Typically, we think of dimension as the number of independent measurements that are required to define a particular object in a particular space. A line can be described using one dimension, an area, such as a square, requires two dimensions, and a
cube floating in space, three. Here, through the application of mathematical concepts, we can see how the dimension of an object can be defined without regard to numbers that we measure independently. This new capacity will enable us to examine and describe
new and fascinating objects - fractals!
- One-dimensional, two-dimensional, and three-dimensional objects behave differently as they scale—that is, as they expand or shrink.
can write an expression for dimension based on scale factor and the number of self-similar copies.
Consider the one-, two-, and three-dimensional worlds that we typically think of. The basic object in each dimension: the line segment, the square,
and the cube, respectively. Now we're going to observe these objects as they undergo a process known as "scaling"; basically, we look at how each object changes as we shrink or enlarge it by a constant factor.
First up, the line segment—here is
a line segment of length one unit.
If we were to triple the size of this object, we would have
a line segment of length three units. We could view this result as three of our original line segments. So, we see that if we scale the line segment by a factor of three, we end up with three copies of the original. Each of these copies is said to be "self-similar"
to the original segment.
Now, let's do the same thing with a square whose sides are each one
unit in length.
To increase the size of this object by a factor of three, we have to lengthen both the horizontal and vertical elements
(or else it won't be a square anymore). When we do this, "scaling up" each segment by three, we get an entirely different relationship than we got with the scaling of the line segment.
Notice that our new shape is not made up of three copies of the original, but rather nine! This is an important property of area: it does not scale linearly with the side length. When we double the side length
of a square from 3 units to 6 units, the area does not just double—it quadruples!
Initial area = 3 × 3 = 9 units2
Final area = 6 × 6 = 36 units2
Ratio of Final Area to Initial Area = = 4
Returning to our example square, notice that if we scale the side length by three, the resulting object is made
up of nine copies of the original. Note that 9 = 32. In words, when a square is scaled, the number of self-similar squares in the resulting square is equal to the scale factor to the second power.
Now, let's look at the basic three-dimensional object, the cube. This time, as we scale the side length by a factor of three, we have to take three perpendicular directions into account.
So, if we increase the side length of a cube systematically by a factor of three, the volume increases by a factor of 3 × 3 ×
3, or 27. This means that volume scales not linearly, and not as the square of side length (as does area), but, rather, as the cube of side length. Furthermore, notice that each of the new cubes generated is self-similar to the original cube. So, we have 27
= 33, verifying that the number of self-similar copies is equal to the scale factor to the third power.
This last point is important for any budding sculptors. If you wish to make a large version of a small figurine, you would do well to
make sure that the figure's legs are strong enough to hold up its disproportionately heavier mass!
Let's organize our results from the scaling of these three objects:
Notice that the exponent in each case is equal to the dimension of the object being scaled. Let's generalize this.
N = number of self-similar copies
S = Scale factor
D = Dimension
So, if we want to develop an equation that yields the dimension of an object when we know how many self-similar copies it has as it scales, we should solve the equation above for D. To bring D out of the exponent position, we can use
the natural logarithm, which comes in quite handy whenever we need to deal with exponents or convert powers to multiplication, or convert multiplication to addition. So, taking the natural logarithm of both sides, we get:
ln N = D ln S
both sides by ln S, we get:
This equation can be used to determine the dimension
of an object based solely on its properties of scaling and self-similarity. Something similar to this definition of dimension was first identified by Felix Hausdorff, a German astronomer and mathematician working in the first quarter of the twentieth century.
The value he identified is commonly known as an object's Hausdorff dimension.
THE KOCH CURVE
- The Koch curve has infinite perimeter in a finite space; this incongruity indicates that it is not simply a 1-D
- The Koch curve has an area of zero, which indicates that it is not a 2-D object.
Now that we have a completely new way to look at dimension, let's consider some strange objects that defy traditional explanation. The first is
the famous Koch curve, or "Koch Snowflake."
This shape can be created by beginning with a
line segment and then iteratively replacing the line segment with the following curve:
look at this curve as if it were a 1-D line. At the outset, its length would be one unit. After the first iteration, its length would be of a unit.
In the second iteration, each line segment is replaced
with a curve that is as long. So, we can multiply the length from the first iteration by the factor of to obtain a length of ()2 units for the second iteration of the Koch curve.
Now, as we repeat the same steps for the third iteration, it should be evident that the new length will be ×× = ()3 units. We can generalize this by saying
that the curve will increase in length by a factor of with each iteration. Thus, we are led to conclude
that the length of the total curve continually gets larger without bound! This curve is infinite in length and yet stays within the confines of the page—very strange indeed! Perhaps this is not a 1-D line but rather a 2-D plane figure.
As we can
see in this progression of images, squares, no matter how small we make them, will "over count" the measurement of the curve. They will never have the resolution that we need to cover only the curve and no extra space.
Let's see what happens if we treat
each line segment as a square. The area of the square each time will be equal to the length of the straight segment times itself.
For the three cases depicted here (plus one thrown in to help show the trend) we have the following information:
It should be evident that the total area of this curve depends on the area of the squares we are using to measure it. In fact, the smaller the squares, the smaller the area. Notice that after the first iteration
the area of the curve has gone from 1 unit2 to less than half of a square unit. After the third iteration, the area has diminished to about a fifteenth of a square unit. It's clear to see that following this trend, the total area of the curve is
headed towards zero!
In summary, measuring the curve as a 1-D object fails miserably, as it generates an infinite length, and measuring the curve as a 2-D object gives us an area of zero, which also classifies as a miserable failure. Let's return to
our equation for the Hausdorff dimension to see if we can get to the root of this conundrum.
- Using the Hausdorff definition of dimension, we find that the dimension of the Koch curve is
some decimal value between 1 and 2.
To find the Hausdorff dimension, we need to know how
the self-similarity of this object relates to how it scales. We see that after one iteration, each line segment is replaced with four copies of itself. Furthermore, we see that each self-similar copy is the length of the original. This means that our scale factor is 3 and our number of self-similar objects is 4.
Substituting these values for S and N in the dimension equation that we derived earlier,
D = ≈ 1.26..
Hence, this object is somewhere between one-dimensional
and two-dimensional! Results like this are fractional, or fractal, dimensions, and the objects themselves are simply called "fractals."
So, our path through the story of dimension has just taken another turn. Not only have we glimpsed the behavior of
dimensions higher than the three to which we are accustomed, but now we have also seen that objects can be described by non-integer dimensions. Put another way, some objects seem to exist in spaces between intuitive dimensions.
Fractals were popularized
by Benoit Mandelbrot in the 1970s when it was found that many objects in nature resemble fractal designs to some degree or another. Indeed, the vast numbers of intricate shapes found in nature are rarely as conveniently geometric as simple lines, squares,
and planes. In fact, natural shapes tend to exhibit intriguing behavior at different scales, and while not always exactly self-similar in the way that the Koch curve is, many natural objects exhibit statistical self-similarity. And as it turns out, this property
can come in quite handy.
WHAT ABOUT NATURAL SHAPES?
In our analysis of the Koch curve, we were fortunate that it behaves so nicely—that is, it lends itself to being measured. Many objects in nature are not
so "nice." They may exhibit properties of self-similarity either only at limited scales (e.g., a fern leaf)—or only in a rough, approximate manner—or both.
Nevertheless, the concept of fractal dimension can generally be used to help describe and analyze naturally occurring phenomena and objects. In order to use this tool, however, we must replace our requirement
of strict self-similarity with a notion of approximate, or statistical, self-similarity. Let's look at a famous example:
HOW LONG IS THE COASTLINE OF BRITAIN?
- Real objects are not exactly self-similar;
rather, they are statistically self-similar.
- The length of a curvy object, such as a coastline, depends on the size of the ruler you use to measure it.
This famous application of fractals was posed as the question: "How Long is the Coastline
of Britain?". This question embodies the fact that the value obtained when measuring the length of a complicated shoreline, such as that of Britain, depends on the length of the "ruler" that is used. Indeed, as with the Koch curve, we can convince ourselves
that the length can be as long as we choose.
Alexandre Van de Sande, HOW
LONG IS THE COAST OF BRITAIN? STATISTICAL SELF-SIMILARITY AND FRACTIONAL DIMENSION (2004). Courtesy of Alexandre Van de Sande at wanderingabout.com
Benoit Mandlebrot saw that, if we view the coastline as a fractal,
we can start to make some sense of its measurement. The problem is that the curve does not repeat its exact shape at different scales, as the Koch curve does. Rather, statistical features repeat at different-length scales. This might include the number of
bays or peninsulas of a certain scale that one finds when measuring with a specific ruler.
might find that one quadrant of the entire curve contains three bays and four peninsulas of length one unit (here we're letting a unit equal the length of one quadrant). If we then look at one-eighth of the curve, our unit becomes smaller, and the larger bays
and peninsulas that showed up in the first view become more-or-less flat. New bays and peninsulas become evident, however, now that we have a more detailed view. We might find that the number of smaller bays and peninsulas (of length ) is similar to before—say, three bays and five peninsulas. So, although the exact shape is not the same at both scales, the number of significant features is about the same. This gives us the idea
that the coastline is approximately self-similar.
We can use these properties to find the dimension of our coastline, but
we need a new technique. The strategy we used previously to find the dimension of the Koch curve won't work in this case, because we do not have exact self-similarity, but, rather, only statistical self-similarity. To find out more about a method that might
work, let's look again at the Koch curve and use rulers of different sizes to measure its length.
Recall that the first time we tried this, we found that the length of the curve approaches infinity as we take closer and closer looks. This time, however,
instead of being concerned with the absolute length, we'll focus on how the length changes with the size of the ruler with which we choose to measure.
We start with a ruler of length one and find that the length of the curve is 4 units. Now, if we measure
with a ruler as long (what might be considered a "more sensitive" ruler), we find that the length is 16
× units. As we use smaller and smaller rulers, the following table begins to take shape:
Notice that nowhere so far are we concerned with finding copies that look exactly like the entire curve—we
care only about how the measured length of the curve changes with the ruler size. Hopefully, it is becoming apparent that this technique will work on curves that are not as uniform as the Koch curve. To find the relationship between these quantities, we can
plot them on a graph.
Notice that the scales with which we are dealing suggest that we should look at
a logarithm graph (log-log) of these data. This kind of plot is often useful when dealing with quantities (like these) that change exponentially. To make the log-log graph, we simply take the logarithm of all the quantities and re-plot the data.
Now, to find out how these two values are correlated, we can look at the slope of the best-fitting line. For simplicity's
sake, we'll just choose the start and end points:
Subtracting this from 1 yields log ,
which is the same expression for dimension that we obtained earlier by looking at self-similar copies.
So, to find the dimension of our original coastline, which will allow us to come up with some sort of meaningful measurement, we can take a set of
data that includes both the length of the ruler we use and the total length that we find. If we then plot the data on a log-log graph, we can find the relationship between the choice of ruler and the total length. This will generate a line (or we can choose
a line of best fit), the slope of which is related to the dimension of the coastline.
that the slope of this line is equivalent to 1 minus the dimension of the coastline—or, alternatively, the dimension of the curve is equal to 1 minus the slope of our line. With this knowledge of the approximate dimension, we can select a unit of an
appropriate size with which to make our measurements. This unit is not a length and not an area, but something in between—call it "larea" for now. Furthermore, it is specific to the coastline with which we are concerned, so it doesn't provide a means
of determining whether a certain coastline is "longer" than another. However, it does enable us to talk about the relative curviness of shorelines. For instance, we would expect a coastline with a fractal dimension close to 1 to be much more featureless than
a coastline whose dimension is closer to 2.
Statistical self-similarity abounds in nature. The surface of a dry landscape has the same features at many different scales. The branching of trees follows similar rules. One of Mandelbrot's great contributions
was seeing how fractals relate to the natural phenomena and rhythms of our world.
(The above explaination of Fractal Dimension was taken from "Mathematics Illuminated", on website: https://www.learner.org/courses/mathilluminated/units/5/textbook/06.php