Category Archives: Fractals & Chaos

Fractals & Chaos Recap for 10/18

If you have a compass (the circle-drawing kind) please bring it with you to class over the next few days.

We discussed Mandelbrot’s Article, then used it to segue into a further discussion of dimension. As we are aware, we still have problems with the Hausdorff dimension formula for calculating dimension of fractals. It can’t handle fractals with stems (i.e., non-iterating segments that never disappear) and with fractals that are not exactly self-symmetric.

Today, we considered a football field, a circle, and a Koch Curve, and looked at how the size of the measuring stick we use to measure the length or perimeter of such things has an impact on the total amount of length we actually calculate. For a football field, the size of the stick makes no difference. We’ll be obtaining 100 yards worth of length even if we use a foot (S = 3) or an inch (S = 36) as our step size.

For a circle, this isn’t the case. Use a measuring stick the length of the diameter, and we can only make two steps before we end where we started. Use a stick the size of the radius (S = 2) and we can make 6 such steps (resulting in a measure of three diameters). Use a half-radius (S = 4), and we wind up with a total length of slightly more than 3 diameters. There is a limit to this, of course: pi*d, which is precisely the formula for the circumference of a circle.

For the Koch Curve, the story is very different. Use a step size the length of the original baseline, and we can make one step. Use a step size of 1/3 the baseline (S = 3), we can make 4 steps, giving a length of 4/3 the base. Use a step size of 1/9 the baseline (S = 9), and we can make 16 steps, for a total length of 16/9 the base. As we shrink the length of the ruler we use, the number of steps increases more quickly, and so the total length increases without bound.

We’ve seen suggestions at this idea before. In the second article we read (The Diversity of Life), we saw that reducing the scale of our perspective dramatically increases the amount of living space we can find. This idea is also found at the center of the coastline paradox, hinted at in the Ants in Labyrinths article (see also this blog post from UK Urban Planner Alasdair Rae)

We will be expanding on this in class tomorrow, including a discussion on what all this has to do with the dimension of what we’re measuring.

Fractals and Chaos Lesson Recap for 10/17

We had our official Fractal Art Show Gallery Opening today!

For tomorrow, 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 10/16

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 10/11

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

Your FractaSketch designs were due today! You should be turning in one entry in three of the following categories:

  • Fern
  • Tree (or shrubs, bushes, weeds, etc.)
  • Spiral
  • Realistic (other natural phenomena)
  • Artistic (patterns, designs, etc.)

On the back of each of your submissions, you should write your name, the category, a notation on which way is up, a title (for the realistic and artistic categories) and a note about whether it is an “official” work, i.e., one of the three expected from all students, or an “additional” work that you would like to be considered in an “Additional Works” category.

In addition, for one of your official submissions, I also need one template. You’ll need to take a screenshot of the template, including its arrows, and print that as a separate page. Use the “snipping tool” in Windows, and leave the image “actual size” (don’t blow it up). My goal is to fit all templates on a single piece of paper for a matching game as a part of the art contest.

Do not print the .png images directly! Instead, I recommend importing your images into a single Google Docs file, one image per page, and printing that document. This allows you to resize your images more convenient and generally makes the resolution of the images less grainy.

Fractals & Chaos Recap for 10/10

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

Your FractaSketch designs are due tomorrow, Friday, October 11. Remember, I expect from each of you one entry in three of the following categories:

  • Fern
  • Tree (or shrubs, bushes, weeds, etc.)
  • Spiral
  • Realistic (other natural phenomena)
  • Artistic (patterns, designs, etc.)

On the back of each of your submissions, you should write your name, the category, a notation on which way is up, a title (for the realistic and artistic categories) and a note about whether it is an “official” work, i.e., one of the three expected from all students, or an “additional” work that you would like to be considered in an “Additional Works” category.

In addition, for one of your official submissions, I also need one template. You’ll need to take a screenshot of the template, including its arrows, and print that as a separate page. Use the “snipping tool” in Windows, and leave the image “actual size” (don’t blow it up). My goal is to fit all templates on a single piece of paper for a matching game as a part of the art contest.

Do not print the .png images directly! Instead, I recommend importing your images into a single Google Docs file, one image per page, and printing that document. This allows you to resize your images more convenient and generally makes the resolution of the images less grainy.

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 Recap for 10/8

We spent the day working on FractaSketch again. Your submissions for the Fractal Art Show are due this Friday, October 11. Remember, I expect from each of you one entry in three of the following categories:

  • Fern
  • Tree (or shrubs, bushes, weeds, etc.)
  • Spiral
  • Realistic (other natural phenomena)
  • Artistic (patterns, designs, etc.)

Again, each student will be submitting three entries, each falling in a separate category. You are welcome to submit more designs if you would like, but they will be placed in a separate “Additional Works” category.

In addition, I will want you to submit the template for one of those official submissions for a template/design matching challenge.

Please feel free to work on your designs outside of class and transfer them to the laptops we’ve been using in class.

Fractals & Chaos Lesson Recap for 10/7

We started class today with a quick visit to H Courtyard to observe and analyze the dimension of the fractals found there, then came back to class and practiced our Generalized Hausdorf Dimension formula on a new batch of fractals. We’ll spend tomorrow working on our FractaSketch designs (remember that they are due on Friday!)

Tonight, take a look at this article from Nature: Fractal secrets of Rorschach’s famed ink blots revealed

Fractals & Chaos Recap for 10/4

We spent the day working on FractaSketch. Your submissions for the Fractal Art Show are due next Friday, October 11. Remember, I expect from each of you one entry in three of the following categories:

  • Fern
  • Tree (or shrubs, bushes, weeds, etc.)
  • Spiral
  • Realistic (other natural phenomena)
  • Artistic (patterns, designs, etc.)

Again, each student will be submitting three entries, each falling in a separate category. You are welcome to submit more designs if you would like, but they will be placed in a separate “Additional Works” category.

In addition, I will want you to submit the template for one of those official submissions for a template/design matching challenge.

Please feel free to work on your designs outside of class and transfer them to the laptops we’ve been using in class.

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:

Annotation 2019-10-01 084135
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 Recap for 9/26

We discussed Ivars Peterson’s Ants in Labyrinths at the start of class, noting some interesting passages and talking about questions we had. In particular, I made a note to remember the part towards the beginning, where Peterson suggests an interesting problem with measuring a particular coastline:

Finer and finer scales reveal more and more detail and lead to longer and longer coastline lengths. On a world globe, the eastern coast of the United States looks like a fairly smooth line that stretches somewhere between 2000 and 3000 miles. The same coast on an atlas page showing only the United States […] seems more like 4000 or 5000 miles. […] A person walking along the coastline, staying within a step of the water’s edge, would have to scramble more than 15,000 miles to complete the trip.

This is an important idea. Remember it! We’ll be revisiting it later in the course.

The rest of our time in class was spent working on our fractal designs in FractaSketch. Don’t forget the expectations for the soon-to-be-announced Fractal Art Show! I expect from each of you one entry in three of the following categories:

  • Fern
  • Tree (or shrubs, bushes, weeds, etc.)
  • Spiral
  • Realistic (other natural phenomena)
  • Artistic (patterns, designs, etc.)

Again, each student will be submitting three entries, each falling in a separate category. You are welcome to submit more designs if you would like, but they will be placed in a separate “Additional Works” category.

Please feel free to work on your designs outside of class and transfer them to the laptops we’ve been using in class.

Homework: Read On Being the Right Size, an essay written by biologist JBS Haldane in 1926. We will discuss this reading tomorrow. As you read, ask yourself this classic question from the Internet: “Which would you rather fight: one horse-sized duck or 100 duck-sized horses?”

Fractals & Chaos Lesson Recap for 9/25

We spent today practicing finding the Hausdorff Dimension for a variety of fractal and template designs, running into some interesting potential problems as well as a few surprising results.

By tomorrow (Thursday), please read the passage Ants in Labyrinths, from Ivars Peterson’s The Mathematical Tourist. As usual, make a note of questions you have and passages you think are significant.

And don’t forget the expectations for the soon-to-be-announced Fractal Art Show! I expect from each of you one entry in three of the following categories:

  • Fern
  • Tree (or shrubs, bushes, weeds, etc.)
  • Spiral
  • Realistic (other natural phenomena)
  • Artistic (patterns, designs, etc.)

Again, each student will be submitting three entries, each falling in a separate category. You are welcome to submit more designs if you would like, but they will be placed in a separate “Additional Works” category.

Please feel free to work on your designs outside of class and transfer them to the laptops we’ve been using in class.

Fractals & Chaos Recap for 9/24

We practiced finding dimension with our new definition for a few additional fractals, including seeing a surprising result with the Dragon Curve, then spent the remaining time working on our fractal designs in lab.

For class on Thursday, please read the passage Ants in Labyrinths, from Ivars Peterson’s The Mathematical Tourist. As usual, make a note of questions you have and passages you think are significant.

Two videos from our conversation in class:

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 9/19

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?