit., one has ΔT(Δx) ∼ (Δx)GH, ΔT(Δy) ∼ (Δy)GH, and ΔT(Δz) ∼ (Δz)H, we find that H<1. It is easily found that for the record of T, DMG=DBG=3 irrespectively of H and G. However, other dimens easily found that for the record of T, DMG=DBG=3 irrespectively of H and G. However, other dimensions of the record, and the dimensions of other objects related to T, usually depend on the object itself, and on H and/or on G.

1, ΔT(Δx) is dominated by ΔT(Δz) ∼ (Δz)GH. Therefore, covering the record of T by boxes of side Δx = Δy = Δz requires (Δx)−3 stacks with ∼ (Δz)GH-1 boxes in each stack. Conclusion: DBL=4-GH.

8.4 Coverings by rectangles, and the x elliptical dimension

In the case E=3, b1=b2 and b3=b1H, op- cit.⁸ gives prominence to the quantity 2+H, which it calls elliptical dimension of space, Del. The motivation is that in the isotropic 3-D case, Del=3, and in the isotropic 2-D case, Del=2; it is therefore natural to regard 2+H as the fractal dimension of this self-affine space.

This attractive motivation does not suffice. It is, moreover, weakened by the second supporting argument, which notes that the case E=2 and b2=b1H with H=1/2 yields Del=1.5, which is the same as the fractal dimension suggested [sic] by me for the record of B(t). We know, however, that for BH(t), one has DHB=2-H rather than 1+H; these two formulas take the same value for H=1/2 because of a numerical coincidence.

The first supporting argument is that the number of eddies of horizontal scale λ may be written as λ−D with D=Del. But considering the vertical scale would give D=1+2/H; why choose the horizontal scale?

A search for a clearcut interpretation of 2+H as dimension has involved private conversations with J. P. Kahane and J. Peyriere, who suggest checking on intrinsic coverings that do not use cubes but affine rectangles b′′−k high and b′−k wide, the radius of a rectangle being its longer side. Local Hausdorff-1ike dimensions of this kind are discussed by Peyriere⁹.

The concrete physical meaning of covering by rectangles is as yet unproven. Its adoption would of course involve additional local and global dimensions, many of which are found to take on very questionable values. For example, the mass dimensions in the case studied in Section 6 become logb′ N globally and logb N locally. Both values yield a very biased and incomplete view of these fractals’ structure. Specifically, take a self-affine Sierpinski carpet with b′=9, and b′′=3 and one big gap leaving in N=20. Its global mean dimension based on intrinsic rectangles is logb′ N = 1.36. The same values continue to apply if b′′ is replaced by any integer from 3 to 9 (inclusive). The carpets obtained in this fashion greatly differ from each other, except from this peculiar viewpoint. To put this dimension in perspective, note that DG=1.81, DML = DBL = logb, N+1-H=1.86 and DMG = DBG = logb′′N+1-1/H=1.72. In op. cit.⁸, however, logb′ N is given (without explanation) as the only fractal dimension of this carpet.

In the limit case N = b′b′′, the main dimensions based on rectangles simplify to 1+2/H locally and 2+H globally. This last value might provide, after-the-fact, an element of motivation for the elliptic dimension. But this motivation is no better than the motivation for the unbounded twin value 1+2/H.

Observe also that, in order to verify the standing of this global 2+H and this local 1+2/H as dimensions, one must know in advance which boxes one should use in the covering, which requires advance knowledge of H. Every dimension based on the common square boxes can be measured by direct algorithms.

9 COMMENT ON PHYSICAL EXTRAPOLATION VERSUS MATHEMATICAL INTERPOLATION

The constructions and procedures that fractal geometry has borrowed from mathematics all involved infinite interpolation. In physics, on the other hand, interpolation cannot proceed without end, and constructions tend to proceed by extrapolation. At the time of the first uses of fractals in physics, in 1980, this contrast was often brought to my attention by physicists. In the self-similar case, and to the mathematicians’ and physicists’ surprise, the infinitesimal techniques extend via power law relationships valid uniformly at all scales. In the self-affine case, the two basic procedures involve different tools.

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*Supported in part by the Office of Naval Research, grant