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 (bkirk@icsd.k12.ny.us) 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.

]]>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

Wow.

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

]]>Also, read over the Chapter 4 Investigative Task, about Auto Safety. Your response to this task will be due on Monday the 23rd.

]]>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.

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.

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