Fractals are often viewed as self-similar shapes. That is, a geometrical shape that repeats itself regardless of how much an spectator zooms into the shape.
This is Von Koch’s snowflake, a fractal made out of four segments that repeat infinitely. Surprisingly, real-life snowflakes are fractals. And they are just one example of fractals in nature.
Although this is true for self-similar fractals, fractals are not always self-similar. In fact, Benoît Mandelbrot, the father of fractal geometry, invented fractals to prove that any real-life object, self-similar or not, can technically be a fractal, as long as its roughness is constant in all magnifications.
An example of constant roughness is Britain’s coastline. The coastline is not geometrical and self-similar, yet it has a constant roughness because, in theory, if an spectator zoomed into the coastline forever, there would be roughness at all levels. Therefore, it is a fractal.
This hypotheses was Mandelbrot’s rebellion against calculus, which assumes that objects tend to get smooth as they are magnified.
Although dividing the area under a curve into rectangles whose width approaches 0 provides a nearly perfect approximation of an integral, Mandelbrot’s point was that the integral is never perfect.
He therefore proposed that objects with constant roughness are fractals, regardless if they are self-similar or not.
But how did he determine the roughness of fractals?
Mandelbrot redefined them. Essentially, fractals have a constant roughness if their fractal dimension that exceeds its topological dimension.
Although this statement sounds eccentric and abstract, this article will break it down.
Let’s define a topological dimension: it is the connectivity of points in a given object, and it is equivalent to everyday dimensions.
In topology, a null set (symbolized by the symbol Ø) is an imaginary space that has a topological dimension of -1. An empty space has a topological dimension of 0. A line has a topological dimension of 1. A plane has a topological dimension of 2. And a volume has a topological dimension of 3. But since dimensions should measure the connectivity of points, how are these numbers determined?
Although it is intuitive that lines, planes, and volumes are labeled as one, two, and three dimensional, topology has a different way to label that intuition. Topological dimensions are “one greater than the dimension of an object that could be used to separate a part of the first space from the rest.”
Let’s think about what that means.
Since it is impossible to actually separate an empty space from itself, topology defines the null set as the only dimension that can do that. Hence, -1, the dimension of the object that can “separate” an empty set, +1, is 0, the dimension of an empty set. Erasing a line separates it, and adding an empty space (0) to the line, +1, is 1, the dimension of a line. Dividing a plane with a line separates it, and adding a line (1) to the plane, +1, is 2, the dimension of the plane. Finally, intersecting a volume with a plane splits it, and adding a plane (2) to the volume, +1, is 3, the dimension of a volume.
Essentially, topological dimensions are the everyday integers used to describe dimensions, but with a slight change that will be relevant to fractals at the end of this article. Fractal dimensions, on the other hand, are trickier.
Fractal dimensions are defined as the way an object fills the space in which it is immersed. Interestingly enough, fractals have a finite area and infinite perimeter, so measuring how it fills space conventionally is not possible.
Let’s take Koch’s Snowflake as an example:
In the snowflake, a triangle intersects another, dividing the shape into three, and this pattern repeats in all the shapes’ edges infinitely. The area accumulates, but it is expected to reach a limit because it is bounded inside a region.
In the first transformation, three triangles of side length ⅓ increase the area by A ⋅ 3 ⋅ (⅓)². In the second transformation, twelve triangles of side length (⅓)² increase the area by A ⋅ 12 ⋅ (⅓)4. In the third transformation, forty-eight triangles of side length (⅓)4 increase the area by A ⋅ 48 ⋅ (⅓)6.The total area is given in the following series:
At = A + (A ⋅ 3 ⋅ (⅓)²) + (A ⋅ 3 ⋅ 4 ⋅ (⅓)4 ) + (A ⋅ 3 ⋅ 4² ⋅ (⅓)6) + …
(equations may appear differently)
The accumulation of the areas creates a geometric series that states 8A/5 is the maximum possible area for a Koch’s Snowflake:
At = A (1 + 39)n0(49)n = A (1 + 13⋅ 11-49) = A (1 + (13⋅95)) = A (1 +35) = 8A5
However, since the length of the perimeter keeps increasing, it is therefore infinite, regardless of its finite area.
Hence, defining fractal dimensions and how they fill space empirically is only possible by measuring mass. And since Mandelbrot was looking for constant roughness, he computed how mass changes in all magnifications.
Let’s take normal shapes as a baseline for his method:
Decreasing a line’s side length by ½ would yield a line that is ½ the mass of the original, or (12)1 = ½. Decreasing a square’s side length by ½ would yield a square that is ¼ the mass of the original, or (½)² = ¼. Dividing a cube of iron into equal parts would yield eight cubes of ⅛ the mass of the original, or (½)3 = ⅛. The pattern is: the scaling factor is a fraction, the exponent is the dimension of the object, and the result is the new object’s mass.
But how does this relate to Von Koch’s Snowflake?
Mandelbrot decreased the snowflake’s side length by ⅓. Since the snowflake has four equal parts, then the magnification would yield a piece that is ¼ the mass of the original. But ⅓ to what power is ¼? In other words, what is the snowflake’s dimension? This is the same as asking log₃(4) ≈ 1.262. So in a sense, Von Koch’s snowflake is 1.262 dimensional.
Although the snowflake lives in a two-dimensional plane, the snowflake is made from a line that could be separated by adding an empty space, and therefore its topological dimension is 1, and not 2. And because its fractal dimension exceeds its topological dimension, Von Koch’s snowflake is a fractal with constant roughness. Fractal dimensions with decimals further away from an integer (like 0.5) have the most roughness, and fractal dimensions of a whole number have the least roughness.
Although all this math might seem meaningless, fractals are more relevant than people initially realize. Fractals are present in snowflakes, lighting bolts, sycamores, seashells, leaves, radio and phone antennas, and whole galaxies, among other things, so learning about them provides a new way of looking at the world.
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