Fractals & Chaos Recap for 10/16

We’ve finalized our fractal designs and everyone has submitted their works. We will have our Fractal Art Gala on Thursday, October 18. Please be sure to bring in some snacks to share!

For Wednesday, you should 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 10/15

Please sign up for snacks for next week’s Fractal Art Show

Your fractal art designs are due! By today, you should have turned in one submission in three of the following categories

  • Fern
  • Tree
  • Spiral
  • Realistic
  • Artistic

All works should have 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 try to make it small. My goal is to fit all templates on a single piece of paper.

Do not print the 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 appropriately and generally makes the resolution of the images less grainy.

Greetings IHS Parents/Caregivers!

Hello and welcome to my website!  As I mentioned during the open house, I use this site to communicate with students, parents, and whomever else might be interested outside of class.  On this site, you can find assignments, announcements, links to instructional videos and online review, copies of handouts I distribute in class, and the occasional mathematical musing.

Please take some time to check out this website.  If you’re looking for a course syllabus or other information, you can find it under Course Information above.  If you’re wondering about how to obtain some extra help for your student, I recommend checking the Contacting me and Getting Help link above.  For more general information about what you can find on this site, check out the welcome post from the beginning of the year.  Thanks for stopping by!

Fractals & Chaos Recap for 10/11

For Monday, read this article from Nature about the connection between fractals and the famous Rorschach inkblot test.

Please sign up for snacks for next week’s Fractal Art Show

A reminder that the due date for our Fractal artwork will be Monday, October 15th.

On Monday, you will turn in one submission in three of the following categories

  • Fern
  • Tree
  • Spiral
  • Realistic
  • Artistic

All works should have 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 will be collecting one template. You’ll need to take a screenshot of the template, including its arrows, and print that as a separate page.

I recommend importing your images into a single Google Docs file, one image per page, and printing that document. This seems to work well.

Fractals & Chaos Recap/Assignment for 10/9

Please sign up for snacks for next week’s Fractal Art Show

We have decided that the due date for our Fractal artwork will be Monday, October 15th.

On Tuesday, you will turn in one submission in three of the following categories

  • Fern
  • Tree
  • Spiral
  • Realistic
  • Artistic

All works should have 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, 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 will be collecting one template. You’ll need to take a screenshot of the template, including its arrows, and print that as a separate page.

Today also starts a series of more recent articles on how Fractals are being used in today’s research. To start this off, check out this article from The Atlantic about why fractals can be so visually soothing

Fractals & Chaos Recap/Assignment for 9/28

We finished our discussion of the first dimension calculation practice sheet with an observation that the S^d = N definition for dimension has some weaknesses, primarily:

  • 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. Think about this as you read Gould’s essay Size and Shape.

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

Fractals & Chaos Recap/Assignment for 9/27

We started class with a writing 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.

Tomorrow, we go back to the computer lab to work on our fractal designs. By Monday, 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/Assignment for 9/25

We discussed the article Ants in Labyrinths, then did some more practice with finding dimension of a variety of fractals. Some of the designs were templates, but some of them were “completed” fractals, which prompted us to discuss how to recreate the template from these designs. We have also already observed that the S^d = N definition for dimension may not be sufficient!

We’ll have some more time to work in the lab tomorrow, and for Thursday you should read On Being the Right Size, an essay written by biologist JBS Haldane in 1926.

Fractals & Chaos 9/21 Recap/Assignment

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.

This weekend, 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.

Fractals & Chaos 9/20 Recap/Assignment

We wrapped up yesterday’s lesson with an explanation of the formula for the Hausdorff Dimension of a fractal and a few examples of finding the dimension for the Sierpinski Triangle, Koch Curve, and a newly designed fractal called the Sierpinski Carpet (essentially the same design as the Sierpinski Triangle, but with a square as the starting shape). We spent the rest of the time in class working with FractaSketch.

Fractals & Chaos Recap/Assignment for 9/19

Today was an important day.

We started with a conversation about dimension, 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.

We ended class by deriving a formula that we could use to measure such dimension, called the Hausdorff Dimension, and given by the formula S^d = N, where S is the scale factor by which the fractal is shrunk when it is repeated and N is the number of times it is repeated at that scale.

For homework: Complete the Sierpinski Carpet drawing we started in class!

Fractals & Chaos Assignment for 9/19

We had a day working with FractaSketch today, which included an announcement of the 2018 Fractal Art Show.

The exact date of the art show will be determined later, but I will expect one entry from each student in three of falling into 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.

Please feel free to work on your designs outside of class. We’re still trying to get the PC mobile lab open, so until then we will continue to work with the MacBook lab.

For tomorrow: continue to think about what the term “dimension” really means, per yesterday’s conversation.

Fractals & Chaos Assignment for 9/17

We saw the Sierpinski Triangle pop up in a surprising place today, arising from bending a single line segment in the same manner as FractaSketch does. How can it be, then, that the same image can arise from an obviously 1D structure as it did from an obviously 2D structure, as we saw before? What is the dimension of the Sierpinski Triangle?

To help us answer this question, we need a clear idea on what we mean by dimension. When we say something is “two dimensional,” what does that mean, exactly? Think about this and try to have some ideas to share on Wednesday.