Fractals & Chaos Recap for 10/16

One more chance to sign up for snacks for tomorrow’s Fractal Art Show!

In class today, we finished up the dimension classwork sheet and practiced how the segments of each fractal design that you measure to calculate its dimension can be used to recreate the fractal in FractaSketch (aka “stealing the template”). Neat!

We also have an important reading assignment: By Friday, October 18, please read Mandelbrot’s revolutionary paper that sparked the recognition of fractals and fractal geometry How Long is the Coast of Britain? You might also want to read this version, where Mandelbrot himself explains how he originally wrote this paper as a “Trojan Horse” to introduce his vision of fractal dimension into the scientific community conversation.

Fractals & Chaos Recap for 10/15

Please sign up for snacks for this week’s Fractal Art Show on Thursday, October 17.

In class today, we worked on our fourth and final dimension classwork sheet (while a handful of stragglers turned in their Fractal Art Show designs) and looked at how the analysis we do to find the dimension of the shapes can be used to recreate them in FractaSketch. Neat!

We also have an important reading assignment: By Friday, October 18, please read Mandelbrot’s revolutionary paper that sparked the recognition of fractals and fractal geometry How Long is the Coast of Britain? You might also want to read this version, where Mandelbrot himself explains how he originally wrote this paper as a “Trojan Horse” to introduce his vision of fractal dimension into the scientific community conversation.

Fractals & Chaos Recap for 10/9

We spent some more time in class working on practicing our Generalized Hausdorf Dimension formula on the second batch of fractals you got on Monday, then started in on a third batch. We’ll spend tomorrow working on our FractaSketch designs (remember that they are due on Friday!)

I would also like you to read Pollack’s Fractals, an article from Discover Magazine about the math underlying Jackson Pollack’s famous paintings.  For some interesting follow-up reading, check out this article from the New York Times about the use of fractal analysis to examine the authenticity of supposed Pollack paintings and this article from the Science Daily blog suggesting that such an analysis is not scientifically valid.

Fractals & Chaos Lesson Recap for 10/3

We continued to practice finding dimension for a variety of new fractal designs, finishing the first sheet and observing that there is still a bit of a problem with stems with this new definition of dimension. Tomorrow will be a FractaSketch lab day.

For Monday, please read this recent piece from The Atlantic (Why Fractals Are So Soothing)

Fractals & Chaos Recap for 10/1

We had a brief discussion on Stephen Jay Gould’s essay “Size and Shape,” seeing some common threads with prior reading assignments. For one last follow-up, I recommend this NPR piece from 2007 – Size Matters: The Hidden Mathematics of Life (and/or this followup to the  Kurzgesagt video from last week).

From there, we moved on to revisit the problems with the current Hausdorff Dimension formula that we identified yesterday, specifically about what to do with fractals that are self-symmetric, but at inconsistent scales. We developed a new, Generalized Hausdorff Dimension formula:

Here, S_1, S_2, …, S_N are all the differing ratios each iterating piece form with the original whole (note that here, each S is a fraction, so a piece that is 1/2 the length of the whole has S = 1/2 instead of S = 2 as it was in the previous definition)

We then practiced this definition with a few more examples, found on this sheet.

Fractals & Chaos Recap for 9/30

I decided to postpone our discussion of Gould’s essay, assigned on Friday, so that we could finish our discussion of the first dimension calculation practice sheet, which we did so with an observation that the S^d = N definition for dimension has some weaknesses. In particular:

• What do you do with fractals with non-iterating stems (e.g., 2 and 13)?
• What if the fractal is not exactly self-similar (like 5)?
• What if it is self-similar, but at differing scales (like 15)?

We will continue to work on our definition of dimension to accommodate these issues. You have also been issued a challenge: use what you’ve learned from the S^d=N definition to design a fractal with a dimension of exactly 1.5.

We spent the rest of the period working on our FractaSketch designs.

Fractals & Chaos Recap for 9/27

We started class with a writing/discussion prompt based on Haldane’s On Being the Right Size, a response to the meme “Would you rather fight one horse-sized duck or 100 duck-sized horses?” The horse-sized duck won the argument, as an application of the central takeaway from Haldane’s article is that artificially scaling a duck up to the size of a horse would increase its weight exponentially faster than its strength could accommodate, resulting in a duck incapable of walking around, let alone flight. We followed this up with a relevant Kurzgesagt (In a Nutshell) video: What Happens if we Throw and Elephant From a Skyscraper (as promised, the follow up — How to Make an Elephant Explode — can be found here).

In the second half of the period, we continued our discussion of the dimension of the various fractals found on our first dimension practice sheet. We found that one of the templates was a cleverly disguised version of the Sierpinski Carpet, and that another–Sierpinski’s Pyramid–has the interesting property of being simultaneously defined as having 1, 2, or 3 dimensions, depending on what type of dimension you use.

Monday, we go back to the computer lab to work on our fractal designs. By Tuesday, read Size and Shape, paleontologist and evolutionary biologist Stephen Jay Gould‘s follow up to Haldane’s essay. If Gould’s name sounds familiar, it’s because he co-developed the idea of (and coined the term for) punctuated equilibrium, a theory of evolution.

Fractals & Chaos Lesson Recap for 9/23

We wrapped up Thursday’s lesson with an explanation of the idea of the Hausdorff Dimension of a fractal. In brief, the Hausdorff Dimension is the solution to the equation S^d = N, where S is the scale by which a fractal is being broken up into pieces, and N is the number of such pieces. So for the Sierpinski Triangle, we cut the original triangle into pieces that are all half the length of the original (S = 2), but keep 3 of those pieces. Solving the equation 2^d = 3 gives d ≈ 1.585, which is the Hausdorff dimension of the Sierpinski Triangle. For the Koch Curve, we get the equation 3^d = 4, producing d ≈ 1.26, and for the Cantor Set, we get 3^d = 2, so d ≈ 0.631.

For homework, you’ve been asked to think about the dimension of the Dragon Curve, and have been reminded that the base template for the curve involves a right isosceles triangle, as well as to design a fractal called the Sierpinski Carpet (essentially the same design as the Sierpinski Triangle, but with a square as the starting shape) and to find the dimension of that.

Fractals & Chaos Recap for 9/20

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.

Fractals & Chaos Recap for 9/19

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.

Wow.

Of course, this raises an immediate question: If the dimension of the Sierpinski Triangle is somewhere between 1 and 2, what is it?

Fractals & Chaos Lesson Recap for 9/17

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…