"In fact, the term fractal was not coined until 1975 by Benoit Mandelbrot. With their introduction, we have a...."
That's right, Benoit Mandelbrot. Again, let me quote from what I wrote last year about Mandelbrot and fractals:
Part II of Benoit B. Mandelbrot's The Fractal Geometry of Nature, "Three Classic Fractals, Tamed," describes, as the title implies, three well-known fractal shapes. This part spans Chapters 5 through 8, and in Chapter 6, "Snowflakes and Other Koch Curves," Mandelbrot discusses self-similarity.
Ordinarily, when referring to self-similarity, one doesn't literally mean that T(F) = F for some particular dilation T. (Just as in the case where T is a translation, we can't have T(F) = F for a dilation T unless F is infinite -- except that F could be a single point, the center of the dilation.) Instead, we usually mean that T(F) is some subset of F -- that is, a part of the object looks just like a smaller version of the whole object.
Mandelbrot writes that there are two kinds of self-similarity: standard and fractal. An ordinary segment or square satisfies self-similarity, since a segment can be divided into several segments similar to itself, and likewise for the square. But notice that if the scale factor of the dilation is 2, we have two copies of the segment, but four copies of the original square. This brings to mind the Fundamental Theorem of Similarity -- found in Lesson 12-6 of the U of Chicago text:
If G ~ G' and k is the ratio of similitude, then:
(a) Perimeter(G') = k * Perimeter(G) or Perimeter(G') / Perimeter(G) = k;
(b) Area(G') = k^2 * Area(G) or Area(G') / Area(G) = k^2; and
(c) Volume(G') = k^3 * Volume(G) or Volume(G') / Volume(G) = k^3.
Of course, we can't help but notice that perimeter is one-dimensional, area is two-dimensional, and volume is three-dimensional. So we can combine all three statements into one, where "D-measure" denotes measure in D dimensions:
D-measure(G') = k^D * D-measure(G) or D-measure(G') / D-measure(G) = k^D
And if we denote the ratio D-measure(G') / D-measure(G) by G, then the equation becomes:
G = k^D
So now we have a way to calculate what Mandelbrot calls the "similarity dimension" of a fractal. We can take a fractal, perform any dilation of scale factor k, and then count how many copies G of the original fractal there are. Once we know G and k, we can solve the equation for the dimension D!
In Chapter 6, Mandelbrot describes a Koch snowflake curve. I could describe it for you, but as a picture is worth a thousand words, let me link to a website where this fractal is described:
http://mathworld.wolfram.com/KochSnowflake.html
Let's find the similarity dimension of the snowflake. We take one side of the snowflake and notice that to make it three times as long, we need four copies, since each iteration of its construction involves converting three segments into four. (The link above writes this as "F" -> "F+F--F+F," where the new "F" is one-third the length of the old "F.") So G = 4 and k = 3. This gives us:
G = k^D
4 = 3^D
In Algebra II, we learn how to solve this equation:
D = log_3(4)
that is, D is the base-3 logarithm of 4. Unfortunately, most calculators don't have a base-3 logarithm button, so we ordinarily use the Change of Base Theorem to write:
D = log(4)/log(3)
where "log" can be to any base -- usually we use base 10 or e so we can divide on a calculator. We find that the dimension is approximately 1.2619 -- Mandelbrot writes that he usually rounds values off to four decimal places.
What we have calculated is the similarity dimension of the Koch snowflake fractal. According to Mandelbrot, similarity dimension is not necessarily the same as Hausdorff dimension -- but for many of the fractals listed in the book, including the Koch snowflake, they are equal.
The other fractals mentioned in Part II are the Peano space-filling curve (a curve of topological dimension 1 and Hausdorff dimension 2) and the Cantor middle-thirds set (or Cantor dust, which has Hausdorff dimension between 0 and 1). Here are links to those fractals:
http://mathworld.wolfram.com/PeanoCurve.html
http://mathworld.wolfram.com/CantorSet.html
Ordinarily, when referring to self-similarity, one doesn't literally mean that T(F) = F for some particular dilation T. (Just as in the case where T is a translation, we can't have T(F) = F for a dilation T unless F is infinite -- except that F could be a single point, the center of the dilation.) Instead, we usually mean that T(F) is some subset of F -- that is, a part of the object looks just like a smaller version of the whole object.
Mandelbrot writes that there are two kinds of self-similarity: standard and fractal. An ordinary segment or square satisfies self-similarity, since a segment can be divided into several segments similar to itself, and likewise for the square. But notice that if the scale factor of the dilation is 2, we have two copies of the segment, but four copies of the original square. This brings to mind the Fundamental Theorem of Similarity -- found in Lesson 12-6 of the U of Chicago text:
If G ~ G' and k is the ratio of similitude, then:
(a) Perimeter(G') = k * Perimeter(G) or Perimeter(G') / Perimeter(G) = k;
(b) Area(G') = k^2 * Area(G) or Area(G') / Area(G) = k^2; and
(c) Volume(G') = k^3 * Volume(G) or Volume(G') / Volume(G) = k^3.
Of course, we can't help but notice that perimeter is one-dimensional, area is two-dimensional, and volume is three-dimensional. So we can combine all three statements into one, where "D-measure" denotes measure in D dimensions:
D-measure(G') = k^D * D-measure(G) or D-measure(G') / D-measure(G) = k^D
And if we denote the ratio D-measure(G') / D-measure(G) by G, then the equation becomes:
G = k^D
So now we have a way to calculate what Mandelbrot calls the "similarity dimension" of a fractal. We can take a fractal, perform any dilation of scale factor k, and then count how many copies G of the original fractal there are. Once we know G and k, we can solve the equation for the dimension D!
In Chapter 6, Mandelbrot describes a Koch snowflake curve. I could describe it for you, but as a picture is worth a thousand words, let me link to a website where this fractal is described:
http://mathworld.wolfram.com/KochSnowflake.html
Let's find the similarity dimension of the snowflake. We take one side of the snowflake and notice that to make it three times as long, we need four copies, since each iteration of its construction involves converting three segments into four. (The link above writes this as "F" -> "F+F--F+F," where the new "F" is one-third the length of the old "F.") So G = 4 and k = 3. This gives us:
G = k^D
4 = 3^D
In Algebra II, we learn how to solve this equation:
D = log_3(4)
that is, D is the base-3 logarithm of 4. Unfortunately, most calculators don't have a base-3 logarithm button, so we ordinarily use the Change of Base Theorem to write:
D = log(4)/log(3)
where "log" can be to any base -- usually we use base 10 or e so we can divide on a calculator. We find that the dimension is approximately 1.2619 -- Mandelbrot writes that he usually rounds values off to four decimal places.
What we have calculated is the similarity dimension of the Koch snowflake fractal. According to Mandelbrot, similarity dimension is not necessarily the same as Hausdorff dimension -- but for many of the fractals listed in the book, including the Koch snowflake, they are equal.
The other fractals mentioned in Part II are the Peano space-filling curve (a curve of topological dimension 1 and Hausdorff dimension 2) and the Cantor middle-thirds set (or Cantor dust, which has Hausdorff dimension between 0 and 1). Here are links to those fractals:
http://mathworld.wolfram.com/PeanoCurve.html
http://mathworld.wolfram.com/CantorSet.html
Returning to 2017, on page 131 Pappas mentions the Koch snowflake curve as well as a higher dimensional equivalent -- one with a dimension between 2 and 3.
Lesson 15-2 of the U of Chicago text is called "Regular Polygons and Schedules." Just like Lesson 13-3 from last month, 15-2 naturally lends itself to an activity, so this will be considered the activity for this week.
The text defines a round-robin tournament as a tournament in which each competitor (or team) plays each other competitor exactly once. True round-robins are rare at the highest levels of sports. The first two rounds of the World Baseball Classic and first round of the World Cup (of soccer) divide the field into groups of four nations each, and each group is played as a round-robin tournament. After the first two baseball rounds or first soccer round, the tournaments become single-elimination instead.
On the other hand, round-robin regular seasons are fairly common in high school sports. During football season, every team in the league plays every other team. It might be easy to schedule the four-team round-robins by hand, but the school from which I graduated is in a seven-team league. So the U of Chicago text provides a method of complete a seven-team round-robin:
1. Let the 7 teams be vertices of an inscribed regular 7-gon (heptagon).
2. Draw a chord and all chords parallel to it. Because the polygon has an odd number of sides, no two chords have the same length. This is the first week's schedule.
3. Rotate the chords 1/7 of a revolution. This is the second week's schedule.
4. Continue rotating 1/7 of a revolution for each week. Because in a week no two chords have the same length, no pairing repeats. In a total of seven weeks, the schedule is complete.
Here is the resulting schedule as given by the U of Chicago text:
1st week: 7-2, 6-3, 5-4, 1 bye
2nd week: 1-3, 7-4, 6-5, 2 bye
3rd week: 2-4, 1-5, 7-6, 3 bye
4th week: 3-5, 2-6, 1-7, 4 bye
5th week: 4-6, 3-7, 2-1, 5 bye
6th week: 5-7, 4-1, 3-2, 6 bye
7th week: 6-1, 5-2, 4-3, 7 bye
In my alma mater's league, from one year to another, the home games become away games and vice versa, and then the schedule itself rotates so that the first week match-ups are contested the second week, the second week match-ups are contested the third week, and vice versa. Many of the stronger teams don't actually take the bye week -- instead they play special games against out-of-state teams.
If there are an even number of teams, then the extra team is placed at the center, and a radius is drawn in addition to the chords -- this replaces the bye week. For example, we can add an eighth team to the list above by having the eighth team play the team with the bye.
Here is the activity. Students should begin by cutting out the circle, drawing the parallel chords, and leaving the numbers where they are, so that the circle can be rotated to match up with the numbers.
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