"Fractals have come to be known as the geometry of nature. Though there are abundant examples of objects from...."
Hold on a minute! This section is "A Closer Look at Fractals and Nature." Well, last year I spend several blog posts discusses fractals. I might as well cut and paste some of what I wrote last year, since this comes directly from the father of fractals, Benoit Mandelbrot:
Part I of Mandelbrot's The Fractal Geometry of Nature is simply an Introduction. Mandelbrot begins the first chapter, "Theme," as follows:
Why is geometry often described as "cold" and "dry?" One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
Mandelbrot writes that the shapes of many real-life objects are better described as fractals. Indeed, it was Mandelbrot himself who first came up with the name "fractal":
I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means "to break:" to create irregular fragments. It is therefore sensible -- and how appropriate for our needs!-- that, in addition to "fragmented" (as in fraction or refraction) fractus should also mean "irregular," both meanings being preserved in "fragment."
The concept of fractal is intimately tied up with the concept of dimension. Indeed, Mandelbrot ultimately defines fractal in terms of its dimension in Chapter 3:
A fractal is by definition a set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension.
I believe that Mandelbrot will define Hausdorff dimension later in the text. But right now, let me define topological dimension. Let's recall what topology is -- as Mandelbrot writes, "Topology ... considers that all pots with two handles are of the same form ...," or as is more popularly stated, it considers a doughnut to be of the same form as a coffee cup.
Now both geometry and topology agree that a point has dimension 0, a line has dimension 1, and a plane has dimension 2. (Yes, this is the Dimension Assumption, part of the Point-Line-Plane Postulate in Lesson 1-7 of the U of Chicago text.) The difference is that the geometrical dimension of a circle is 2, while its topological dimension is 1. This is because even though a circle fits on a plane, on the circle itself one can only travel one way or the other, just like on a line. Likewise, a sphere may have three dimensions in geometry, but it's a two-dimensional manifold in topology.
Throughout Part I of the book, Mandelbrot provides many pictures of fractals. One picture shows us two artificial fractal "flakes." One of these has Hausdorff dimension 5/2, the other dimension 8/3. As we can see, these lie between two and three dimensions. These computer-generated fractals remind me of a recent episode of Futurama. In "2-D Blacktop," the main characters suddenly find themselves smashed into two dimensions. Fortunately, the Professor has previously installed "Dimensional Drift" on his spaceship, which allows the crew to return to the third dimension. On the way from dimension 2 to dimension 3, several fractals appear in the background. These fractals, which look just like the fractal flakes in Part I of the book, have dimension between 2 and 3. Recall that many of the writers on Futurama were professional mathematicians, and they made sure that the dimensions of the fractals shown on the episode gradually increased from 2 to 3 as the crew made its slow return back to its home dimension.
Back to 2017, we now reach the final chapter of the U of Chicago text, "Further Work with Circles." I didn't cover this chapter in great detail the first two years of this blog, since by the time we reached it we were pressed up against the PARCC and SBAC tests. And besides, the only section that's really tested on PARCC is Lesson 15-3, the Inscribed Angle Theorem.
This year, I follow the digit pattern and so we cover Chapter 15 on Days 151-159. But for those who start PARCC next week and need to complete Lesson 15-3, we'll reach it on Friday. (Recall that my own class is testing SBAC this week.)
Lesson 15-1 of the U of Chicago text is on Chord Length and Arc Measure. The key theorem of this lesson is:
Chord-Center Theorem:
a. The line containing the center of the circle perpendicular to a chord bisects the chord.
b. The line containing the center of the circle and the midpoint of a chord bisects the central angle determined by the chord.
c. The bisector of the central angle of a chord is perpendicular to the chord and bisects the chord.
d. The perpendicular bisector of a chord of a circle contains the center of the circle.
Proof:
Each part is only a restatement of a property of isosceles triangles.
a. This says the altitude to the base is also a median.
b. This says the median to the base is also an angle bisector.
c. This says the angle bisector is also an altitude and a median.
d. This says the median to the base is also an altitude. QED
Why didn't the text just say "diameter" instead of "the line containing the center of a circle"? I assume it's because a diameter is a segment, but bisectors are lines. Here is the other key theorem:
Arc-Chord Congruence Theorem:
In a circle or in congruent circles:
a. If two arcs have the same measure, they are congruent and their chords are congruent.
b. If two chords have the same length, their minor arcs have the same measure.
The U of Chicago text points out that we can't use the terms "are congruent" and "have the same measure" interchangeably. Two angles are congruent if and only if they have the same measure, but two arcs with the same measure aren't necessary congruent. A 50-degree arc of a tiny circle is nowhere near congruent to a 50-degree arc of a large circle. The theorem tells us that an additional condition is needed -- the circles must be congruent also.
But what does it mean for two circles to be congruent? The U of Chicago text proves that two circles are congruent if and only if their radii are equal. Recall that in Common Core Geometry, we can only show two figures congruent by showing that some isometry maps one to the other:
Lemma:
Two circles are congruent if and only if they have equal radii.
Proof of Lemma:
If two circles X and Y have equal radii, then one can be mapped onto the other by the translation mapping X to Y. So they are congruent. Of course, if they do not have equal radii, since isometries preserve distance, no isometry will map one to the other. QED
Proof of Part a of Arc-Chord Congruence:
In circle O, you can rotate Arc AB about O by the measure of Angle AOC to the position of CD. Then the chord
Part b is left in the text as an exercise. A hint is given -- the measure of an arc equals the measure of its central angle. This suggests that we could use a traditional two-column proof via SSS:
Given: AB = CD in Circle O
Prove: measure Arc AB = measure Arc CD
Proof:
1. AB = CD 1. Given
2. AO = CO, BO = DO 2. All radii of a circle are congruent.
3. Triangle AOB = COD 3. SSS Congruence Theorem
4. Angle AOB = COD 4. CPCTC
5. Arc AB = CD 5. Definition of arc measure
The text warns us that in circles with different radii, arcs of the same measure are not congruent -- they are similar. This isn't proved in the text, but notice that one of the Common Core Standards directs students to "prove that all circles are similar." So let's do so right here:
Theorem:
All circles are similar.
Proof:
If the two circles are concentric, then their common center is also the center of a dilation, with the scale factor obviously R/r, with r the smaller radius and R the larger radius. If the two circles have different centers, then we can translate one of the circles so that its center matches the other, then perform the dilation. QED
Actually, a single dilation will work if the centers aren't the same, but it's difficult to locate the center of this dilation, so it's easier just to translate first.
Here is my first newly created worksheet for Chapter 15:
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