Mathematics and real life have not been very good friends throughout history. Even the Greeks, supposed geniuses of all things geometrical, were content to deal with undefined "points" and "planes," compensating for their apparent weirdness by making up things called "triangles" and forcing a guy named Pythagoras to come up with a theorem about the three-cornered pests which would supposedly make life better for everyone. It did, but only because it meant the mathematicians had something to talk about and were thus kept off the streets and out of trouble. Regardless, the division between a supposedly perfect, measureable triangle and the gritty, rough, and at times outrageously uncouth world was distinct from the start.
Math became a lot like philosophy, but without the fun stuff that had to do with talking about leaves or basketweaving. In lieu of that sort of nonsense, they substituted their own new brand of the same called arithmetic, which had a lot to do with those triangles and things like "log," "dy/dx," or "x," depending on the temperment of the mathematician at the time. Most mathematicians dealt in arithmatic, and geometry, which was considered by some people to be (gasp) slightly useful, was relegated to a role in a weird, perfect, whole-dimensional fantasy, where it would hopefully remain safe from inquiring minds.
Math was quite content in its dream world, preferring to split with all things Earthly by dealing with lines and objects and curves that didn't really exist until mathematicians said so. That was the way it was, and there was no reason to change. Normal people found problems with this, wondering how one could use math in real life situations where perfectly straight lines and absolutely accurate whole-number measurements were not easy to come by (or, in fact, impossible, but we're getting ahead of ourselves), but the mathematicians would merely laugh their cocky little laughs, turn their heads, and the normal person would walk away frustrated with mathematics, looking for a larger hammer (the traditional way of solving "real world situations").
This old math was centered around the idea of dimensions. Nobody except the really elite math geeks knew exactly what a dimension was, but it was generally accepted that figures either existed in 0 dimensions like points, 1 dimension like lines, 2 dimensions like squares and sections of planes, or 3 dimensions like cubes and spheres. Later, mathematicians would find out why these numbers had been selected and make it look like there was reason to it in the beginning, but there was no reason to argue about it because that was all things could be.
The entrenched math monarchy, however, had some problems in keeping normal people away from their odd creations. Early in this century, a normal guy disguised as a mathematician named Cantor decided to pull a little trick on mathematicians everywhere. He challenged them to imagine a thing constructed in a curious, recursive manner.
Cantor took a line segment of length x.
He decided then it would be fun to take out the middle third as his first step in building his little object.
Then, he divided the remaining segments by three as well, taking out the middle thirds of those.
After that was all done, he got tired. "Keep doing that forever," he said, and went to get some aspirin for the headache all the thinking had given him. The problem with infinite things is that we can't actually calculate each step, but theoretically the end result would be point-like things clustered around a few areas.
Mathematicians, not exactly brain surgeons if you catch my drift, were their cocky selves, not getting the point and laughing at poor Cantor. Then they were shocked as two startling facts became apparent:
Mathematicians did the wise thing and shut Cantor up as quickly as possible, denouncing his creation as a useless oddity.
A fellow named Sierpinski, not to be outdone, created his own little figure made from a solid, two-dimensional triangle
with the triangle formed by the midpoints of its sides taken out
and so on
and so on.
This was equally troublesome.
This figure, despite its pleasing look and the fun involved in approximating it to greater accuracy during a boring class, was also swept under the mathematical carpet. The world of zeroes, ones, twos, and threes was safe again.
Then, in 1924, came Benoit Mandelbrot, a Polish-born son of Lithuanian Jewish parents who turned the mathematical world on its head.
Mandelbrot's uncle Szolem liked to hang out with a bunch of mathematicians called the "bourbaki" in Paris. Mandelbrot's family sensed what would develop into grand-scale anti-Semitism and took him to France, leaving Mandelbrot with little to do but drink wine and eat cheese. He did not learn multiplication tables past fives or the alphabet, and maintains that he cannot use phone books to this very day.
His experiences with the bourbaki were worthwhile in that they led to his matriculation in elite Parisian institutions. In these schools, however, he astonished his instructors by not being able to show work for many of his solutions. His curious study habits were not the results of cheating (as far as we can tell) as much as they were the product of his mind: Mandelbrot solved traditional equations by turning them into pictures in his visually-oriented mind. On an entrance exam for a class, he passed with the highes score despite being able to do virtually no algebra.
His mind was clearly capable of mathematical thought on a high level, but it was a different type than the Old World bourbaki style. He soon realized he would have to conceal his true nature until after obtaining a degree, after which he fled to the United States.
He began work at an IBM research center and began to formulate his own definition of dimensionality, sparked mostly by the idea of self-similarity over an infinite scale, which had been inspired by the ideas of the philospher Liebniz and immortalized in the following verse by Jonathan Swift in 1733.
So, Nat'ralists observe, a Flea hath smaller Fleas that on him prey, And these have smaller Fleas to bit 'em, And so proceed ad infinitum.
The idea behind self-similarity as applied to dimensions was that one could take a little bit of a bigger mathematical figure that, when enlarged or duplicated, would exactly resemble the original figure. It is this relationship between enlargement and duplication that determines the dimensionality of an object.
Let us first consider the one-dimensional line of length x.
and let's divide x by two and see what sort of things we get.
If we took the line on the left and wanted to make a copy of the original object, we could enlarge it by a factor of 2 (2 * x/2 = x) or add 2 copies of it together (x/2 + x/2 = x).
With this conveniently left hanging for a second, let's look at the two dimensional square of side length x.
Let's divide this x by two and see what sort of things we get.
If we took one of the small boxes and wanted to build our original box again, we could enlarge it by a factor of 2 (x/2 * 2 = x) or add 4 copies of it together, letting the areas add up to that of the original square (4(x2/4) = x2).
Now let's look at a cube of side length x.
And divide its side length by two.
If we wanted to construct the original cube from a part of side length x/2, we could once again just double its side length (magnify it) by 2. If we wanted to add boxes together, however, we would need 8 of them.
Let's look at the relationship between enlargement and duplication in these three examples:
|square||2||4 (2 * 2)||2|
|cube||2||8 (2 * 2 * 2)|| 3|
Extension of this would reveal that enlargement by n results in duplication by n for a one-dimensional object, n2 for a two dimensional object, and n3 for a three-dimensional thing. Dimensionality looks like the power you raise the enlargement factor to in order to get the duplication factor, or as nerds know it, the logarithm with the base of the enlargement amount of the duplication factor. It's time for a little algebra to take over, letting d = the dimension, b = enlargement factor, and c = duplication amount. Looking at our data, it would look like logb c = d. We know how to deal with odd bases (the property used is: logb c = logn c / logn b, or the log of an odd base of a number is the log of the number in a more friendly base divided by the friendly log of the odd base), making the expression log c/log b = d. It would appear as if the dimensionality is the logarithm of the duplication amount divided by that of the enlargement factor. For whole-number-dimensional things, notice how the expression is easily simplified.
|line:||log (21)/log (2) = 1 log(2)/log(2) = 1 (= dimension 1)|
|square:||log (22)/log (2) = 2 log(2)/log(2) = 2 (= dimension 2)|
|cube:||log (23)/log (2) = 3 log(2)/log(2) = 3 (= dimension 3)|
For the above to make sense we must remember that the logarithm of anything raised to a power is the same as the logarithm of that thing times a coefficient numerically equal to the power, or logb an = n logb a.
With this new concept in mind, let's look at the Cantor dust again-- a Cantor dust created out of a line segment of length n, simplified to an extent we can easily deal with (it has only been "middle-thirded" once).
Let's take one of those dust segments of length n/3 (keep in mind its middle third would also be missing, and the middle thirds of what's left over, and so on in an actual representation of the dust). If we wanted it to look like the original, we'd multiply its length by a factor of three. If we wanted to get it like the length n dust through addition of n/3 dusts, we'd only have to add two of them together.
So our dimensionality? It cannot be simplified using our property of logs with exponentiation. It is merely log(2)/log(3), or about 0.631 -- a "fractional dimension" existing somewhere between 0 and 1, as befits something a little more than a point (dimension 0) but a little less than a line (dimension 1).
Our Sierpinski triangle's mysteries can be revealed in the same way.
As we can see, for each enlargement by two, we have to duplicate our thing by three. Its properties will be neither infinite nor zero in dimension log(3)/log(2) or dimension 1.585 (approximately).
Mandelbrot's study of this led to his proving that a fourth dimension exists-- a fourth dimensional composed of all the fractional ones in between.
The obvious question arises: how do infinite triangles or something that looks like a line passed through a Salad Shooter have anything more to do with real life than "perfect" objects, despite the obvious Salad Shooter metaphor?
The answer is in a somewhat uncommon idea: we are existing in these fractional dimensions (which are all under the fourth-dimension umbrella). "Real life" as we know it is in the fourth dimension, as we are fourth dimensional creatures, traveling at a constant rate and direction in the fourth-dimensional "space-time continuum." Time passes for us, and time passes for nature -- and nature has a lot more to do with infinite triangles then one might think at first.
Take the atlantic shoreline of the United States. On a globe, it could look anywhere from two to four thousand miles long, depending on how smooth the coastline was forced to be due to the low detail level of a world map. A smaller desk map of the Atlantic seaboard might show some of the larger bays, making the coastline much longer though the actual distance between the northern and southern boundaries of the coast remained the same. A human being walking along could not leave out most major inlets or curves where the water meets the sand, and would thus travel an even greater distance than indicated on the desk map. An ant would have to walk around every small tiny inlet as he could not walk over them, making many more turns and direction changes-- resulting in an even longer voyage (in terms of distance, though I'm sure it would take him a while longer as well). Once again, the distance between point a and b has not changed, but the level of detail has.
An electron traveling the same path would have to maneuver his way around land atoms on one side and sea atoms on the other making thousands of millions of tiny turns and shifts, and could not trace over the coast as easily as the ant had supposedly done tediously only a moment before. What about something that is an electron to an electron, a flea's flea as spoken of in the verse? Assuming that nature is infinitely detailed (which it is), the shoreline between point A and point B is infinite, though the area bounded by the water is finite. A shoreline exists in a fractional dimension.
There is no reason for dimensionality to be constant, either. A roll of tinfoil exhibits fractional behavior-- when a sheet is at first torn off, its dimensionality is a little above 2.0. If crumpled and then flattened out, its new complexity results in a dimensionality closer to 3.0. This is another property of dimensionality, defined as how "space-filling" an object is. If our Cantor dust fills .63 of a 1-dimensional plane, we can assume that something of a greater closeness to one would fill the space a little better, getting us closer to 1.0.
Interesting things result from fractional dimensions and self-similarity. Euclidean geometry, concerned mainly with perfect abstracts nonexistent in nature, had no way of describing items in our everyday lives.
Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
-- Benoit Mandelbrot
Many self-similar items (easily described and manipulated in fractional geometry) do exist in our world, however-- the figure of a mountain or ridgeline is often reflected in smaller formations of rock along its base. The bronchi of the human lung exhibit self-similarity over at least 15 levels. Ferns frequently exhibit self-similarity, as a branch may have fronds exactly resembling small versions of itself. Many Asian styles of art thrive on repetition of simple, self-similar shapes, often creating patterns of harmonious beauty. A tree, for example, can be minimalistically yet realistically implied by drawing with only small arcs of varying-radius circles. Peaks and bends in the path of a long river are echoed in the sand formations along a single bank. It is suggested that the secret to encoding so much information in DNA is in self-similarity.
Mandelbrot put this new infinite-dimensional consciousness to work. He found that seemingly random errors and failures in data-transmitting cable at IBM, a potentially dangerous problem, occurred in time according to the fractional dimension described by the Cantor dust. In a now-famous study of the fluctuations of cotton prices (the only commodity on which reliable records had been kept for years), he realized that seemingly random jumps and drops in prices followed a large pattern, which was then repeated over a greater scale. While patterns of this sort are common to statistics, their application to economics and their fractal-dimension nature were truly revolutionary, leaving economists speechless and baffling even Mandelbrot himself for quite some time.
Mandelbrot's adaptation of the dimension into a flexible entity has contributed a great wealth of information to the progress of mathematics as we know it. In the newer forms of math dealing with the fourth dimension and all the fractional ones around it, we see a way to describe the chaotic elements of nature and our universe.
Fractal geometry is not just a chapter of mathematics, but one that helps Everyman to see the same world differently.
-- Benoit Mandelbrot
This paper was written by a friend of mine in 1995. Since his current work is completely unrelated, he has asked to have his name removed.