Lesson 6. How holey is the Swiss cheese;
Fractal dimension
Objective: We say a line segment is one-dimensional, a triangle or square is two-dimensional, and a pyramid or cube is three-dimensional. This is because intuitively dimension has something to do with the number of distance measurements needed to specify the size of an object in the Euclidean world we live in. But, alas, what is the dimension of a fractal object that is fractured and scattered in space. We therefore resort to a definition of dimension based on the concept of capacity, that is, how much space an object actually takes up in reality. First, the capacity definition is applied to a line, triangle, and cube to recover the Euclidean dimensions 1, 2, and 3, respectively. We then find that fractal dimension d is not necessarily a whole integer, but can take on any value between the integers, as shown.
Here is the summary of dimensions for the objects in this lesson.
Object / DimensionCantor set / 0.63
Line / 1
Sierpinski triangle / 1.58
Cantor gasket / 1.89
Sierpinski pyramid / 2
Triangle and square / 2
Menger sponge / 2.73
Cube and solid pyramid / 3
Scary words: Fractal dimension, Logarithmic function, log-log plot.
Last update: 7 Jan 2003
Lesson
Euclidean geometry
For a straight line one measures the distance by two end points. But, the two distance measurements of base and height are needed for the area of a triangle, and another measurement of height is required for the volume of a pyramid. Therefore, our intuitive definition is that dimension is the number of distance measurements needed to specify how much room an object takes up in space. Put it differently, a one-dimensional line segment has only the length, a two-dimensional triangle covers an area in plane, and a three-dimensional pyramid occupies a volume in space. Hence, the larger the dimension of an object, the more space it occupies in the Euclidean world, so that dimension is a measure of how an object takes up space to exist. But, what about the Cantor set, Sierpinski triangle and pyramid, Cantor gasket, and Menger sponge of Lesson 5? They are so fractured or broken up in space that it is not even intuitively clear how to define their dimension, let alone quantifying it.
For being fractured, Bernoit Mandelbrot first coined the word fractal from the Latin fractus. He is the founder of fractal geometry, educated in France and did most of his work at the IBM laboratory in NY. For fractal objects such as, the Cantor set, Sierpinski triangle, etc., we cannot come up with dimension by simply counting the number of distance measurements. It is therefore necessary to have an alternate definition of dimension that can be applied to any objects, be they Euclidean or fractal. It is based on the notion of capacity, that is, how an object takes up space to fill in, which was first suggested by the Russian mathematician Andrei Kolmogorov (1903-1987). As a preliminary requisite, we must first show that the capacity definition gives correctly the known dimensions 1, 2, and 3 when we have a straight line, square, and cube, respectively.
Line: Your job is to paint yellow centerline on the highways. Since the painted strip is only 6 inches wide, we may consider the yellow line as a theoretical line with zero width when it extends over a mile (63,360 inches) as in figure 1. The yellow paint comes in a large barrel container and it takes one barrel to paint 1-mile long yellow line. It then requires 2 barrels to paint 2-mile long yellow line, 3 barrels to paint 3-mile long yellow line, and so on, as listed in table 1.
Figure 1. Yellow centerline on highways
Table 1. Painting yellow centerline on the highways
Mile of yellow strip (x)
/ 1 / 2 / 3 / 4 / 5Barrel of paint (y) / 1 / 2 / 3 / 4 / 5
Now, here is a helpful trick. We plot the relationship of x and y in table 1 on a special graph paper called the log-log plot of figure 2. As you see, unlike the regular graph paper which is linear-linear plot, both axis values for x and y are piling up toward 10 in figure 2. This is because log(1) = 0, log(2) = 0.301, log(4) = 0.602, log(8) = 0.903, and log(10) = 1, as may be checked by the - key for the natural logarithmic function ‘log’ on a pocket calculator. Figure 2 shows the plot of x versus y gives a straight line, so that we can compute its slope by rise/run of the straight line. In fact, the slope is dimension d
. (1)
The last equality in the above expression follows from log(1) = 0. Even without the use of formula (1), we see at once that d = 1 because the straight line plot is a 45diagonal in figure 2. We have thus reconfirmed that dimension d =1 for a line segment.
Figure 2. Log-log plot of table 1
Square: You are covering the floor of 10ft 10ft room with 1ft1ft tiles. One tile covers a 1ft1ft area. But, you need not 2 but 4 tiles to cover a 2ft2ft area, 9 tiles to cover a 3ft3ft area, and so on, as summarized in table 2. By plotting x versus y in figure 3, we find the dimension for a square, as expected.
Table 2. Covering a floor with 1ft x 1ft tiles
Side of square area in ft (x)
/ 1 / 2 / 3 / 10Number of tiles (y) / 1 / 4 / 9 / 100
Figure 3. Log-log plot of table 2
Cube: A customer buys a few tropical fish and a small cubic fish tank of side 1dm (=10cm) holding 1 liter of water. After the fish have grown, he returns to buy a mid-size fish tank of side 2dm. Later on, a large fish tank of side 3dm is purchased to add the undersea sculptures. As shown in table 3, the mid-size fish tank holds not 2 but 8 liters of water, and the large fish tank 27 liters of water. From the plot of x versus y in figure 4, the slope turns out the dimension of a cube, as we already know.
Table 3. The holding capacity of fish tanks
Side of fish tank in decimeter (x)
/ 1 / 2 / 3Number of liters (y) / 1 / 8 / 27
Figure 4. Log-log plot of table 3
Fractal dimension
The formula (1) has worked out correctly for a line, square, and cube to recover the Euclidean dimension 1, 2, and 3, respectively. So, we now proceed to compute dimension of the fractal objects introduced in Lesson 5.
Cantor set: We again consider the scenario of painting the yellow centerline but on a highway that is infrequently traveled. The highway maintenance department has noticed that, although the traffic is busy within a 1-mile radius of towns, the highways in between towns that are farther apart are traveled lightly most of the time. Specifically, the longer the distance between towns, the less the traffic measured in terms of the number of two cars passing each other in opposite directions. So, the following Cantor painting strategy is proposed: (1) Paint a continuous yellow centerline when the towns are 1 mile apart. (2) We leave out the yellow centerline in the middle 1-mile stretch when the towns are 3 miles apart. (3) However, for the 9-mile highway we leave out the yellow centerline in the middle 3-mile section and then use the 3-mile painting strategy (2) for the highways next to the town. This is illustrated in figure 5. Again, assuming one barrel of yellow paint for a mile of centerline strip, table 4 presents paint requirement for the Cantor painting strategy.
Figure 5. The Cantor painting strategy
Table 4. Paint requirement for the Cantor painting
Mile of highway (x)
/ 1 / 3 / 9Number of yellow paint barrels (y) / 1 / 2 / 4
Figure 6. Log-log plot of table 4
The plot of x versus y gives a straight line in figure 6, but the slope is less than one because it lies below the diagonal (see, figure 2). By formula (1) we find that 0.631, which is not an integer dimension. A fractal object can have either an integer or non-integer dimension, as we shall see later. Here, intuitively, that dimension is less than one makes sense because the Cantor painting requires less paint than the solid centerline painting. Hence, the fractal dimension of is a measure of brokenness in the yellow centerline.
Sierpinski triangle: A local lumberyard has teak wood floor inlays in equilateral triangle of side 1ft. Since it is too costly to cover the entire floor, a plan is to accent the floor by laying them sparingly in the Sierpinski triangle pattern, as shown in figure 7. With the triangular inlay of side 1ft, it requires 3 inlays for the first Sierpinski triangle of side 2 ft and 9 inlays for the second Sierpinski triangle of side 4 ft, as listed in table 5.
Figure 7. Triangular inlays
Table 5. Floor covering by triangular inlays of side 1ft
Side of the triangle in feet (x)
/ 1 / 2 / 4Number of inlays (y) / 1 / 3 / 9
From the straight-line plot of x versus y in figure 8, we compute the dimension by formula (1). Again, it is a non-integer dimension and less than d = 2 for a solid triangle.
Figure 8. Log-log plot of table 5
Sierpinski pyramid: In Project a of Lesson 5, a large Sierpinski pyramid is assembled from basic building-block pyramids that are made of cardboard paper. We propose here an alternate construction with candle wax. To do this requires a mold for the basic pyramid with three equilateral triangle sides of 1 inch, which we assume to hold 1 oz of candle wax. As shown in figure 9, there are 4 basic (building block) wax pyramids in the Sierpinski pyramid of side 2 inches and 16 basic wax pyramids in the Sierpinski pyramid of side 4 inches, as listed in table 6.
Figure 9. Candle wax Sierpinski Pyramids
Table 6. The Sierpinski pyramid with the basic mold of 1 oz
Side of triangle in inch (x)
/ 1 / 2 / 4Number of ounce (y) / 1 / 4 / 16
From the straight-line plot of x versus y in figure 10, we find dimension by formula (1). It is interesting to point out that the fractal dimension turns out an integer d =2. Although the Sierpinski pyramid appears a three-dimensional object, it only has dimension 2. This means that, theoretically speaking, if one were to flatten out a wax Sierpinski pyramid into a very (infinitely) thin sheet, it will cover the outer triangular surfaces of the Sierpinski pyramid.
Figure 10. Log-log plot of table 6
You can now estimate fractal dimensions of the Cantor gasket (Project a) and Menger sponge (Project b) by following the line of reasoning we have presented thus far.
Project a – Fractal dimension of the Cantor gasket
The Cantor gasket is built up with small squares of side, say, 1cm, as shown below (see, Lesson 5)
You first make a table for the side of Cantor square in centimeter (x) and the number of squares (y), and the log-log plot of x versus y gives a straight line.
,
What is fractal dimension of the Cantor gasket?
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Project b – Fractal dimension of the Menger sponge
The first three generations of the Menger sponge are shown here (see, Lesson 5)
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You first make a table for the side in centimeter (x) and the volume in cubic centimeter (y), and again the log-log plot of x versus y gives a straight line.
What is fractal dimension of the Menger sponge?
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