Synopsis Ahmer On Fractals

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FEDERAL URDU UNIVERSITY Of Arts, Science and Technology Gulshan-e-Iqbal Campus

Department Of Physics

Synopsis On:

“Fractals”

STUDENT NAME:

s.m.hasan.ahmer

SUPERVISOR NAME:

PROF. DR. V.E. ARKHINCHEEV

M. Phil 2006

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INTRODUCTION A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced/size copy of the whole [1]. The term was coined by Benoit Mandelbrot 1975 and was derived from the Latin fractus meaning broken or fractured. Beautiful fractals, the butterfly effect and unpredictable systems were the images that chaos conjured up in my imagination before I sat down and read about it. Within the incredible diversity of chaotic systems; and the diversity is remarkable is presented and explained. It is staggering to see the picture unfold, the gradual realization that the scientific statement of the eighteenth century; that the universe runs according to a set of immutable laws is unable to explain much of the behavior in even the simplest of classical systems. Nature has laws and we can find them. Unfortunately, although mathematics allows us to calculate the solutions to many difficult problems, we are still left in an unordered world, where apparently simple motions, on closer inspection, become unpredictable and hence unexplainable in the language of mathematics. It is appropriate at this point to introduce the nature of chaos. Stewart is quick to point out that since this branch of mathematics is still in its formative stages giving it a precise definition is not possible or wise More roughly speaking, random behavior in a system governed by laws. Where is the dividing line between order and chaos? The chapter 'The Laws of Error, introduces another field of mathematics, Probability theory the mathematics of chance. Mathematicians had found that analyzing the detailed workings of large systems was too involved and complex. Probability theory grew out of a need to simulate detail without actually having to examine it. As Stewart states: "Mathematicians could calculate the motion of a satellite of Jupiter, but not that of a snowflake in a blizzard." Fractals are important part of chaos that joins the discussion at this point. They present us with a language to describe what we see happening with chaos. A fractal, generally speaking, is a geometric object, which continues to exhibit detailed structure over a wide range of scales. Self-similarity exhibited again. Interestingly the method of describing the detail level of a fractal is by allocating it a dimension, known as its Hausdorff dimension. These dimensions tend to be fractional (hence fractal).

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A fractal as a geometrical object generally has the following features: • • • • •

Fine structure at arbitrarily small scales Is too irregular to be easily described in traditional Euclidean geometric language. is self-similar (at least approximatively or stochastically) Has a Hausdorff Dimension is greater than its Topological this requirement is not met by space-filling curves such as the Hilbert Curve. Has a simple and recursive definition

Due to them appearing similar at all levels of magnification, fractals are often considered to be infinitely complex. Obvious examples include clouds, mountain ranges and lightning bolts. However, not all self-similar objects are fractals, for example the Real (a straight Euclidean Line) is formally self-similar but fails to have other fractal characteristics.

BACKGROUND HISTORY

A Koch Snowflake the limit of an infinite construction that starts with a triangle and recursively replaces each line segment with a series of four line segments that form a triangular bump. Each time new triangles are added (an iterations, the perimeter of this shape grows by a factor of 4/3 and thus diverges to infinity with the number of iterations. The length of the Koch snowflake's boundary is therefore infinite, while its area remains finite. For this reason, the Koch snowflake and similar constructions were sometimes called "monster curves."

444 Objects that are now described as fractals were discovered and described centuries ago. Ethnomathemetics like Ron Eglash's African Fractals describes pervasive fractal geometry in indigenous African craft work. In 1525, the German Artist Albrecht Durer published The Painter’s manual, in which one section is on "Tile Patterns formed by Pentagons". The Durer’s pentagon largely resembled the Sierpiski carpet, but based on pentagon instead of squares. The idea of "recursive self-similarity" was originally developed by the philosopher Leibniz and he even worked out many of the details. In 1872, Karl Weierstrass found an example of a function with the nonintuitive property that it is everywhere continuous but nowhere Differentiable — the graph of this function would now be called a fractal. In 1904, Helge von Koch, dissatisfied with Weierstrass’s very abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch Snowflake. In 1915, Walclaw Sierpinski constructed his triangle and one year later his carpet. Actually, these fractals were described as curves, which is hard to realize with the well-known modern constructions. The idea of self-similar curves was taken further by Paul Pierre Levy who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole, described a new fractal curve, the Levy’s curve. Georg Cantor gave examples of Subset of the real line with unusual properties — these Cantor sets are also now recognized as fractals. Iterated functions in the complex plane had been investigated in the late 19th and early 20th centuries by Henri Poicare, Felix Klein, Pierre Fatou and Gaston Julia. However, without the aid of modern computer graphics they lacked the means to visualize the beauty of many of the objects that they had discovered. In the 1960s Benoît Mandelbrot started investigating self-similarity in papers such as How long is the coast of Britain? Statistical self-similarity and Fractal dimension. This built on earlier work by Lewis fry Richardson. In 1975, Mandelbrot coined the word fractal to denote an object whose Hausdorff Dimension is greater than its Topological. He illustrated this mathematical definition with striking computer-constructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term fractal.

EXAMPLES

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A Julia set, a fractal related to the Mandelbrot set A relatively simple class of examples is given by the Cantor sets, Sierpinski triangle and carpet, Mengor sponge, dragon curve, space-filling curve, Koch curve. Additional examples of fractals include the Lyapunov fractal and the limit sets of Kleinian groups. Fractals can be deterministic (all the above) or stochastic (that is, non-deterministic). For example the trajectories of the Brownian motion in the plane have Hausdorff Dimension. Chaotic dynamical systems are sometimes associated with fractals. Objects in the phase space of a dynamical system can be fractals. Objects in the parameter space for a family of systems may be fractal as well. An interesting example is the Mandelbrot set. This set contains whole discs, so it has the Hausdorff dimension equal to its topological dimension of 2 —but what is truly surprising is that the boundary of the Mandelbrot set also has the Hausdorff dimension of 2 (while the topological dimension of 1), a result proved by M. Shishikura in 1991. A closely related fractal is the Julia set.

SELF-SIMILARITY DIMENSION The self-similarity dimension is a simplification of the Hausdorff dimension which can be applied to exactly self-similar objects.

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Sierpinski Triangle

The following analysis of the Koch Snowflake suggests how self-similarity can be used to analyze fractal properties. The total length of a number, N, of small steps, L, is the product NL. Applied to the boundary of the Koch snowflake this gives a boundless length as L approaches zero. But this distinction is not satisfactory, as different Koch snowflakes do have different sizes. A solution is to measure, not in meter, m, nor in square meter, m², but in some other power of a meter, mx. Now 4N(L/3)x = NLx, because a three times shorter steplength requires four times as many steps, as is seen from the figure. Solving that equation gives x = (log 4)/(log 3) ≈ 1.26186. So the unit of measurement of the boundary of the Koch snowflake is approximately m1.26186. More generally, suppose that a fractal consists of N identical parts that are similar to the entire fractal with the scale factor of L and that the intersection between parts is of the Lebesgue measure 0. Then the Hausdorff dimension of the fractal is example, the Hausdorff dimension of

•

the Cantor set is

•

the Sierpinski gasket is

. For

, ,

777

•

the Sierpinski carpet is

,

and so on. Even more generally one may assume that each of N parts is similar to the fractal with a different scale factor Li, i = 1...N. Then the Hausdorff dimension can be calculated by solving the following equation in the variable s:

TECHNIQUES FOR GENERATING FRACTALS

Three common techniques for generating fractals are: Escape-time fractals — These are defined by a recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set, Julia set, the Burning ship fractal and the Lyapunov fractal. Iterated function systems — These have a fixed geometric replacement rule. Cantor set, sierpinski carpet, Sierpinski gaset, peano curve, Koch snowflake,Harter-Heighway dragon curve, T-square, menger sponge, are some examples of such fractals. Random fractals — Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, levy flight, fractal landscapes and the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, diffusion –limited aggregation or reaction-limited aggregation clusters.

CLASSIFICATION OF FRACTALS

Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in fractals:

Exact self-similarity This is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact self-similarity.

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Quasi-self-similarity This is a loose form of self-similarity; the fractal appears approximately (but not exactly) identical at different scales. Quasi-self-similar fractals contain small copies of the entire fractal in distorted and degenerate forms. Fractals defined by recurrence relations are usually quasi-self-similar but not exactly self-similar.

Statistical self-similarity This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of "fractal" trivially imply some form of statistical self-similarity. (Fractal dimension itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar.

FRACTALS IN NATURE

A fractal fern computed using an Iterated function system Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snowflakes, mountains, river networks, cauliflowers or broccoli and systems of blood vessels. Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive algorithm. This recursive nature is obvious in these examples — a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. The surface of a mountain can be modeled on a computer by using a fractal: Start with a triangle in 3D space and connect the central points of each side by line segments, resulting in 4 triangles. The central points are then randomly moved up or down, within a defined range. The procedure is repeated, decreasing at each iteration the range by half. The recursive nature of the algorithm guarantees that the whole is statistically similar to each detail.

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In mathematics, the Hausdorff dimension is an extended non-negative real number (that is a number in the closed infinite interval [0, ∞]) associated to any metric space. It was introduced in 1918 by the mathematician Felix Hausdorff.. Many of the technical developments used to compute the Hausdorff dimension for highly irregular sets were obtained by Abram Samoilvitch Besicovitch. For this reason, Hausdorff dimension is sometimes referred to as Hausdorff-Besicovitch dimension. Less frequently it is also called the capacity dimension or fractal dimension. Hausdorff dimension gives another way to define dimension, which takes the metric into account.

Sierpinski. A space having fractional dimension ln 3 / ln 2, which is approximately 1.58 To define the Hausdorff dimension for X, we first consider the number N(r) of balls of radius at most r required to cover X completely. Clearly, as r gets smaller N(r) gets larger. Very roughly, if N(r) grows in the same way as 1/rd as r is squeezed down towards zero, then we say X has dimension d. In fact the rigorous definition of Hausdorff dimension is somewhat roundabout, since it first defines an entire family of covering measures for X. It turns out that Hausdorff dimension refines the concept of topological dimension and also relates it to other properties of the space such as area or volume.

101010 The Hausdorff dimension gives an accurate way to measure the dimension of an arbitrary metric space; this includes complicated sets such as fractals. Suppose (X,d) is a metric space. As mentioned in the introduction, we are interested in counting the number of balls of some radius necessary to cover a given set. It is possible to try to do this directly for many sets (leading to so-called box counting dimension), but Hausdorff's insight was to approach the problem indirectly using the theory of measure developed earlier in the century by Henri Lebesgue and constantin caratheodory. In order to deal with the technical details of this approach, Hausdorff defined an entire family of measures on subsets of X, one for each possible dimension s ∈ [0,∞). For example, if X= R3, this construction assigns an s dimensional measure Hs to all subsets of R3 including the unit segment along the x-axis [0,1] × {0} × {0}, the unit square on the x-y plane [0,1] × [0,1] × {0} and the unit cube [0,1] × [0,1] × [0,1]. For s = 2, one would expect

• • •

The above example suggests that we can define a set A to have Hausdorff dimension s if its s-dimensional Hausdorff measure is positive and finite; in fact we have to modify this slightly. The Hausdorff dimension of A is the cutoff value s where below s the sdimensional Hausdorff measure is ∞ and above s it is 0. It is possible for the s dimensional Hausdorff measure of an s dimensional set to be 0 or ∞. For instance R has dimension 1 and its 1-dimensional Hausdorff measure is infinite. To carry this construction of this measure, we use a theory of measure which is appropriate for metric spaces. Define a family of metric outer measures on X using the Method II construction of outer measures due to Munroe and described in the article outer measures. Let C be the class of all subsets of X; for each positive real number s, let ps be the function A → diam(A)s on C. Hausdorff outer measure of dimension s, denoted Hs is the outer measure corresponding to the function ps on C. Thus for any subset E of X

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Where the infimum taken over sequences {Ai}i which cover E by sets each with diameter ≤ δ. Then

We can succinctly (though not in a very useful way) describe the value Hs(E) as the infimum f all h > 0 such that for all δ > 0, E can be covered by countably many closed sets of diameter ≤ δ and the sum of the s-th powers of these diameters is less than or equal to h. The function s → Hs(E) is non-increasing. In fact, it turns out that for all values of s, except possibly one Hs(E) is either 0 or ∞. We say E has positive finite Hausdorff dimension if, and only if, there is a real number 0 d, then Hs(E) = 0. If Hs(E)=0 for all positive s, then E has Hausdorff dimension 0. Finally, if Hs(E)=∞ for all positive s, then E has Hausdorff dimension ∞ In other words,

The Hausdorff dimension is a well-defined extended real number for any set E and we always have 0 ≤ d(E) ≤ ∞. It follows from the Lipschitz property of Hausdorff measure that Hausdorff dimension is a Lipschitz invariant. Its relation to topological properties is outlined below.

RESULTS The Hausdorff outer measure Hs is defined for all subsets of X. However, we can in general assert additivity properties, that is

for disjoint A, B, only when A and B satisfy some additional condition, such as both being Borel sets (or more generally, that they are both measurable sets). From the perspective of assigning measure and dimension to sets with unusual metric properties such as fractals, however, this is not a restriction. Theorem. Hs is a metric outer measure. Thus all Borel subsets of X are measurable and Hs is a countably additive measure on the σ-algebra of Borel sets.

121212 Clearly, if (X, d) and (Y, e) are isomorphic metric spaces, then the corresponding Hausdorff measure spaces are also isomorphic. It is more useful to note however that Hausdorff measure even behaves well under certain bounded modifications of the underlying metric. Hausdorff measure is a Lipschitz invariant in the following sense: If d and d1 are metrics on X such that for some 0< C < ∞ and all x, y in X,

then the corresponding Hausdorff measures Hs, H1s satisfy

for any Borel set E. Note that if m is a positive integer, the m dimensional Hausdorff measure of Rn is a rescaling of usual m-dimensional Lebesgue measure λm which is normalized so that the Lebesgue measure of the m-dimensional unit cube [0,1]m is 1. In fact, for any Borel set E,

The Euclidean space Rn has Hausdorff dimension n. The Circle S1 has Hausdorff dimension 1. Countable sets have Hausdorff dimension 0. Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological. For example, the Cantor set (a zero-dimensional topological space) is a union of two copies of itself, each copy shrunk by a factor 1/3; this fact can be used to prove that its Hausdorff dimension is ln2 / ln3, which is approximately 0.63 Sierpinski is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln3 / ln2, which is approximately 1.58. Space filling curves like the Peano and the Sierpinski by definition have Hausdorff dimension 2. The trajectory of Brownian motion in dimension 2 and above has Hausdorff dimension 2 almost surely.

HAUSDORFF DIMENSION AND TOPOLOGICAL DIMENSION

131313 Let X be an arbitrary separable metric space. There is a notion of topological dimensions for X which is defined recursively. It is always an integer (or +∞) and is denoted dimtop(X). Theorem. Suppose X is non-empty. Then

Moreover

where Y ranges over metric spaces homeomorphic to X. In other words, X and Y have the same underlying set of points and the metric dY of Y is topologically equivalent to dX. These results were originally established by Edward szpilrain (1907-1976). The treatment in Chapter VIII of the Hurewicz and Wallman reference is particularly recommended.

SELF-SIMILAR SETS Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set E is self-similar if it is the fixed point of a setvalued transformation ψ, that is ψ(E) = E, although the exact definition is given below. The following is Theorem 8.3 of the Falconer reference below: Theorem. Suppose

are contractive mappings on Rn with contraction constant rj < 1. Then there is a unique non-empty compact set A such that

This follows from Banach’s contractive mapping fixed point theorem applied to the complete metric space of non-empty compact subsets of Rn with the Hausdorff distance. To determine the dimension of the self-similar set A (in certain cases), we need a technical condition called the open set condition on the sequence of contractions ψi which is stated as follows: There is a relatively compact open set V such that

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where the sets in union on the left are pair wise disjoint.

Theorem. Suppose the open set condition holds and each ψi is a similitude that is a composition of an isometry and dilation around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is s where s is the unique solution of

Note that the contraction coefficient of a similitude is the magnitude of the dilation. We can use this theorem to compute the Hausdorff dimension of the Sierpinski triangle (or sometimes called Sierpinski gasket). Consider three non-collinear points a1, a2, a3 in the plane R2 and let ψi be the dilation of ratio 1/2 around ai. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket and the dimension s is the unique solution of

Taking natural logarithms of both sides of the above equation, we can solve for s, that is:

The Sierpinski gasket is self-similar. In general a set E which is a fixed point of a mapping

is self-similar if and only if the intersections

151515 where s is the Hausdorff dimension of E. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally: Theorem. Under the same conditions as the previous theorem, the unique fixed point of ψ is selfsimilar.

MAKING A FRACTAL

The Sierpinski Triangle

Step One Draw an equilateral triangle with sides of 2 triangle lengths each. Connect the midpoints of each side.

Shade out the triangle in the center. Think of this as cutting a hole in the triangle.

Step Two Draw another equilateral triangle with sides of 4 triangle lengths each. Connect the midpoints of the sides and shade the triangle in the center as before.

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Notice the three small triangles that also need to be shaded out in each of the three triangles on each corner - three more holes. Step Three Draw an equilateral triangle with sides of 8 triangle lengths each. Follow the same procedure as before, making sure to follow the shading pattern. You will have 1 large, 3 medium, and 9 small triangles shaded.

Step Four How about doing this one on a poster board? Follow the above pattern and complete the Sierpinski Triangle.

The Koch Snowflake A really interesting characteristic of the Koch Snowflake is its perimeter. Ordinarily, when we increase the perimeter of a geometric figure, we also increase its area. If you have a square with a huge perimeter, it also has a huge area. But wait till we see what happens here! Remember the process: 1. Divide a side of the triangle into three equal parts and remove the middle section. 2. Replace the missing section with two pieces the same length as the section we removed. 3. Do this to all three sides of the triangle. Let's investigate the perimeter below.

171717 Question 1: If the perimeter of the equilateral triangle that we start with is 9 units, what is the perimeter of the other figures?

Perimeter = 9 units

Perimeter = ? Units

Perimeter = ? Units? Hint: Think of the original triangle with sides of three parts, and the next figure with sides of four parts. Question 2: Is there a pattern here? The perimeter of each figure is ___ times the perimeter of the figure before. Question 3: If the original triangle has a perimeter of 9 units, How many iterations would it take to obtain a perimeter of 100 units? (Or as close to 100 as you can get.) Question 4: Now think of doing this many, many times. The perimeter gets huge! But does the area? We say the area is bounded by a circle surrounding the original triangle. If we continued the process oh, let's say, infinitely many times, the figure would have an infinite perimeter, but its area would still be bounded by that circle. An infinite perimeter encloses a finite area... In the Koch Snowflake, an infinite perimeter encloses a finite area. The perimeter of the Koch Snowflake gets bigger and bigger with each step. But what about the area? Imagine drawing a circle around the original figure. No matter how large the perimeter gets, the area of the figure remains inside the circle. Could we actually calculate the area?

181818 If we don’t like to draw and experiment with the figures, we may print and use this triangle grid paper to help us with the drawing. Remember the process: 1. Divide a side of the triangle into three equal parts and remove the middle section. 2. Replace the missing section with two pieces the same length as the section we removed. 3. Do this to all three sides of the triangle. Let's investigate the area below.

Notice the second iteration of the Koch Snowflake above. Notice that the original triangle (yellow) is still contained in the Koch Snowflake with three smaller triangles (red) added in the first iteration, and twelve even smaller triangles (blue) added in the second iteration. So finding the area of the Koch Snowflake is just an addition problem. we find the area of the original triangle, add the area of the three red triangles for the first iteration, and add the twelve blue triangles for the second iteration. Let's use the triangles of our grid to measure the area. The original yellow triangle has 81 triangles inside. So we'll say its area is 81. What is the area of each red and each blue triangle? Let's organize all our data into a table. Iteration No. -1

Area of 1 triangle 9

No. of triangles added 3

Amount of area added 27

Total Area 81 108

191919 2 1 12 12 120 Each "Area of one triangle" is 1/9 the previous one. Each "No. of triangles added" is 4 times the previous one. Let's predict the next steps from the rule we notice above. Area of 1 No. of triangles Amount of area Iteration No. Total Area triangle added added -81 1 9 3 27 108 2 1 12 12 120 3 1/9 48 5.33 125.33 4 1/81 192 2.37 127.7 5 1/729 768 1.05 128.75 6 1/6561 3072 .4682 129.21 7 1/59049 12288 .2081 129.43 8 1/531441 49152 .0924 129.522

Observe what is happening. The Snowflake is still growing in area but more and more slowly. It is converging on some number, getting closer and Closer to the number, but will never go above it

.

The Anti-Snowflake

Let's make another fractal. It's an interesting variation on the Koch Snowflake. Directions: Step One. Start with a large equilateral triangle. If you use the triangle grid paper, make the sides of your triangle 9 grid triangles long (or some other multiple of 3). Step Two. Make a pinwheel: 1. Divide one side of the triangle into three parts and remove the middle section.

202020 2. Replace it with two lines the same length as the section you removed, just like in the Koch Snowflake. But this time, instead of turning the section out to form a snowflake, turn them inside the triangle 3. Do this to all three sides of the triangle. Step Three. Repeat the process with the "triangles" inside the pinwheel. Want to take a slow and careful look below?

Original Triangle

First Iteration

Second Iteration

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Once more... And one more time...

Third

Fourth Iteration

FRACTAL BENEFITS Fractal Dimension A point has no dimensions - no length, no width, no height. That dot is obviously way too big to really represent a point. But we'll live with it, if we all just agree what a point really is. A line has one dimension - length. It has no width and no height, but infinite length.

Again, this model of a line is really not very good, but until we learn how to draw a line with 0 width and infinite length, it'll have to do. A plane has two dimensions - length and width, no depth.

It's an absolutely flat tabletop extending out both ways to infinity. Space, a huge empty box, has three dimensions, length, width, and depth, extending to infinity in all three directions.

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Obviously this isn't a good representation of 3-D. Besides its size, it's just a hexagon drawn to fool you into thinking it's a box. Fractals can have fractional (or fractal) dimension. A fractal might have dimension of 1.6 or 2.4. How could that be? Let's investigate below.

Just as the images above weren't very good pictures of a point, line, plane, or space, the drawing meant to be the Sierpinski Triangle has limitations. Remember as we continue that fractals are really formed by infinitely many steps. So there are infinitely many smaller and smaller triangles inside the figure, and infinitely many holes (the black triangles). Let's look further at what we mean by dimension. Take a selfsimilar figure like a line segment, and double its length. Doubling the length gives two copies of the original segment. Take another self-similar figure, this time a square 1 unit by 1 unit. Now multiply the length and width by 2. How many copies of the original size square do you get? Doubling the sides gives four copies.

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Take a 1 by 1 by 1 cube and double its length, width, and height. How many copies of the original size cube do you get? Doubling the side gives eight copies. Let's organize our information into a table. Figure Dimension

No. of Copies

Line segment

1

2 = 21

Square

2

4 = 22

Cube

3

8 = 23

Do you see a pattern? It appears that the dimension is the exponent - and it is! So when we double the sides and get a similar figure, we write the number of copies as a power of 2 and the exponent will be the dimension. Let's add that as a row to the table. Figure

Dimension

No. of Copies

Line Segment

1

2 = 21

Square

2

4 = 22

Cube

3

8 = 23

Doubling Similarity

d

n = 2d

We can use this to figure out the dimension of the Sierpinski triangle because when you double the length of the sides, you get another Sierpinski Triangle similar to the first. Start with a Sierpinski triangle of 1-inch sides. Double the length of the sides. Now how many copies of the original triangle do you have? Remember that the black triangles are holes, so we can't count them.

242424 Doubling the sides gives us three copies, so 3 = 2d, where d = the dimension. But wait, 2 = 21, and 4 = 22, so what number could this be? It has to be somewhere between 1 and 2, right? Let's add this to our table. Figure

Dimension

No. of Copies

Line Segment

1

2 = 21

Sierpinski's Triangle

?

3 = 2?

Square

2

4 = 22

Cube

3

8 = 23

Doubling Similarity

d

n = 2d

So the dimension of Sierpinski's Triangle is between 1 and 2. Do you think you could find a better answer? Use a calculator with an exponent key (the key usually looks like this ^ ). Use 2 as a base and experiment with different exponents between 1 and 2 to see how close you can come. For example, try 1.1. Type 2^1.1 and you get 2.143547. I'll bet you can get closer to 3 than that. Try 2^1.2 and you get 2.2974. That's closer to 3, but you can do better. That's how fractals can have fractional dimension.

BIBLIOGRAPHY 1. Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. W. H. Freeman and Company. 2. Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd., xxv. 3. Barnsley, Michael F., and Hawley Rising. Fractals Everywhere. Boston: Academic Press Professional, 1993. 4. Falconer, Kenneth. Techniques in Fractal Geometry. John Willey and Sons, 1997. 5. Jürgens, Hartmut, Heins-Otto Peitgen, and Dietmar Saupe. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992. 6. Benoit B.MedelBrot The Fractal Geometry of Nature. New York: W. H. Freeman and Co., 1982. 7. Peitgen, Heinz-Otto, and Dietmar Saupe, eds. The Science of Fractal Images. New York: Springer-Verlag, 1988. 8. Clifford A pickover, ed. Chaos and Fractals: A Computer Graphical Journey - A 10 Year Compilation of Advanced Research. Elsevier, 1998. 9. Jesse Jones, Fractals for the Macintosh, Waite Group Press, Corte Madera, CA, 1993.

252525 10. Hans Lauwerier, Fractals: Endlessly Repeated Geometrical Figures, Translated by Sophia Gill-Hoffstadt, Princeton University Press, Princeton NJ, 1991., cloth. paperback. 11. Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University.