Lesson 4-7 of the U of Chicago text is on Reflection-Symmetric Figures. Symmetry is one of the more important Common Core topics, as there is a question on the PARCC Practice Exam about the idea that a transformation can map a polygon to itself.
OK, I've decided what I will do about Lesson 4-6 being skipped. Today I will post both sides of the 4-6 worksheet, but only the second side of last year's 4-7 worksheet -- for two reasons. The first is that I want to look again at my proof of the Converse of the Perpendicular Bisector Theorem. Recall that earlier this week, I said how the Houghton Mifflin Harcourt Integrated Math I text gave a proof of the Perpendicular Bisector Converse that required the Pythagorean Theorem. As it turns out, the converse does have a neutral proof. Indeed, I mentioned in last year's post that it should follow from the Isosceles Triangle Symmetry Theorem of Lesson 5-1 -- since after all if P is equidistant from A and B (and P isn't the midpoint, which leads to a triviality), then PAB is an isosceles triangle. So I'll delay the Perpendicular Bisector Converse to Lesson 5-1.
(Speaking of neutral proofs, notice that a non-neutral idea slipped into yesterday's lesson. One of the questions involves a translation, but translations are highly dependent on parallel lines. Translations do exist in neutral geometry, but they look so different from pure Euclidean translations. But the fact that the point (x, y) rotated 180 degrees about the origin is (-x, -y) holds in neutral geometry. As soon as we teach what rotations are, we can show that the composite of an x-axis reflection and a y-axis reflection really is the 180-degree rotation about the origin, and so the proof follows directly from the theorems about such reflections that I posted last week.)
The second reason is that last year's 4-7 worksheet didn't come out to well. Indeed, no proof of the Perpendicular Bisector Converse is visible on that page. And so this works out well -- by omitting that proof, 4-6 and 4-7 can be condensed into a manageable lesson about the reflections of polygons.
This is what I wrote last year about today's lesson -- dropping the discussion about proofs that I'm no longer including in this lesson:
Lesson 4-7 of the U of Chicago text deals with reflection-symmetric figures. A definition is in order:
A plane figure F is a reflection-symmetric figure if and only if there is a line m such that r(F)=F. The line m is a symmetry line for the figure.
In other words, it's what one usually means when one uses the word "symmetry." Some geometry texts use the term "line-symmetric" instead of "reflection-symmetric." Some geometry and algebra texts use the term "axis of symmetry" instead of "symmetry line" -- especially Algebra I texts referring to the axis of symmetry of a parabola. Some biology texts use the term "bilateral symmetry" instead of "reflection (or line) symmetry" - in particular, when referring to symmetry in animals. As animals are three-dimensional, instead of a symmetry line there's a sagittal plane.
Indeed, it is this last topic that makes symmetry most relevant and interesting. Most animals -- including humans -- have bilateral symmetry. I once read of a teacher who came up with an activity where the students look for the most symmetrical human face. The teacher blogged about how students who are normally indifferent to geometry suddenly came fascinated and engaged to learn about the relationship between symmetry and human beauty. Unfortunately, this was over two years ago, and I can't remember or find what teacher did this activity -- otherwise I'd be posting a link to that teacher's blog right here!
(And unfortunately no, I still haven't found a link to the symmetry beauty lesson as of this year.)
In the Common Core Standards, symmetry is first introduced as a fourth grade topic:
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
Later on, symmetry appears in the high school geometry standards:
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Notice that if a reflection over a line carries a polygon to itself, then that line is a symmetry line. But symmetry lines for polygons formally appears in Chapter 5 of the U of Chicago text. Right here in Chapter 4, we only cover symmetry lines for simpler shapes -- segments and angles. The text reads:
"In the next chapter, certain polygons are examined for symmetry. All of their symmetries can be traced back to symmetries of angles or segments."
In this lesson, we found symmetry lines for simple figures such as segments and angles. But can we find symmetry lines for the simplest figures? As it turns out, a point has infinitely many lines of symmetry -- any line passing through the point is a symmetry line. But a ray has only one line of symmetry -- the line containing the ray.
Finally, does a line have a line of symmetry? This is exactly the answer to Question 25 of this lesson, in the Exploration/Bonus Section. A line -- considered as a straight angle -- contains more than one symmetry line. This is because any point on the line can be taken as the vertex of that straight angle. Since straight angles measure 180, their angle bisectors must divide them into pairs of 90-degree angles. Therefore, any line perpendicular to a line (straight angle) is a symmetry line of the given line. This is what I called the Line Perpendicular to Mirror Theorem. It implies that a line (straight angle) has infinitely many symmetry lines. (Of course, the line has one more symmetry line that I didn't mention -- namely the line itself.)
I included Question 24, even though it appears to mention corresponding and same-side interior angles formed by two lines and a transversal. But nowhere in the question does it mention anything about the two lines being parallel.
I left out Questions 16 and 17, which give the construction of an angle bisector. I finally plan on going to constructions sometime next week (and of course, I already introduced some constructions, including this one, in last week's Euclid: The Game). But here's another video from Square One TV, where doctors have to perform a "bisectomy" on an angle. (Unfortunately, only the entire 30-minute show is available on YouTube -- the "bisectomy" doesn't begin until the 11-and-a-half-minute mark.)
Today's lesson is about figures that are reflection-symmetric. As we will see in Chapter 5, the isosceles triangle and trapezoid are reflection-symmetric. There exist figures that have rotational, rather than reflection, symmetry -- the best known is the parallelogram. In order for a figure to have translation symmetry, the figure must be infinite, like a tessellation. A figure that has glide reflection symmetry must also have translation symmetry, and so is also infinite. In every case, we say that for a transformation T, a figure F has T-symmetry if and only if T(F) = F.
But we may notice that we haven't covered all of the possible transformations for T. We've mentioned the cases where T is a reflection, rotation, translation, or glide reflection. Yet there's one case that we haven't mentioned -- what if T is a dilation?
Technically speaking, only isometries can be symmetries, so if T is a dilation, we actually can't say that any figure is T-symmetric. Instead, we can say that any figure F such that T(F) = F for some dilation T is similar to itself, or self-similar, since a dilation is a similarity transformation. And so we wonder, what sort of shape can we describe as being self-similar.
The answer is -- a fractal.
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:
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: