" It is the glory of geometry that from so few principles, fetched from without, it is able to accomplish so much."

Sir Isaac Newton [1]

Calculating Fractal Dimension

In classical geometry, shapes have integer dimensions. A point has a dimension of , a line has a dimension of , an area has a dimension of  and volume has a dimension of . From these elements--points, lines, areas and volume--we derive the basic shapes of traditional geometry: triangles, squares, circles, cones, cubes and spheres .

Figure 4.1 Traditional dimensions point, line, square and cube.

We can use non-spatial dimensions--time, color and perspective--to add dimension to otherwise static objects. For example, you may have seen an image on a -dimensional computer screen that appears -dimensional because of perspective. Using computer-generated perspective, architects can visually walk through a entire building's design before construction even begins see Figure 4.2 and Color Plate 22. Using color changes, scientists can display thermodynamic heat fluctuations that can be used to indicate venerable points with low heat dissipation. By plotting sound frequencies over time, acoustical engineers use visual representations to show how different harmonic patterns interact. See Figure 4.3 and Color Plate 23 to see how changing colors can add information associated with dimension.

Figure 4.2 Computer rendering of a -dimensional view.

Figure 4.3 An added dimension represented by color (shade) changes [2] .

By using non-spatial dimensions, mathematicians and scientists have explored spaces beyond traditional -dimensions.  Examples include -dimensional 'cubes' called tesseracts. These cubes when viewed in -dimensions appear as seven cubes that share common sides with each other see figure 4.4. While no tesseract can physically be built, understanding these structures can offer insights into real world problems such as optimizing a network's path to follow the shortest distance.   Physicists and mathematicians, routinely formulate even higher dimensions, known as ordered states, to manipulate complicated equations that would be much harder to work with at a lower dimension. It is not uncommon for atomic physicists to work with a dimensional space in the teens in order to keep track of all possible states of particles found at the subatomic level.

Figure 4.4 Tesseract, a -dimensional drawing of a -dimensional object.

It is sometimes difficult to imagine these higher dimensions. Here we have taken an excerpt from the book Flatland [3] a romance of many dimensions, where Pointland, Lineland and Spaceland all see the same world differently to aid in visualizing different dimensions.

Excerpt from Flatland

4.5 Map of lands with different dimensions from the book Flatland.

" §  1.-Of the Nature of Flatland

I call our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space.

Imagine a  vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows - only hard and with luminous edges - and you will then have a pretty correct notion of my country and countrymen.  Alas, a few years ago, I should have said "my universe":  but now my mind has been opened to higher views of things.

In such a country, you will perceive at once that it is impossible that there should be anything of what you call a "solid" kind; but I dare say you will suppose that we could at least distinguish by sight the Triangles, Squares, and other figures, moving about as I have described them.  On the contrary, we could see nothing of the kind, not at least so a to distinguish one figure from another.  Nothing was visible, nor could be visible, to us, except Straight Lines; and the necessity of this I will speedily demonstrate.

Place a penny on the middles of one of your tables in Space; and leaning over it, look down upon it.  It will appear a circle.

But now, drawing back to the edge of the table, gradually lower your eye (thus bringing yourself more and more into the condition of the inhabitants of Flatland), and you will find the penny becoming more and more oval to your view; and at last when you have placed you eye exactly on the edge of the table (so that you are, as it were, actually a Flatlander) the penny will then have ceased to appear oval at all, and will have become, so far as you can see, a straight line.

The same thing would happen if you were to treat in the same way a Triangle, or Square, or any other figure cut out of pasteboard.  As soon as you look at it with your eye on the edge on the table, you will find that it ceases to appear to you a gure, and that it becomes in appearance a straight line.  Take for example an equilateral Triangle - who represents with us a Tradesman of the respectable class.  Fig. 1 represents the Tradesman as you would see him while you were bending over him from above;  figs. 2 and 3 represent the Tradesman, as your would see him if your eye were close to the level, or all but on the level of the table; and if your eye were quite on the level of the table (and that is how we see him in Flatland) your would see nothing but a straight line."

(1)        (2)        (3)

Edwin A. Abbott

•••

Now lets briefly look at mathematical tools developed over the centuries that help us to understand fractals.

Infinite Lengths and Scale Ability a prelude to fractals.

Many of the principles found in fractal geometry [4] have origins in earlier mathematics. For example scale ability and line lengths have long been associated with geometrical structures. In Elements, Euclid ( 330- 275 B.C. ) proposed lines with infinite lengths to illustrate the concept of parallel lines, there he also used self-similar triangles to show the congruency of triangles see Figure 4.6. Archimedes (287-212 B.C.) used spirals to illustrate repeating transformations. Later the mathematician Jacob Bernoulli (1654-1705) expanded this idea to show that some spirals could be drawn with an infinite length, of which the logarithmic spiral is the most famous see Figure 4.7. Another well known spiral with infinite length is the golden mean spiral derived from the ancient Greek's golden ratio  see Figure 4.8. This spiral closely resembles the sea creature, nautilus seen in Figure 3.89 in Chapter 3.

Figure 4.6 Parallel lines of infinite lengths and triangles within triangles.

Figure 4.7 Spirals with finite and infinite lengths.

Figure 4.8 Golden ratio spiral.

Along with geometrical transformations seen in scaling, mathematical equations too can be transformed. An example of this is seen here with logarithms and with iterated function systems in Chapter 5.

Logarithms as Mathematical Transformation

John Napier (1550-1617) originally invented logarithms (represented as log) to simplify multiplication. By converting two numbers to their corresponding log values, adding them together, and then reversing the process, he showed how a close approximate product of two numbers could be relatively easily calculated. Before computers, logarithms were an invaluable method for multiplying large numbers, along with solving division problems, calculating powers and roots. Here is a list of logarithm transformations:

Figure 4.9 Table of Logarithm Transformations.

A logarithm's base can have any positive value.  The most common values are  and   [5] denoted by .

Centuries later these logarithms would be crucial in the calculation of fractal dimension as seen in the following sections.

The Hausdorff-Besicovitch Dimension (top)

" As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain they do not refer to reality."

Albert Einstein

Felix Hausdorff (1868-1942) and Abram Besicovitch(1891-1970) revolutionized mathematics by proposing dimensions with non-integer values. They demonstrated that though a line has a dimension of  and a square a dimension of ,  many curves have an "in-between" dimension related to the varying amounts of information they contain. We refer to such in-between dimensions as the Hausdorff-Besicovitch dimension.

Figure 4.10 Dimensions caught in the middle.

To explore the Hausdorff-Besicovitch dimension, we will look first at traditional dimensions ( lines, area and volume) and then explore expanded dimensions of fractals using three methods of calculations:

• 1. The exactly self-similar method for calculating dimensions of mathematically generated repeating patterns.

• 2.  The Richardson method for calculating a dimensional slope.

• 3. The box-counting method for determining the ratios of a fractal's area or volume.

" One must say-  instead of points, straight lines and planes - tables chairs and beer mugs"

David Hilbert

To calculate a fractal's dimension, we simply extend the formula for calculating traditional dimensions, so let's begin with this basic formula. Look at the dimensional relationship of how a line, a square and a cube are linked together dimensionally in Figure 4.11.

Figure 4.11. Illustration of line segments,  and scaling factors,  raised to the appropriate dimension .

The number of line segments  of a unit, is equal to the inverse of the scaling factor , raised to the appropriate dimension .

• The general equation is represented by:

The first three integer dimensions are  = for a line, = for a square, and = for a cube

Figure 4.12 Basic construction of lines, squares and cubes of unit lengths , , .

The list below shows the number of pieces  in relationship to  vs. , for a given line, square and cube.

values:    line   square   cube

Figure 4.13. Table of the number of segments n in relationship to  vs. .

Though non-spatial dimensions expanded the range of dimensions we can explore, these dimensions are still whole, or integer, dimensions.  Not until recently, with the advent of fractal geometry, have we begun to explore partial, or fractional, dimensions. In this section, we will look at these fractal dimensions.

Calculating Dimensions of Self-Similar Fractals

Calculating the fractal dimension of exactly self-similar shapes is fairly straightforward. This approach, which is limited to fractals whose structure can be predetermined mathematically, produces precise values. These are the linear fractals we saw in Chapter 3.

Figure 4.14 Exactly self-similar fractal.

Recall that the basic equation for calculating dimension is:

Although there is no rule that dimension  has to have an integer value, this has been the convention in traditional geometry. Here we will carry out numeric calculations,  for many different values of  and , where   is not an integer dimension, but rather a 'fractional' or 'partial' dimension.

Given:

the familiar case.

take of both sides.

factor the exponent  out of the scale factor.

divided by  to set the equation equal to .

We are left with the equation:

In a true mathematical fractal, replacing the line segments with seeds is a never ending process.

Therefore the general case is written , where  represents the level the fractal "seed" has been replicated.

Which gives    where = the level of replication.

However since  is found in both the nominator and the denominator of the equation it can be neatly factored out, so we are still left with the basic equation:  where the exponent  is factored out .

The formal fractal dimension equation for  is given as the .

Calculating the Dimension of the Koch Curve

Now lets apply the formula that we have derived in the previous section to the Koch curve. This curve makes a good example because its construction is uniform and we can calculate its dimension with relative ease.

Figure 4.15 Koch curve showing  levels for  and  values.

From Chapter 3, we know the number of line segments in the Koch curve is  and that each line segment is replaced by a replica of the original, reduced in scale by .  To calculate the dimension:

• Use the basic dimension equation .

• Replace  with , for the number of unit line segments and  with , for the scale factor. The equation now becomes , or simply .

• To find , take the log of both sides.  Simplify and you are left with .

Now lets use what we just learned to calculate the dimension for the Koch Snowflake.

The Dimension calculated for the Koch Snowflake

The Koch Snowflake can be created using two distinctive techniques. One method used assembles the Koch Snowflake from perimeter components of the Koch curve, the other method produces the entire the Koch Snowflake from a single generated curve.

Perimeter

Figure 4.16 The Koch snowflake constructed by connecting edges of the Koch curve shown with .

The first technique uses three Koch curves which are joined together at the edges as seen in Figure 4.16 to form the Koch snowflake. Since the construction of these curves involves calculations only involving the outer boundary, the fractal dimension calculated will be that of its perimeter, the Koch curve.

Inner dimension

Figure 4.17 Two different seeds producing the Koch snowflake.

The second technique involves the construction of the whole Koch snowflake from a single seed. Many varying seeds can be used as seen in Figure 4.17. Since the entire snowflake is formed from a single seed the calculated dimension will be that of the entire curve.

Now lets look at the dimension of one variety call the 7 Snowflake Sweep. As seen from the curve not all replacement components have the same length, so the basic equation for calculating the exact dimension can not be used here. So then how do we  calculate its dimension?  We begin by looking at how the overall snowflake curve is formed. The basic seed has 7 line segments with 6 reduced by  scale and 1 reduced by  scale. If all the line segments were reduced by the scale  the dimension equation would be simply . However, since at least one segment is reduced by a scale of a lesser amount, namely , the dimension of the curve has to be greater and a ratio relationship has to be establish between the different scaling values.

The equation we use  to develop this ratio relationship is given by .

For the Snowflake Sweep's exact calculation is given by .  Calculating the dimension form this equation we find .

This value corresponds to the observed value, dimension , of the Snowflake Sweep curve that covers the area of a plane. This region is referred to as its inner dimension.

As we have just seen the Koch Snowflake actually has two types of dimensions: one for its perimeter and one for its inner region. In the following section we will see how to numerically calculate other exactly self-similar fractals with varying  scaling values.

Other curves with a perimeter and inner dimension.

All fractal curves with area ( inner dimension ) also have a perimeter dimension. Here we will look at two more examples: the Gosper island and later the Dragon curve. Some fractals with an inner dimension of less than  also have an alternate perimeter dimension, as we will see with the Monkey Tree fractal. As shown with the Koch Snowflake, dimensions calculated relate to the specific seed used to construct it. For example if a perimeter curve is used to create a fractal, that curve's dimension relates to its boundary, accordingly if a curve's construction is that of a complete fractal the dimension calculated will be that of its entire structure.

Figure 4.18 Gosper island with both perimeter and inner dimension.

As with the Koch snowflake, the Gosper island [6] too can be created by placing perimeters together or from a single seed. The calculated perimeter dimension is given by  segment lengths each reduced by a scale s of . This results in a perimeter dimension that is calculated to be . Since entire Gosper curve fills an area it has an inner regional dimension of . This can be seen in Figure 4.21 that shows a numerically calculated value of .

Figure 4.19 Calculating the Gosper island in the plane dimension.

Now lets look at calculating an exact fractal with different segment lengths whose inner dimension is less than .

Figure 4.20 The Monkey Tree's seed generator and its next growth level.

The Monkey Tree [7] seed uses two different segment lengths for , each with a different scaling value for  to generate its fractal, six with =0.333 and five with =0.186. Its over all dimension is given by the segment's ratio where  or .

Monkey Tree Maze Game.

Figure 4.21 Monkey Tree Maze.

The Monkey Tree curve makes for a visually appealing maze with its many turns and empty regions. Since the Monkey Tree is created from one continuous curve that never overlaps, a path can be made from any region that escapes to the outside. To see this more clearly randomly choose a section within the curve in Figure 4.21 and see how fast you can find a way out. If you get stuck you can follow the curve path for the exit.

Calculating the Dimensions of the Varying Dragon Curves

Figure 4.22 Component lengths and completed form of the Dragon curve.

We begin our calculation of the dragon curve's dimension with its most basic construction, in which  two seed segments of length  replaces a single line segment with length .  This results in a scale reduction, , of . The dimension is calculated to be . This curve is contained and fills a region with a confined area as seen in figure 4.22.

The dragon curve comes in variety of forms with differing dimensions. In the following section we will see some of these variations.

Now lets look at what happens if we lower the fractal dimension of the seed for the basic dragon curve.

Lower dimensions, its area disappears.

If line segments were shortened the fractals dimension would be reduced and it would no longer cover an area. This can be done by shortening seed segments at the same time keeping the starting and ending points constant or by increasing the distance between the beginning and end points. For illustrations on dragon curves with dimensions less than dimension- see figure 4.23.

Figure 4.23 Dragon curves with dimension less than .

Now lets look what happens to a dragon curve whose component lengths (generator) are increased in relationship to its overall length (initiator).

Higher Dimension Seeing How Some Fractals Can Grow Forever.

If we decrease the Dragon Curve seed's segment angle its calculated dimension would also increase. This can be done by lengthening seed segments at the same time keeping the starting and ending points constant or by reducing the distance between the beginning and end points of the seed. Unlike other fractals we have seen earlier, which are limited to an confined space,  fractals with 'calculated' dimension greater than  can continue to expand outwardly from the center forever. If a seed segment line is longer than the distance between the starting point and the ending point of the seed the fractal will diffidently continue to do so, see Figure 4.24.

Figure 4.24 Dragon curve with scaling value greater than .

Now what would happen when we use two line replacements whose lengths are greater than . The new 'calculated' dimensions will fall in 3 parts:

I.          scale , where the dragon structure grows continuously in relation to its scaling factor and its calculated dimension range is .

II.         scale , a special case where all 'seed' replacements over lap causing a growing triangular grid with uniform density, see Figure 4.25.

III.       scale where 'calculated' dimensions have negative values and the fractal will always continue to expand for higher levels of the curve.

Figure 4.25 Dragon curve with a scaling length equal to segment length.

Figure 4.26 Four, 2 segment replacements with continually longer segment lengths and their growth.

Does this mean that our new fractal is greater than -dimensional? In the physical and mathematical world, the answer is no because it is constrained to a -dimensional plane. In theory, however, the answer 'could be' yes, a somewhat philosophical answer.  The dragon curve contains more mathematical information than is needed to fill a plane. If we could find a way to liberate this fractal from its confined space, its high dimensional curves could fill up a volume space (similar to the way the twisted Hilbert curve segments do in Chapter 3). In fact, some could even conceivably fill up a -dimensional space or greater if we could find one for it to exist in. These principles hold true for other linear fractals too as we will shall see using FractaSketch.

Let use FractaSketch to build an ever expanding fractal.

Using FractaSketch to Build an ever-growing fractal.

Figure 4.27 Dragon with line segments greater than the its seed length.

Here we will use FractaSketch to grow fractals whose mathematical dimensions are calculated to be greater than . A dragon seed works nicely, but any other fractal seed following the same instructions should also work.

Step 1: Open a drawing pallet and create a small fractal with at least one line segment that is longer than the distance between the beginning and end points, as in Figure 4.27.

Step 2: Click on levels 2-10 to watch the fractal grow.  Notice that the fractal continues to take up more and more area on the computer screen until the screen in completely covered see Figure 4.28. The rate of growth depends on the length of the seed's segment used. If the growth rate is very rapid you can use the "Reduce" feature from the "Scale" menu as often as needed to get a more complete view.

Step 3: Repeat the growth process with seed lines of various lengths.

Figure 4.28 A growing fractal 'seed' in FractaSketch that will increase in size at each higher level.

FractaSketch gives fractals that continue to grow uncontrollable an uncertain calculated value of "??" in its menu bar, see Figure 4.29.

Figure 4.29 Menu bar with uncertain dimension.

Now lets use what we have learned in pervious sections to calculate the dimensions and apply these principles to objects that reside in a volumetric space. Here we will look at two such examples: the Menger sponge and the Sierpinski tetrahedron  or pyramid.

Calculating Dimensions for the Menger sponge and Sierpinski Pyramid or tetrahedron

Figure 4.30 Components of the Menger sponge.

The Menger sponge consists of a primary cube divided equally into 27 smaller cubes, each a 1/3 scale copy of the original.  Then the center cubes are removed from all 6 sides and the middle, leaving 20 smaller cubes. By using the exact method formula we get a  calculated dimension for the Menger sponge of . An object residing in a 3-dimensional space whose components take up no volume.

Note, if you take a line connecting any corner diagonal points of the Menger sponge, their intersection would be that of the Cantor set for that length.

Figure 4.31 The Cantor set found in the Menger sponge.

Figure 4.32 Calculating a Sierpinski pyramid (tetrahedron) dimension.

The Sierpinski pyramid consists of a primary pyramid that is replaced by  smaller pyramid each  the original scale. The calculated dimension is equal to . The Sierpinski pyramid gives an example of a  dimensional object without area. If such a pyramid existed, it would reside in a volumetric space with its physical structure not taking up any volume space, even though its residing space would be defined.

How are these Fractals Alike ?

Can you guess what the fractals is Figure 4.33 have in common ?

Figure 4.33 Assortment of fractal images.

Answer: They all have a fractal dimension of roughly 1.5.  If you look carefully you can see that though each fractal has a different form, they all cover a similar amount of the -dimensional plane.

Next we will play the Totem Pole game in which we will place fractals in order of their dimension.

The Fractal Totem Pole Game

There are a number of objects in Figure 4.35  below, and each has been generated using a number of equal line segments. Then the segments have been replaced with a small replica of the whole object. This was repeated 4 times (level 4). The goal of the game is to put the objects in order of their fractal dimension, and to find their underlining seed-shape.

Hints: 1) divide the object into halves or thirds, 2) look at the ends for a recognizable seed-shape of the whole, 3) check the density of area being covered. Generally, the more solid the area is, the higher the fractal dimension. Also notice disproportionately long lines, as they generally indicate lines that have not been transformed.

Figure 4.34 Images to put in order of their fractal dimension.

Now use FractaSketch to see if you can create totem pole fractals, if you haven't done so already.

Answers to the Totem Pole Game showing a list of their exactly calculated fractals can be seen on the next page.

Figure 4.35 The Totem Poles seeds and their corresponding dimensions.

For examples of other fractals with exactly calculated dimensions see Gallery of Fractals in Appendix B.

In the next section we will use another method to calculate fractal dimension. This technique called the Richardson method is generally used when an object has a fractal structure that is not exactly repeating, such as a coastline, and therefore the exact method can not be applied.

The Richardson's Method of varied measured lengths.

" Big whorls have little whorls,

Which feed on their velocity;

And little whorls have lesser whorls

And so on to viscosity

(in a molecular sense). "

- Lewis Fry Richardson [8]

Lewis Fry Richardson (1881-1953),  an English meteorologist, pioneered a process for calculating dimensions with varied measurements. Using this technique an object's perimeter is measured with rulers of different lengths, then by graphing its slope the corresponding dimension is calculated. His work compared the dimensional slopes of coasts, such as Great Britain, that remain jagged at many levels of magnification, to non-fractal boundaries like circles that remain smooth [9] .

Figure 4.36 Dimensions of coastlines vs. a circle, Richardson's calculations. Illustration from The Fractal Geometry of Nature, © 1982 Benoit Mandelbrot.

Figure 4.37 Measured segments for the coast of Britain [10] .

Though Richardson
[11] had spent many years in researching boundaries, it was not until 1961, eight years after his death, that the results of his experiments were published. In his paper, Richardson  points out that countries with common borders often report different border lengths as he notice from examining various encyclopedias. For example, Spain claimed its boarder with Portugal was 987 km, whereas Portugal claimed it was 1214 km . Similarly, Holland claimed its border with Belgium was 380 km, whereas Belgium claimed it was 449 km.

Through graphs, Richardson tries to explain what accounts for these relatively large differences in measurements, often varying as much as 20%. [12] He suggests that the different 'measuring sticks' [13] used by one country might be disproportionally shorter, maybe even by an  factor of , than what an other country uses.  Also, a small country might take more care in measuring its boarder than a large country that has a greater perimeter to cover.

Figure 4.38 Borders of Spain and Portugal and borders of Holland and Belgium.

This method that Richardson developed, also known as varying slope dimension or compass dimension, is an effective technique for measuring an object's perimeter fractal dimension. Although not as precise as the method used for exactly self-similar objects, this procedure enables us to calculate the dimensions of real-world objects, which are not perfectly self-similar.

To calculate the dimension of perimeters with the Richardson Method, you use 'rulers' of differing lengths. Then by comparing the measured lengths of an objects perimeter to corresponding variations in ruler lengths and plotting the results, a graph can be made with logarithmic scales, from whose slope an object's dimension can be readily calculated. Logarithmic scales are used because the exponent values associated with dimension, translate easily  to linear values that define a slope. It is the object's structural changes, weather it becomes more detailed or smoother, at various scales that is the object's fractal dimension.

Figure 4.39 Measure lengths on regular graph vs. logarithmic graph.

As you reduce the length of your measuring "sticks",  the precision of your measurement increases, generally resulting in a greater apparent length for objects with fractal dimension. For different objects, you should use an appropriate measuring stick. For example, in measuring the coastline of Britain you might measure it in units of 1000 km, 500 km, 100 km, 50 km, 20 km. (It is easier, of course, to use maps with a corresponding scale.) For the circular parameter of a wheel, your measurement might be in  meters 1m, .5m, .1m, .05m, .02m. You might use a measuring tape as a final measurement, which works well for wheel because there are not large deviations on the surface.

Figure 4.40 Measuring perimeters using different measuring sticks.

Richardson's Dimension Equation.

Let's look closer into the Richardson's method for calculating fractal dimensions using varying measurement lengths. To calculate a dimension, we first have to establish a logarithmic relationship between the object's over all 'measured' length and the length of the 'ruler' used to measure it. We do this by graphing various values of , where  is the length of the perimeter, against the corresponding values of , where  is the scaling factor (the rulers length), used in measuring the perimeter. The resulting slope of the ( / ) graph will be the related fractal dimension of the measured object.

For a smooth or linear object, smaller rulers will measure similar lengths as do larger ones, so the slope will be horizontal.  A graph with a horizontal slope corresponds to the dimension . As the fractal dimension of an object increases, so does the severity of the slope.

Figure 4.41 Different slopes of / corresponding to different fractal dimensions.

The formula for the Richardson slope method is given by the relationship , where  is the calculated slope dimension of the  graph added to the dimensionally of a line with value , and where  is the calculated standard dimension.

Constructing the Equation in a Familiar Form

Now lets formulate the Richardson equation we have just seen. Later on we will compare the results with values calculated from the exact method using the Koch curve.

Figure 4.42 Scaling values and perimeter values

Select , the scale value equal to the ratio between the segment's replacement length and the length of the corresponding seed shape. To plot different points of the slope, various scales values will be used. Here the scale measurement values are calculated for higher levels of replacement (a decrease in ruler length) with the formula . By using unit levels , scaling values can be easily calculated that correspond to dimension .

Calculate the perimeter length. The length is given by .

Set the two equations equal in terms of the exponent value , so    .

Simplifying we are left with  and .

• Now by setting both equations equal to  and then solving, we are left with a equation , with the  constant equal to which we set to .

• Rewriting the equation we get  or simply the perimeter equation , where  is the slope dimension giving in the form .

Now lets see how to form the relationship .  Remember  and  are interchangeable values.

• First lets use the perimeter equation . Taking the logarithm of both sides we are left with .

• Next take the equation for the number of line segments  and proceed to put it in the logarithmic form .

• Take the perimeter length given by , where perimeter length is equal to the number of pieces multiplied by the scale. Now putting it in its logarithmic form, we derive .

• Now by substituting  for  and  for  we are left with ,  which  simplifies to .

Now lets construct the Koch curve using  the Richardson method and compare the results with its exact value calculated in the previous section.

Illustrating the Richardson method using the Koch curve.

Figure 4.43 Components of calculating the Koch curve.

Set the scale value of the Koch curve to be , .

The perimeter length of the Koch curve is given by , .

Setting the two equations in terms of t  results in .

Simplifying, results in  and .

• Setting both equations equal to  and solving we are left with:

or

.

Then by using an appropriate logarithmic value for calculating the constant value, in the Koch curve's case , the results simplify to form a nicely factored equation [14] . This leaves the slope for the Koch curve to be  as seen in Figure 4.44.

Figure 4.44 The Koch curve's slope plotted on a / graph.

So the calculated value with the Richardson method for the Koch curve is   . This value is in close agreement with the exact method's calculation from the previous section.  Curiously, this calculated value is similar to the fractal dimension of Britain's west coast. In the next section we are going to look at the fractal dimensions of different states in the United States of America.

The United States of America and Their Dimensions.

Figure 4.45 Colorado, Hawaii and Kansas seen in -dimension plane.

We begin by looking at the different fractal structures of states in the United States of  America. Our first comparison is between Colorado and  Hawaii. In -dimensions you can see that Colorado and Hawaii have perimeters that are quite different. Colorado's  boarders  are formed by four straight lines while Hawaii has a rugged coastline. Here we see that a smaller state can be the one with a longer coastline. In - dimensions the two states formations are quite similar with their mountainous terrain, whereas another state like Kansas, which is basically flat would a have considerably lower dimension.

Figure 4.46 Colorado, Hawaii and Kansas seen in -dimensional space.

Now what happens if we look at California, in -dimensions it has boarder properties of both Colorado and Hawaii, with coastal properties on its west side and straight boarders on its east side. In -dimensions it shares traits with all three states, by having the mountainous terrain of Colorado and Hawaii in regions such as the Sierra Nevada Mountains and the flat plains found in Kansas in its Central Valley. Having more than one distinctive fractal characteristic puts it in the classification of a multi-fractal. In a multi-fractal calculation,  an object's varying dimensions are measured and a record is kept of how much of each dimension is found. Multi-fractals are discussed further in Chapter 6.

Figure 4.47 California the multi-fractal

In the following game we are going to look at and then measure objects for their different fractal dimension.

Measurements, calculating the dimensions of different objects.

How long is anything anyway? How long is the coast of Britain, the shore of Santa Cruz or the length of the Amazon River? As we have seen earlier the answer depends on what scale you measure them. We typically would not use the same device to measure a rock as a mountain. For this fractal experiment you will measure things around you. The measurement sticks you will need are a tape measure, a foot ruler and a yard stick. If these things are not available or you want to be unconventional you can use other things too like a baseball card, sticks of varying lengths, a shoelace, or almost anything you can find to measure around an object. You can make up you own units for them, for example: stiff baseball card units, chop stick units or even 'Fractal Exploration' book units. For units of measurement are just items with lengths assigned to them. One note: for doing your calculations, if you use your own units you will have to calculate a length relationship between the different 'object rulers' you use, such as three stiff baseball cards are the length of a shoe.

Step 1: Find objects with different dimensions to measure. For a low dimensional object a flat table is a good choice. It will serve as a reference source for a standard Euclidean object. For objects of higher dimension you can use contoured chairs, house plants, a kid brother or sister, or any other objects with parts sticking out requiring measurements to be made around those parts. One thing to be careful of is, since you are measuring a perimeter length you must  follow the same path for each measurement.

Step 2: Take each object and measure the same perimeter for each of your different ruler lengths, recording the results on your chart as seen in Figure 4.48. This will show you how different rulers measure different lengths.

Figure 4.48 Chart to record the measured lengths of objects from different sized 'rulers'.

Warring: this section is part of a more advanced exercise and therefore may be skipped. The results vary widely do in part to your ruler's limited accuracy and scaling range. The main propose here is to give you an idea on how to physically calculate fractal dimension using the Richardson method.

Step 3: Take the recorded data you have recorded of perimeter vs. measured length, convert the data to its logarithmic values and plot it on a piece of graph paper.  You will find graph paper in Appendix C. Now calculate the slope, this should roughly correlate to the slope dimension . Now place  into the equation  to calculate the objects full perimeter dimension.

We have look at the Richardson Method to calculate the fractal dimension of an objects perimeter. In the next section we will use the Box Counting Method of counting area and volume to calculate an object's fractal dimension.

The Box Counting Method (top)

Figure 4.49 Box counting method using divided segments, grid for a plane space and cube lattice for a volumetric space.

The Box counting ratio method ,also known as the Brute force method or Grid Method is an estimation procedure for calculating the fractal dimension of complicated objects. It is most effectively used when you cannot calculate an objects dimension with numerical formulas or accurately determine a slope dimension of an irregular shapes. For example, it would  be difficult to calculate scribbles, dust, ocean waves, or clouds using other methods. This accounts for its popularity in spite of its counting resolution's limited accuracy.

Figure 4.50 Some of the many objects best calculated using the box method.

This method is popular because it is straightforward and adaptable to many situations. If you can contain an object within squares or boxes, then you can perform a statistical analysis to determine its physical dimension.  You can use this method to calculate dimensions for a very small object like Cantor's dust or for a very large object like a mountain range. You could even calculate the fractal dimension of the universe--at least what we know of it.

Dimensions Found in a Plane Space

To calculate the fractal dimension of images in a plane, you begin by covering an

area with grids of different mesh sizes. Then, you compare the grid sizes and the

number of squares containing at least a part of the image. The ratio of grid sizes to number of grids containing the object establishes the dimension.

Figure 4.51 Measuring a fern's dimension with different grid sizes.

Note: Even though this process generally produces accurate results, there are limitations with this method. For example, if tiles are used to cover a measured area, the calculated dimension can not exceed 2,  because a fractal's dimension can not exceed the dimension of the units used to measure it.

Dimensions Found in a Volumetric Space

To calculate dimensions with volumetric space, you use a similar method as in calculating the dimension of a plane, only instead of using tiles you use boxes.  These boxes are mapped out to form lattices of varying sizes ( a - dimensional grid ) called an array see Figure 4.52. Then by counting the boxes containing at least part  of the object a ratio is established between the box size and its corresponding count. This ratio at different scales determines the object's dimension.

Figure 4.52 Box counting method for volume a -dimensional lattice.

Actual Calculations

Now lets examine how the grid method is used to measure the fractal dimension of an image found in a plane.

• First, for each grid mesh size, count the number of grids that contain the image.

• Now pair all combination of  counted values and place them in the standard equation for the Box Counting Method. The dimension is calculated by the equation , where  is the number of squares containing the image and is its grid scale. Now average your results to get a good estimate of the objects fractal dimension. It should be noted that generally finer grids produce a more accurate measured dimension. This is due in part to amount of variations that can arise in counting regions dependent on how a grid is placed see Figure 4.53.  Placing grids in the same position with increased subdivisions adds in to decrease fluctuations see fern in Figure 4.51.

Figure 4.53 Variation in grid placement.

Alternative method:

• Plot the values of  vs.  on a  graph. Then take the averaged slope as the calculated fractal dimension, see Figure 4.54.

Figure 4.54 Measuring an objects dimension by comparing values,  vs. , plotted on a  graph.

You can use the same method for volumetric measurements of dimension by using cube boxes, as seen in Figure 4.52.

Now lets do the calculations for the Koch curve, and compare the results to previous methods.

Measuring the Fractal dimension of the Koch curve Using the Box counting Method.

We begin by counting the number of squares that contain part of the Koch curve for each grid. Here we use three grids with ratios 1 : 1/2 : 1/4 with counts 18, 41 and 105 as seen in Figure 4.55, Figure 4.56 and Figure 4.57 respectively.

Figure 4.55 The Koch curve with unit 1 grid size, with 18 containing the curve.

Figure 4.56 The Koch curve with unit 1/2 grid size, with 41 containing the curve.

Figure 4.57 The Koch curve with unit 1/4 grid size, with 105 containing the curve.

I.          .

II.         .

III.       .

Figure 4.58 Calculating dimensions from three different grids using the Box Counting Method.

Now taking the data from our three grids and placing them into the Box Counting Equation, we are left with three approximate calculated dimensions with sizable fluctuations in value see Figure 4.57. Their average of  is a reasonably close result to the actual value of , if you were to do more calculations with different grid sizes you should expect most values to fall within a certain range. If you were to refine your measurements further with considerably smaller grids an increase in accuracy should be reflected in the answer.

In Figure 4.59 we show the dimension of the Koch curve by counting grid boxes containing the object and graphing the results on a  plot.

Figure 4.59 Using the Koch curve data to formulate a slope dimension.

Now we are going to look at different areas of fractal dimension using a garden vegetable, the Broccoli Romanesco. The Broccoli Romanesco is an ideal candidate because its fractal structure can be seen in different ways at several levels of magnification.

The Broccoli Romanesco, a fractal seen in three different dimensions.

The Broccoli Romanesco exhibits fractal structure in its perimeter, planar and volumetric dimensions. Lets look at them now.

Figure 4.60 Measuring the Broccoli Romanesco jagged perimeter.

If we look at the perimeter of the Broccoli Romanesco we can see an outline that bares a 'rough' resemblance to the Koch curve, see Figure 4.61. Notice the continuos jaggedness seen at different levels of magnification even at a close distance. The measurement of the jaggedness of this irregular boundary is its perimeter dimension. We can measure its dimension by comparing the number of squares containing its perimeter at different grid sizes.

Figure 4.61 A cross sectional view of the Broccoli Romanesco.

In  figure 4.61 we see a Broccoli Romanesco as it looks when it is cut in half. This revealing cross sectional slice gives a view of it planar dimension.  By measuring the number of squares that are filled at different grid sizes a comparison can be made that corresponds to its planar dimension. Notice its self-similar branching structure that can be seen to at least 4 levels.

Figure 4.62 Close-up views of the Broccoli Romanesco at different levels of magnification.

In Figure 4.62 we see a two close-up views of the Broccoli Romanesco, notice the difficulty in gauging its true size without a scale reference. In Color Plate 24 we see a Broccoli Romanesco as it looks in its entirety. This form gives a view of its volumetric dimension. If you look closely at its top regions you can see the same type of spiral cones that you see from a distance only smaller. This self similar structure can be seen to at least 3 levels of magnifications. By creating lattices at different sizes and counting the number of regions that contain at least part of the object, calculations can be made that corresponds to its volumetric dimension. This procedure can be greatly simplified if the structure's form can be entered and stored as an array of  numbers in a computer. With this alternate way, a computer can create a mathematical lattice that could partition the object's spatial values without having to physically divide it.

It is the Box Counting method we use to calculate the  fractal dimension of most things found in nature, see Color Plate 25 to visually compare the dimensions of different clouds. Now lets use grids to measure planar dimension.

Grid Game for Measuring Dimensions.

In the grid game, you can use the different size grids found in Appendix C to calculate dimensions of different exact fractals like we did with the Koch and see how well their dimensions correlate to their known dimensions. It might come in handy to copy these grids onto transparencies to use in your calculations if you do not want to mark your fractals with grid lines.  Also included in Appendix C are various grid maps for Great Britain and the Monkey tree that can be used to calculate their dimension.

Figure 4.63 Three different size grids for Great Britain: scale , scale  and scale .

Step 1: Find a fractal to measure whose dimension you know and can be measured by the grid sheet you are using. This fractal can be one you find in a book ( for example this one ) or one that you create and printout in FractaSketch. In FractaSketch the dimension value can be read from the menu bar of the program.

Step 2: Take your fractal and divide it with evenly distributed grid squares with known grid values. A good technique is to place different grid size transparencies over the object  and count the contained squares as seen with Britain in Figure 4.63. For a basic exercise you might want to start by carry out calculation using these grids .

Step 3: Count the contained squares for the different grid sizes, as we did with the Koch curve earlier in this chapter, and record their values.

Step 4: Place the recorded values into the formula  and calculate the image's fractal dimension.

After you feel reasonably comfortable with the accuracy of your calculated dimensions, you can use this technique to examine other fractals whose dimensions you do not known exactly--such as a fern, counties on a map or even a newspaper. Remember due to the limited accuracy of our measurements, dimensional values within 10% of the correct value are consider quite reasonable.

Grid Game for Measuring Dimensions on the Computer.

In the same way you can calculate dimension by using grids on paper, you can also calculate the dimension of images using a Macintosh paint or graphics program.

Step 1:  Paste an images from FractaSketch, MandelMovie or even an image of a scanned photograph onto the pallet on your paint program. Save the image now, so you can recall it for every new grid placed if you need to, you might want to make a backup copy of your picture in case accidentally alter it or save it with a grid on it.

Step 2:  If your paint or graphics program does not include an option for creating grids with varying mesh sizes, make the grids first in the program and paste them in from the "�" menu in your "Scrapbook".  Make 5 grids in sizes of 25, 20, 15, 10 and 5 pixels square. You can make alternate size grids if you want. It is important when pasting the grids over the object that the white regions not be included. This is done by setting the white regions to be transparent in your paint or graphics program.  This will allow you to see the object after the grids have been placed. Instructions on how to do this should be found in your paint or graphics program.

Step 3:  Place a grid divided into squares of equal size (say 25 pixel square) and count the number of squares that contain part of the curve. Repeat the experiment by counting how many squares contain the object when the grids are 20 pixels square, then 15 square , then 10 square, then 5 square. If your program does not generate grids you will have use five duplicate picture files or reopen the picture file each time without saving the changes of the added grid.

Step 4: Count and record the number of squares containing the object for the different grid sizes.

Step 5: Place the recorded values into the formula  and calculate the image's fractal dimension.

Step 6:  If you know the fractals dimension in FractaSketch, compare the results and see how close they correlate.

The fractal dimensions calculated here are averages of the whole structure and can not discern between the dimensions of its different regions. This type of work deals with multi-fractals. [15] which we will see with greater detail in Chapter 6.

We have looked at 3 major ways of calculating fractal dimension.

• Calculations of exactly self-similar dimension, using mathematical formulas.

• Richardson's dimensional measurements of divided segments. Here fractal dimensions correspond to the slope correlating to the length of the "ruler" used and its overall measurement.

• Box-counting dimension, calculating the containment of an object at different measurements.

In the next chapter we will look at ways to generate fractals with equations. This procedure is responsible for producing many of the colorful fractals that almost defy description.  In Chapter 6 we will look at fractals with the similar shapes and dimensions to the ones we have calculated in this chapter. These fractals are generated by taking points and mapping them to other regions, we call this technique Iterated Function Systems.

 [1] From Newton's work, Philosophie Naturalis Principia Mathematica, Praefat. [2] Mathematica 3D graph of Plot3D[Sin[x y],{x,0,Pi}, {y,0,6},PlotPoints -> 30]. [3] From the revised 2nd edition of Flatland published 1884. [4] Initially fractal geometry was classified as a field in topology. However, as it evolved it became heavily in twinned into other areas of mathematics, most notable dynamical systems. Fractal geometry is now used in analyzing many of the structures found in chaos theory. [5] Often is referred to as the natural log and is derived from the limit  given by . [6] In The Fractal Geometry of Nature, 1982 page 46, Benoit Mandelbrot refers to The Gosper Island as The Koch Island or Lake. Originally the curve was named " The Gosper's  flowsnake" for its discoverer William Gosper  in an article " Monsters" by Martin Gardner published in December 1976 Scientific American article pages  124-133. [7] Also called " snowflake halls". [8] Written in 1922 as an annotation to his work in turbulence. According to Benoit Mandelbrot's book The Fractal Geometry of Nature, 1982 page 402 this passage is often quoted anonymously due in part to " ... their kinship to some classics." See a poem on fleas,  from which this poem was parodied also on page 402. [9] It was Richardson's work that prompted Mandelbrot and others to ask the now famous question 'How Long is the Coast of Britain ?'. Benoit Mandelbrot wrote a well known paper 'How Long is the Coast of Britain ?' in which he proposes many original ways of discussing fractal geometry (1975). [10] These measurements were made by scanning in a country's map ( this case Britain )  and then using the draw program's built in segment length measuring tool as the method for  making measurements. [11] It is believed that Richardson noticed many discrepancy in territorial measurements when he was trying to build models on why nations have conflicts that result in war. Richardson spent a great deal of his life's work on peace and conflict studies  mainly on why nations go to war. This is quite understandable considering he lived his life in England experiencing both World War I & II. [12] Page 27 Mandelbort's The Fractal Geometry of Nature. The problem of contiguity: an appendix of statistics of deadly quarrels, General Systems Yearbook 6 (1961) pages 139-187 [13] You will find the term epsilon used. [14] We can write  alternatively as . [15] Benoit Mandelbrot, An introduction to multi-fractal distribution functions, found in: Fluctuations and Pattern Formation, H.E. Stanley and N. Ostrowsky, Kluwer Academic, Dordrecht, 1988. J. Feder, Fractals, Plenum Press, New York 1988. K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, Wiley, New York, 1990.