Due to how many students were absent from class today, I decided to delay the derivation of how to find the dimension of the Sierpinski Triangle to Monday. Instead, we spent the full period working with FractaSketch.
Today was an important day.
We continued a thought started yesterday, acknowledging the difference between intrinsic dimension—the dimension of an object itself—and extrinsic dimension—the dimension of the space the object occupies. For example, the edge of a circle is one dimensional, since there is only one axis of movement around the circle, but the circle exists in two dimensional space. We further observed that the extrinsic/containing dimension of an object must necessarily be greater than or equal to the intrinsic dimension of the space.
We also discussed how the dimension of an object affects the dimension with which we can measure it. We cannot find the volume of a square, and we can’t find the area of a cube. Put another way, if we tried to use a cubic centimeter measuring device to measure the area of a circle, we’d end with a measure of zero. And if we tried to use a square centimeter measuring device to measure the volume of a cube, we’d get a measure of infinity. In general, too small a dimension of measure results in a measure of infinity, and too large a dimension results in zero.
Finally, we considered the implications of this to the Sierpinski Triangle. When we attempted to measure the area of the Sierpinski triangle, we resulted in a measure of zero (an infinity of triangles each with area zero results in a total area of zero). When we attempted to measure its length, we resulted in a measure of infinity (an infinity of segments each with length zero results in a total length of infinity). Thus, 2 is too high a dimension to describe the Sierpinski Triangle and 1 is too low. It must therefore mean that the dimension of the Sierpinski Triangle is strictly between 1 and 2, and is therefore fractional.
Of course, this raises an immediate question: If the dimension of the Sierpinski Triangle is somewhere between 1 and 2, what is it?
We had a short conversation about the nature of dimension in class today, after yesterday‘s wild results. In a quest to identify “true” two-dimensional entities in our world (since the conventional example of a piece of paper still does have some thickness), we observed that there may be a difference between “intrinsic” dimension, that is a characteristic of an object versus the “extrinsic” dimension of the space it occupies. For example, a desk is clearly a three dimensional object, but the surface of the desk could be thought of as 2D and the desk’s height could be considered 1D.
We resolved to think on this some more for a further conversation tomorrow, and transitioned to working some more with FractaSketch. I demonstrated how to make a fern using the program and made available some basic templates that you could use as inspiration for the upcoming 2019 Fractal Art Show.
The exact date of the art show will be determined later, but I will expect from each of you one entry in three of the following categories:
- Tree (or shrubs, bushes, weeds, etc.)
- Realistic (other natural phenomena)
- Artistic (patterns, designs, etc.)
Again, each student will be submitting three entries, each falling in a separate category.
Please feel free to work on your designs outside of class and transfer them to the laptops we’ve been using in class.
For tomorrow: continue to think about what the term “dimension” really means, per yesterday’s conversation.
Today something weird happened.
We’ve already seen the Sierpinski Triangle appear in two ways, by iterating the process of breaking a triangle into four equal pieces and removing the central one, and by coloring the even elements of Pascal’s triangle. Today we found a third way, by Iterating a Curve.
By starting with a solid, filled in triangle, and step-by-step removing stuff from that area, we reach the same exact figure as we get by starting with a simple line segment and lengthening/bending it. The area clearly starts as 2-dimensional, the line segment clearly starts as 1-dimensional. Yet both processes have the same end result. What does this mean for the dimension of Sierpinski’s Triangle? Is it 2D or 1D? It certainly can’t be both, so the only option is… neither?
When I first took this course (back in the days of Mr. Drix), this was the first moment where I realized all of this talk about fractional dimension may be more than nonsense. Maybe there’s something to it after all…
Your homework: think critically about what dimension really means. What does it mean to say something is 1-dimensional or 2-dimensional. We’ll dig into this in the next few days…
We looked briefly at Pascal’s Triangle today, and some of the neat patterns that can be found there. I hinted at some hidden fractals that could be found by removing numbers from the triangle, so your homework is to fill in circles in this smaller version that would represent removing every even number from the triangle (remember, we observed that two filled in circles create a filled in one, two empty circles create a filled in one, and an empty and filled circle create an empty one).
We wrapped up class by playing with FractaSketch some more (linked at left). Before everybody left, I also handed out the next assigned reading for the course: this Science News article from 1997 (Fractal past, Fractal future) and this supplementary article from a 1997 issue of Popular Science about the Heartsongs album mentioned in the first one.
We finished our discussion of the Sierpinski triangle, noting that just like the “final” version of the Koch Curve is “nothing but angles,” this geometric oddity is “nothing but edges,” as the area of the triangle converges to zero as the iterations continue. This conclusion also presented an interesting contradiction. For the Koch curve, we argued that an infinite number of segments, each of length zero, resulted in an infinite perimeter (effectively, ∞ * 0 = ∞). Here, we have an infinity of triangles, each again with an area of zero, resulting in an area of zero (effectively, ∞ * 0 = 0).
What this reveals is that the expression “∞ * 0” is what is called an Indeterminate Form, an expression the defies definition. We can create a reasonable argument that defines it as infinity, and we can create a just-as-reasonable argument that defines it as zero. Therefore, it must be defintionless.
We finished the day by opening up the PC laptop mobile lab and downloading FractaSketch to each device. We’ll be using this software extensively over the next few weeks!
We finished our discussion about the Cantor Set, noting that infinite set of endpoints that are left over with each segment removal are countably infinite. If the claim is that the cantor set is actually uncountable, that requires there to be other elements of the set that are not segment endpoints. And there are such points, uncountably infinitely many of them. See this post for more information on how this works.
We went on to draw a picture of the Koch Snowflake, a figure devised by Swedish mathematician Helge von Koch as an example of a continuous curve with no tangents. As the fractalization process continues, the number of segments that make its perimeter increases without bound, but the length of each segment shrinks to zero. Mathematically, however, the total perimeter also increases, resulting in a figure with finite area and infinite perimeter!
We ended the period by starting to draw an image of another fractal, called the Sierpinski Triangle. Start with an equilateral triangle (connect the dots in the worksheet), then bisect and connect all three sides. Fill in that middle triangle, effectively removing it and leaving you with three triangles at the three corners. Then do that process again for each of the three triangles: bisect the sides, connect them, and fill in the middle triangle. Do that as many steps as you can fit. Don’t cheat and look up what the final result looks like!
We continued our study of the Cantor set by spending some time thinking about its properties, in particular how it has a length of zero yet still has an (uncountable) infinity of points contained inside.
That the length is zero is fairly easy to see from the fact that we remove 1/3 of the set in the first step, 2/9 in the second, 4/27 in the third, 8/81 in the fourth, and so on. That sum 1/3 + 2/9 + 4/27 + 8/81 + … forms an infinite geometric series, the sum of which is 1. And since the length of the original segment is also 1, the length of the “final” version of the Cantor Set is 0.
Yet it clearly contains an infinity of points! With each stage, we create endpoints of segments that never get removed, and an infinite number of stages produces an infinite number of endpoints. But not only that, I claimed the Cantor set is uncountably infinite, which required some explanation of the realization that some infinities are bigger than other infinities.
We added to our list of fractals and discussed some observations from Wilson’s excerpt. We observed that dimension is tied to measurement, with lengths being one-dimensional, areas being two-dimensional, and volumes being three-dimensional. This suggests that any notion of fractional dimension must correspond to measurements that don’t fit within those three categories.
Furthermore, the article illustrated an interesting point (one that we will come back to soon): shrinking our scale of measurement increases the quantity of measurement we can find, disproportionate to the shrinking scale. The amount of space found at progressively smaller scales increases rapidly, and with it biodiversity.
After a brief distraction with a mouse, we introduced our first mathematical fractal: the Cantor Ternary Set, developed in 1883 by German mathematician Georg Cantor to challenge contemporary ideas of infinity and length. In particular, the set has three seemingly contradictory properties:
- Every member of the set is a limit point
- It has an uncountably infinite number of points
- Yet its Length is zero
We will explore these ideas in the next few days.
We finished our discussion of the ideas inspired by the Jurassic Park excerpt, including looking at a few theories of using fractals to predict financial markets (see the silver and bitcoin articles here if you’d like to read them more closely).
From there, we discussed last night’s assigned reading and used it to form some properties about fractals. In particular:
- They demonstrate self-symmetry or self-similarity (each part could be viewed as a scaled-down version of the whole thing)
- They are Non-Euclidean (for a Euclidean curve, no matter how wiggly it is, zooming in far enough eventually makes it look linear, but for a fractal zooming in just reveals the same level of detail).
- They have fractional dimension (unlike one-dimensional lines, two-dimensional squares, or three-dimensional cubes, fractals live in a space between and could have a non-integer dimension)
This last idea is, of course, pretty wild, and if you feel skeptical about it, you should. Hold on to that skepticism! Let me convince you.
We finished the day making a brief list of ideas of fractals, including snowflakes, ferns, feathers, trees, and river deltas. Your homework is twofold:
- Continue to think of examples of fractals in the world around you, and
- Read over the excerpt from Edward O Wilson’s book The Diversity of Life: Living Labyrinths. As before, make a note of 2-3 passages that seems significant or questions you have.
We started out today with a discussion about course expectations. This is a pass/fail course but I still expect you to take it seriously. I will often ask you to read an article or do some math work at home, and I expect that work will be done the next day and ready to be discussed. I expect everyone to actively participate in class discussions and engage in the work we do during class time. This is not a study hall, so please don’t bring other work to do during this class or I will ask you to leave.
After a discussion of some of the pre-existing notions on what fractals are, we read an excerpt from Michael Crichton’s Jurassic Park and discussed some of the big ideas to be found there (you can see a clip of this scene from Steven Spielberg’s 1993 film version here).
For your first reading assignment, I’m asking you to read the article distributed in class: Fractals: Magical Fun or Revolutionary Science, from the March 21, 1987 issue of Science News. Take some notes as you read, jotting down the 2 or 3 major points of the article. Pay particular attention to how this article defines the term “fractal”.