# 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

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

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

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 10/8

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 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 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 9/20

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 Lesson Recap for 9/18

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:

• 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 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.

# Fractals & Chaos Recap for 9/16

After a brief discussion of the weekend’s reading, we started today sharing the images we created by coloring in the even elements (i.e., multiples of 2) of Pascal’s triangle, finding that it produces the same Sierpinski’s Triangle pattern we observed only a few days ago. Wow!

This sparked our curiosity about what other patterns can be found by coloring in other multiples in Pascal’s triangle, and we got into groups to examine the same patterns for 3, 4, 5, 6, and 9. Your homework is to finish these pictures.

We spent the rest of the period playing with FractaSketch,

# Fractals & Chaos Recap for 9/13

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.

# Fractals & Chaos Recap for 9/12

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!