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Polyominoes, Polygons, and Polyhedra Discovering Geometry through Exploration

Teresa D. Magnus

Section 4.4 Compositions of Reflections

Subsection 4.4.1 Compositions

Each of the transformations we have studied so far (reflections, translations, and rotations) performs an action on the points of the plane. In fact, a transformation is a function where the inputs and outputs are geometric objects instead of numbers. In algebra, you learned about the “add three” function, written \(f(x)=x+3\text{,}\) and the “multiply by four” function, \(g(x)=4x\text{.}\) You also learned that you could perform two functions in succession. For example, you could perform the “add three” function on the number \(1\) and then perform the “multiply by 4” function on the result.
\begin{equation*} 1 \overset{x+3}{\rightarrow} 4 \overset{4x}{\rightarrow} 16 \end{equation*}
In function notation, we may write \(g(f(1))=g(1+3)=g(4)=4(4)=16\) where we perform the inner function first. We may also perform these operations on a generic value \(x\text{:}\)
\begin{equation*} g(f(x))=g(x+3)=4(x+3)=4x+12. \end{equation*}
We note the importance of order. If we multiply by 4 first and add 3 second, we get a different result:
\begin{equation*} f(g(x))=f(4x)=(4x)+3=4x+3. \end{equation*}
We now explore what happens when we compose geometric transformations. We could compose any two transformations, but we will first focus on reflections. What happens when we perform two or more reflections in succession? Do we get another reflection? Does it matter whether the lines are parallel? Does the order in which we perform the reflections matter?

Definition 4.4.1.

A composition of two transformations is a sequence of two transformations, \(s\) and \(t\text{,}\) performed in succession. To find the image of a point \(P\) under the composition, \(t(s(P))\text{,}\) first find the image \(P'\) of \(P\) under the action of \(s\text{.}\) Then find the image \(P''\) of \(P'\) under \(t\text{.}\)
The process of performing the two processing in succession can be written
\begin{equation*} P \overset{s}\rightarrow s(P) \overset{t}\rightarrow t(s(P)) \end{equation*}
or more simply as
\begin{equation*} P \overset{s}\rightarrow P' \overset{t}\rightarrow P''. \end{equation*}
Note that in the composition \(P \rightarrow s(t(P))\text{,}\) we perform the inside function \(t\) before the outer function \(s\text{.}\)

Exploration 4.4.1. Reflecting across Parallel Lines.

(a)
First we explore what happens when you reflect an object first across line \(g\) and then across line \(h\) where \(g\) and \(h\) are parallel.
Figure 4.4.2. An interactive GeoGebra applet for reflecting across parallel lines.
(i)
Use the ‘Reflect across Line’ tool to reflect pentomino \(ABCDEF\) across line \(g\) in Figure 4.4.2 to create pentomino \(A'B'C'D'E'F'\text{.}\)
Hint.
To use the ‘Reflect about Line’ tool, select the icon that shows two dots across a diagonal line from each other. Click the interior of the original figure and then click the mirror line.
(ii)
Then reflect the new pentomino \(A'B'C'D'E'F'\) across line \(h\) to form pentomino \(A''B''C''D''E''F''\text{.}\)
Finally, check the box that says ‘Show image’. Does this line up with your final image? Save a screen shot that shows the intermediate step.
(iii)
To help you focus on \(ABCDEF\) and \(A_2B_2C_2D_2E_2F_2\text{,}\) click on the reset button to hide your work. Then check the ‘Show image’ box to reveal the image of the composition.
(A)
What type of isometry takes \(ABCDEF\) to \(A_2B_2C_2D_2E_2F_2\text{?}\) Is it a reflection, a translation, a rotation, or something else?
(B)
Draw the segments \(\overline{AA_2}, \overline{BB_2}, \ldots, \overline{FF_2}\) and measure the lengths \(AA_2, BB_2, \ldots, FF_2\text{.}\) What do you notice?
(C)
What do you notice about the angle between each segment, \(\overline{AA_2}, \overline{BB_2}, \ldots, \overline{FF_2}\) and the original two lines \(g\) and \(h\text{?}\)
(D)
Measure the distance between the original pair of reflecting lines \(g\) and \(h\text{.}\) To do this, reveal segment \(\overline{TU}\) using the checkbox. Then use the ‘Distance or length’tool to measure \(\overline{TU}\text{.}\)
Hint.
For a smaller font, measure the segment, not the distance between points.
(iv)
Each type of isometry has a special object or two that helps to determine it precisely. As shown in Table 4.3.9, a reflection has a reflecting line, a translation has a vector, and a rotation has a center and angle. Identify and draw in the specific line, vector, or center and angle for this transformation. How does this object relate to the parallel reflecting lines \(g\) and \(h\text{?}\) Verify your answer using the appropriate transformation tool (Reflect across Line, Translate by Vector, or Rotate around Point) in geogebra.
Hint.
These transformation tools appear in the same menu in this applet.
(v)
A second copy of the GeoGebra app for reflecting across parallel lines is provided in Figure 4.4.3. This time reflect \(ABCDEF\) first across line \(h\) and then reflect the result across line \(g\text{.}\) How does the result compare to what happened when we reflected across g before h? Where is \(A''B''C''D''E''F''\text{?}\) Have any of the distances or angles changed?
Figure 4.4.3. An interactive GeoGebra applet for reflecting across parallel lines.
Hint.
The ‘Show image’ checkbox reveals the image of the first composition, not this one.
In the next exploration, we will perform a similar experiment reflecting an object across two intersecting lines in succession.

Exploration 4.4.2. Reflecting Across Intersecting Lines.

(a)
Now we explore what happens when you reflect an object first across line \(g\) and then across line \(h\) where \(g\) and \(h\) intersect.
Figure 4.4.4. An interactive GeoGebra applet for reflecting across intersecting lines.
(i)
Use the ‘reflect across line’ tool to reflect pentomino \(ABCDEF\) across line \(g\) in Figure 4.4.4 to create pentomino \(A'B'C'D'E'F'\text{.}\)
(ii)
Then reflect the new pentomino \(A'B'C'D'E'F'\) across line \(h\) to form pentomino \(A''B''C''D''E''F''\text{.}\)
Now check the ‘Show image’ box to check your work.
(b)
Refer to the original pentomino \(ABCDEF\) and its image \(A''B''C''D''E''F''\) created by reflecting across intersecting lines \(g\) and \(h\) in Figure 4.4.4. You may choose to hide \(A'B'C'D'E'F'\) (and the duplicated \(A''B''C''D''E''F''\)) after saving a screenshot of your work. Use the reset button and ‘Show image’ checkbox.
(i)
What type of isometry takes \(ABCDEF\) to the final image \(A_2 B_2C_2 D_2 E_2 F_2\text{?}\) Is it a reflection, a translation, a rotation, or something else?
(ii)
Draw the segments \(\overline{AA_2}, \overline{BB_2}, \ldots, \overline{FF_2}\text{.}\) What, if anything, do you notice?
(iii)
Create the perpendicular bisector of each segment \(\overline{AA_2}, \overline{BB_2}, \ldots, \overline{FF_2}\text{.}\) What do you notice about the perpendicular bisectors?
(iv)
Each type of translation has a special object or two that helps to determine it precisely. Identify and draw in the specific line, vector, or center and angle for this transformation, referring to Table 4.3.9 as needed. How does the object(s) relate to the intersecting reflecting lines \(g\) and \(h\text{?}\)
(v)
Create point \(P\) where the original two lines intersect. What is the measure of \(\angle A_2 PA\text{?}\) \(\angle B_2 PB\text{?}\) \(\angle C_2 PC\text{?}\) \(\angle D_2 PD\text{?}\) \(\angle E_2PE\text{?}\) \(\angle F_2PF\text{?}\) What does this tell you about the transformation taking \(ABCDEF\) to \(A_2B_2C_2D_2E_2F_2\text{?}\)
(vi)
Measure the angles created by the original pair of intersecting lines \(g\) and \(h\text{.}\) What do you notice?
(vii)
Measure the lengths \(AP\text{,}\) \(A_2P\text{,}\) \(BP\text{,}\) \(B_2P\text{,}\) etc. What do you notice?
(c)
A second copy of the GeoGebra app for reflecting across intersecting lines is provided in Figure 4.4.5. This time reflect \(ABCDEF\) first across line \(h\) and then reflect the result across line \(g\text{.}\) How does the result compare to what happened when we reflected across g before h? Where is \(A''B''C''D''E''F''\text{?}\) Have any of the lengths or angles changed?
Figure 4.4.5. An interactive GeoGebra applet for reflecting across intersecting lines.
Hint.
The ‘Show image’ checkbox reveals the image of the first composition, not this one.
Next we will explore what happens when we reflect across three lines in succession. Once again, we will consider the case where the lines are parallel and when they intersect. Will we get the same transformations as before or might these be something different?

Exploration 4.4.3. Reflecting across three lines.

(a) Orientation.
Recall that a reflection across a single line reverses the orientation.
(i)
What happens to the orientation when you reflect an object across two lines in succession?
(ii)
If you were to reflect the object across a third line, what would you expect to happen to the orientation?
(b) Reflecting across three parallel lines.
Reflect the triangle in Figure 4.4.6 across lines \(d\text{,}\) \(e\text{,}\) and \(f\) in succession.
Figure 4.4.6. An interactive GeoGebra applet for reflecting across three parallel lines.
(i)
What type of single transformation (reflection, translation, rotation, or something else) takes the original triangle to the final image?
(ii)
Each type of translation has a special object or two that helps to determine it precisely. Identify and draw in the specific line, vector, or center and angle for this transformation. How, if at all, is this object related to the three lines \(d\text{,}\) \(e\text{,}\) and \(f\text{?}\)
Hint.
As of this writing, the author has not found a relationship between this defining object and the three lines. If you discover one, please let her know. You may be recognized in a future edition!
Unlabeled points on lines \(d\text{,}\) \(e\text{,}\) and \(f\) allow the lines to be moved while staying parallel. Experiment to see whether your claims continue to hold. You can also experiment with reflecting across the three lines in a different order; however, the ‘Show image’ will not work.
(c) Reflecting across three intersecting lines.
Finally, we reflect the triangle across three lines, at least some of which intersect. Use the ‘Reflect about Line’ tool to reflect \(\Delta ABC\) across \(d\text{,}\) then reflect its image \(\Delta A'B'C'\) across line \(e\text{,}\) and finally reflect \(\Delta A''B''C''\) across \(f\text{.}\) After creating \(\Delta A'''B'''C'''\text{,}\) check the result using the checkbox ‘Show image’. Record your work.
Figure 4.4.7. An interactive GeoGebra applet for reflecting across three intersecting lines.
(i)
You may reset your work to focus on the initial triangle \(\Delta ABC\) and final image \(\Delta A'_1 B'_1 C'_1\text{.}\)
(A)
Is the orientation of the figure the same or changed under the single transformation that takes \(\Delta ABC\) to \(\Delta A'_1 B'_1 C'_1\text{?}\)
(B)
Explain how you know that this transformation is not a reflection.
(C)
Explain how you know that this transformation is not a translation.
(D)
Explain how you know that this transformation is not a rotation.
(E)
Hide lines \(d\text{,}\) \(e\text{,}\) and \(f\) by unchecking the ‘Show mirrors’ box. Plot the midpoint of each of the line segments, \(\overline{AA'_1}\text{,}\) \(\overline{BB'_1}\text{,}\) and \(\overline{CC'_1}\text{.}\) What do you notice about these midpoints?
Hint.
GeoGebra allows you to plot the midpoint without drawing the segments.
(F)
Draw the line that passes through the midpoints of \(\overline{AA'_1}\) and \(\overline{BB'_1}\text{.}\) Then reflect \(\Delta ABC\) across this line. What do you notice about the relationship between this new triangle \(\Delta A'B'C'\) and \(\Delta A'_1 B'_1 C'_1\text{?}\)
(G)
Describe a two-transformation process that takes \(\Delta ABC\) to \(\Delta A'_1 B'_1 C'_1\text{.}\)

Subsection 4.4.2 The Four Isometries

In this chapter, we’ve explored the behavior of a particular type of transformation, called an isometry. This term comes from the Greek words “iso-” meaning “equality” and “metria” meaning “measure”.

Definition 4.4.8.

An isometry is a function taking points on the plane to points on the plane which sends collinear points to collinear points and preserves the distance between points.
There are four types of isometries. Through our explorations, we have seen that the orientation of a figure is reversed each time it is reflected. Thus, orientation is changed under an odd number of reflections and maintained (flipped back) under an even number of reflections. Hence, the composition of two reflections in succession will be either a translation (when the reflecting lines are parallel) or a rotation (when the reflecting lines intersect). When we reflect across three lines, not all of which are parallel, we get an isometry which reverses the orientation of figures but is not a reflection. Instead we obtain what is known as a glide reflection.

Definition 4.4.9.

A glide reflection is defined as a composition of a reflection across a line and a translation along a vector parallel to (or on) the reflecting line. In Figure 4.4.10, note how the pentomino is first reflected across the line and then translated along vector \(\overrightarrow{GH}\text{.}\) Any of the parallel vectors, like \(\overrightarrow{A'A''}\) or \(\overrightarrow{CC''}\text{,}\) could be used as the translation vector, but the reflecting line here is \(\overleftrightarrow{GH}\text{,}\) not \(\overleftrightarrow{CC''}\text{.}\)
L-pentamino is reflected across line GH. Then its image is slid using vector GH. A and C are two vertices on the original.
Figure 4.4.10. Glide reflection of a pentomino.

Checkpoint 4.4.11. Reflective Question.

Will a glide reflection have any fixed point(s)? How do you know?
We now update Table 4.3.4 to include glide reflections.
Table 4.4.12. Defining Objects for Three Isometries
Isometry Defining Objects
Reflection Reflecting Line (or mirror)
Translation Translation Vector
Rotation Center and Rotational Angle
Glide Reflection Mirror and Translation Vector
 1 
The translation vector will either lie on the mirror or be parallel to the mirror.

Exercises 4.4.3 Exercises

Building Our Toolbox

1.
Add the information about glide reflections to complete Table 4.4.13. You should have already filled in the other rows in Table 4.3.9.
Table 4.4.13. Defining Objects for Four Isometries
Isometry Orientation Fixed
Points		 Defining
Object(s)		 Properties of
Def. Object(s)
Reflection Reverses Points on
Mirror		 Reflecting Line
(or mirror)		 Mirror is the
perpendicular
bisector of each \(\overline{PP'}\)
Translation Translation
Vector



Rotation Center and
Angle		 Perp. bisector
of each \(\overline{PP'}\)
passes through
center
Glide
Reflection		 Mirror and
Vector		 Midpoint of
each \(\overline{PP'}\) lies
on mirror

Skills and Recall

2.
Add some.

Extending the Concepts

3.
Add

Writing Prompts

4.
Add
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