Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Up until this point, we have been thinking of dimension as the number of independent measurements that are required to define a particular object in a particular space. We will now, through the application of mathematical concepts, 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 that would otherwise baffle us.
Let's return to the one-, two-, and three-dimensional worlds that we explored earlier. Recall 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'll explore how each object changes as we shrink or enlarge it by a constant factor.
First up, the line segment—let's look at a 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 segment. 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 units^{2}
Final area = 6 × 6 = 36 units^{2}
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 = 3^{2}. 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 = 3^{3}, 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
N = S^{D}
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
Dividing both sides by ln S, we get:
D =
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.
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:
Let's first 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 ×× = ()^{2} 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 unit^{2} 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.
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, we get:
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. As it turns out, this property can come in quite handy, as we shall see in the next section.
Next: 5.7 Fractal by Nature