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.
Read the What If? and What Can Go Wrong sections from Chapter 4 (pages 93-95).
Over the weekend, write your response for the Chapter 4 Investigative Task (Auto Safety). As before, you may use your textbook and notes to help you write your response, and you may use the Internet to access Google Docs and Stapplet or whatever other graph-making tools you want to use, but you may use no other resources other than those. Do not go online looking for more information, and definitely no working with other students on this assignment.
Your response should be completed in a Google Document and submitted to me (email@example.com) by the start of class on Monday, the 23rd. Please name your file appropriately; it should have the format “LastName.FirstName.Ch4InvTask”. For example, mine would be “Kirk.Benjamin.Ch4InvTask“.
If you would like to rewatch or explore the data of the documentary from in class, you can find it here: The Fallen of WWII. Try the interactive version to explore the data more thoroughly.
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?
Start reading some of chapter 4, pages 83-90, then do exercises 20, 26, and 27.
Also, read over the Chapter 4 Investigative Task, about Auto Safety. Your response to this task will be due on Monday the 23rd.
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.
We will again be referring back to the Student Survey Data in class today.
Tonight, finish reading Chapter 3 (pages 60-71). From the exercises on pages 74-78, do 11, 21, 23, 25, 37. Question 21 has you calculating the standard deviation by hand. I recommend doing a few by hand just to get a feel for the process, but don’t spend the time doing them all.
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 had a quiz in class today, and your homework tonight is a pair of puzzle sheets (one about a Donkey, the other about Hawaii). You must do one side for a homework credit, and you can choose which one you do. But if you do both, we’ll give you an extra credit point on your quiz.
Continue reading chapter 3, pages 53-60 (up to “What about Spread? The Standard Deviation”)
From the exercises on pages 74-78, do 14, 18, 36, and 41
Be sure to bring in your Chromebooks tomorrow, fully charged!
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,
In class today, we will be referring back to the Student Survey Data from last week.
Tonight, please read pages 43-53 in your textbook (stop at the section “Spread: Home on the Range”). From the exercises, do 5, 9, 43, 44, 47
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.
Over the weekend, write your response for the Chapter 2 Investigative Task (Race and the Death Penalty). As noted in the document, you may use your textbook and notes to help you write your response, and you may use the Internet to access Google Docs and Stapplet or whatever other graph-making tools you want to use, but you may use no other resources other than those. Do not go online looking for more information, and definitely no working with other students on this assignment.
Your response should be completed in a Google Document and submitted to me (firstname.lastname@example.org) by the start of class on Monday, the 16th. Please name your file appropriately; it should have the format “LastName.FirstName.Ch2InvTask”. For example, mine would be “Kirk.Benjamin.Ch2InvTask“.
Because some folks have asked, you can find instructions on taking screenshots using a Chromebook here. I recommend the Ctrl+Shift+Window Switch shortcut to take a screenshot of only the particular items you want to use.