Snowflakes, product logos, rose windows, and mandalas: these are just a few examples of symmetry in our world. While the applications to art and architecture may be most obvious, scientists observe symmetry as they study the structure and behavior of chemicals, bacterium, plants, and animals.
Subsection4.6.1Two Types of Symmetry
Finite two-dimensional designs, such as snowflakes and logos, may have reflectional symmetry and/or rotational symmetry. Figure 4.6.2 displays a design with both reflectional and rotational symmetry.
Definition4.6.1.
A design is said to have reflectional symmetry if there is a line such that reflecting the design across that line causes the design to align perfectly with itself. This line is called a line of symmetry for the design. Each point on one side of the line of symmetry reflects to a corresponding image point on the other side of the line and the image of a point reflects back across the line to the original point.
Reflectional symmetry is also called mirror symmetry and a line of symmetry may be called a mirror. It is possible for a design with mirror symmetry to have multiple lines of symmetry.
Draw the four lines of symmetry in Figure 4.6.2. Verify that the design stays the same when reflected across each line.
Definition4.6.4.
A design is said to have rotational symmetry if there is a rotation about a point with a positive angle of less than 360 degrees that maps the design onto itself. The image of the design will look identical to the original in shape, location, and angle 1
Designs may have color symmetry in which identical colors are required. Sometimes we focus only on symmetry of form and allow differences in color.
. The number of positions to which the figure can be rotated and still look the same is called the order of symmetry. In calculating the order of symmetry, we count a full 360-degree turn, but not a 0-degree rotation. If the order of symmetry of a figure is \(k\text{,}\) we say that the figure has \(k\)-fold symmetry.
Checkpoint4.6.5.Check your understanding.
Refer to the design in Figure 4.6.2 as you answer the following:
(a)
Where is the center of the rotation for the design in Figure 4.6.2? How is it related to the lines of symmetry?
(b)
What is the smallest positive angle of symmetry for the design?
(c)
The design has \(k\)-fold symmetry for what value of \(k\text{?}\) Explain.
Subsection4.6.2Exploring Symmetry of Polygons
In this section, we will focus on the symmetry of polygons and observe why regular polygons have greater symmetry than nonregular ones. Once again, we look for consistant patterns that might generalize.
We begin our exploration with triangles: As you complete the activities in Exploration 4.6.1, consider the following questions: What types of symmetry do triangles have? How does the shape of a triangle affect its symmetry?
Exploration4.6.1.Symmetry of Triangles.
An equilateral triangle is a regular triangle. We explore its symmetry before looking at other triangles.
(a)Symmetry of an Equilateral Triangle.
The GeoGebra applet Figure 4.6.6 provides an equilateral triangle along with some construction tools.
Figure4.6.6.An Equilateral Triangle
(i)
Draw a line of symmetry for this triangle. Then use the reflect across line tool to verify that your line works.
Hint.
Browse the tool menus of the GeoGebra applet. Tools for midpoint, perpendicular line/bisector, and angle bisector are provided.
(ii)
Does the equilateral triangle have any other lines of symmetry? If so, draw all lines of symmetry. Use the reflect across line tool to check your work as needed.
(iii)
An equilateral triangle also has rotational symmetry. Identify the center of rotational symmetry, drawing it if it does not already exist.
Hint.
GeoGebra will only recognize a point at intersection if you construct it. Use the point tool or intersection tool.
(iv)
What is the smallest positive angle of rotational symmetry for this triangle? If you rotate in a counterclockwise direction, where does vertex \(A\) move to under this rotation?
Hint.
Selecting the points in the reverse order will give you the measure of the other angle formed by the rays.
(v)
What other angles of symmetry are possible? What do you notice?
Hint.
Rotational angle measures may be greater than 180 degrees.
(b)Symmetries of Other Triangles.
Figure 4.6.7 shows a picture of an isosceles triangle and a scalene triangle.
Figure4.6.7.An isosceles and a scalene triangle
(i)
Does an isosceles (nonequilateral) triangle have reflective symmetry? If so, draw the line(s) of symmetry for the isosceles triangle in Figure 4.6.7. If not, explain.
(ii)
Does an isosceles (nonequilateral) triangle have rotational symmetry? If so, give the center and angle of symmetry for the isosceles triangle in Figure 4.6.7. If not, explain.
(iii)
Does a scalene triangle have either reflective or rotational symmetry? Justify your answer.
Perhaps you noticed that the symmetry of a triangle depends on the number of congruent sides it has. Since reflections and rotations preserve angle measure and length, each side and angle must map to a congruent object of the same type. Thus, an equilateral triangle has both rotational and mirror symmetry, an isosceles triangle has only mirror symmetry, and a scalene triangle has neither type of symmetry. In the next exploration, we look at the symmetry of regular \(n\)-gons.
Exploration4.6.2.Symmetry of Regular Polygons.
Figure 4.6.8 shows pictures of a regular quadrilateral, a regular pentagon, and a regular hexagon.
Figure4.6.8.GeoGebra applet with regular polygons
(a)Mirror Symmetry of Regular Polygons.
(i)
How many lines of symmetry does a square have? Draw the lines of symmetry for the square in Figure 4.6.8.
(ii)
How many lines of symmetry does a regular pentagon have? Draw the lines of symmetry for the pentagon in Figure 4.6.8.
(iii)
How many lines of symmetry does a regular hexagon have? Draw the lines of symmetry for the hexagon in Figure 4.6.8.
(iv)
How many lines of symmetry will a regular \(n\)-gon have? Does it matter whether \(n\) is odd or even? Explain your reasoning.
(b)Rotational Symmetry of Regular Polygons.
(i)
List all the positive angles of rotation up to and including 360 degrees for a square. What is the order of symmetry for a square?
(ii)
List all the positive angles of rotation up to and including 360 degrees for a regular pentagon. What is the order of symmetry for a regular pentagon?
(iii)
List all the positive angles of rotation up to and including 360 degrees for a regular hexagon. What is the order of symmetry for a regular hexagon?
(iv)
What is the order of symmetry for a regular \(n\)-gon?
(v)
What is the smallest positive angle of rotational symmetry for a regular \(n\)-gon?
(vi)
For a regular polygon, how is the smallest positive angle of rotation related to the order of symmetry?
(c)Notice and Wonder.
What did you notice about the lines of symmetry and centers of rotation as you were completing this activity? What do you wonder about symmetry of finite designs? What questions remain? What conjectures can you make?
As with triangles, we can find quadrilaterals (and other \(n\)-gons) that possess less symmetry.
Exploration4.6.3.Symmetry of non-regular quadrilaterals.
The following tasks may be done on paper or using the blank GeoGebra canvas provided in Figure 4.6.9.
(a)
Sketch a quadrilateral that has exactly two lines of symmetry. Does your quadrilateral also have rotational symmetry? If it does, locate the center and state the lowest positive degree angle of rotation.
(b)
Sketch a quadrilateral that has no lines of symmetry and no rotational symmetry. Explain why your example satisfies these conditions.
(c)
Sketch a quadrilateral that has 2-fold rotational symmetry but no mirror symmetry. What is the smallest positive degree angle of rotation. Explain why your example satisfies these conditions.
(d)
Sketch a quadrilateral that has a line of symmetry but no rotational symmetry. Explain why your example satisfies these conditions.
Subsection4.6.3Symmetry Transformations
Definition4.6.10.
A symmetry transformation for a design is an isometry that sends the design perfectly onto itself. The appearance of the design remains the same after the action of a symmetry transformation.
We also may use just the word symmetry to refer to a symmetry transformation. The context of how the word appears will help us decide whether ‘symmetry’ refers to the property of the design or the transformation that preserves the design.
In Section 4.4, we saw that when we performed a sequence of reflections in succession, the composition was the same as a single isometry, either another reflection, a translation, a rotation, or a glide reflection. The type of isometry depended on the number of reflections and whether the reflecting lines were parallel. With a finite design such as the one in Figure 4.6.2, note that any lines of symmetry must intersect and their point of intersection will be in the center of the figure. If the figure also has rotational symmetry, then the center of the rotation will be at this central point.
Checkpoint4.6.11.Reflective Questions.
Explain why a translation and a glide reflection cannot be symmetry transformation of finite designs or polygons. Do you think translations and/or glide reflections could be symmetry transformations of infinite tilings or infinite wallpaper borders? Explain in words and/or pictures.
We now focus on the symmetries of a single equilateral triangle, investigating what happens to the labeled vertices when we compose symmetry transformations.
Exploration4.6.4.Symmetries of an Equilateral Triangle.
Throughout this exploration, you may use the GeoGebra applet Figure 4.6.12 as needed. For a tactile exploration, label and cut out an equilateral triangle that you can flip and rotate.
(a)Rotations of an Equilateral Triangle.
As previously discovered, an equilateral triangle has 3-fold symmetry.
Figure4.6.12.GeoGebra applet for exploring the symmetries of an equilateral triangle
(i)
What is the smallest positive degree angle of symmetry?
(ii)
Suppose we use the label \(r_{120}\) to represent a counterclockwise rotation by a 120-degree angle about the center of this triangle. What is the effect of performing \(r_{120}\) twice in succession? Is it the same as a reflection or rotation? Specify the reflecting line (if it is a reflection) or rotation angle (if it is a rotation) of the composed transformation
(iii)
What happens if you rotate the original triangle using \(r_{120}\) three times in succession?
(iv)
A 0-degree rotation can be represented by \(r_0\text{.}\) Describe in your own words what happens when you apply \(r_0\text{.}\) How does it compare to \(r_{360}\text{?}\)
(v)
A clockwise 120-degree rotation is the same as a counterclockwise rotation of what degree?
(b)Images of Symmetries.
As we record the result of performing symmetries on the triangle in Figure 4.6.12, we will use the notation \(r_0\text{,}\)\(r_{120}\text{,}\)\(r_{240}\) to represent counterclockwise rotations of 0 degrees, 120 degrees, and 240 degrees about the center of the triangle.
We will also use the notation \(m_n\text{,}\)\(m_p\text{,}\) and \(m_v\) to denote the reflections across line \(n\) with negative slope, line \(p\) with positive slope, and the vertical line \(v\text{,}\) respectively.
Because these are symmetries, the actions will always result in the same equilateral triangle. What changes is the locations of vertices \(A\text{,}\)\(B\text{,}\) and \(C\text{.}\) Note that the image of the triangle after performing \(r_0\) is the same as Figure 4.6.13. We call this the Home Position. When using the GeoGebra applet Figure 4.6.12, we can click on the icon in the upper right that looks like a circle with clockwise-oriented arrows to return to the Home Position.
Figure4.6.13.Home Position of Equilateral Triangle
Being sure to start in the home position each time, draw and save a sketch that shows where the vertices \(A\text{,}\)\(B\text{,}\) and \(C\) land after each of the following symmetry transformations. You will need to refer to these sketches in tasks that follow.
(i)
\(r_{120}\)
Hint.
\(r_{120}\text{:}\)\(C\) is on top; bottom left to right is \(BA\text{.}\)
(ii)
\(r_{240}\)
Hint.
\(r_{240}\text{:}\)\(A\) is on top; bottom left to right is \(CB\text{.}\)
(iii)
\(m_n\)
Hint.
\(m_n\text{:}\)\(A\) is on top; bottom left to right is \(BC\text{.}\)
(iv)
\(m_p\)
Hint.
\(m_p\text{:}\)\(C\) is on top; bottom left to right is \(AB\text{.}\)
(v)
\(m_v\)
Hint.
\(m_v\text{:}\)\(B\) is on top; bottom left to right is \(CA\text{.}\)
It is not an accident that each of these five transformations results in a different arrangement of the vertices, \(A\text{,}\)\(B\text{,}\) and \(C\text{,}\) none of which will be the same as \(r_0\text{.}\)
(c)Setting the Stage.
To prepare for the final task, we will remind ourselves of what a multiplication table is. Complete Table 4.6.14 using your knowledge of whole number multiplication.
Table4.6.14.Multiplication Table
\(\times\)
1
2
3
4
5
1
1
2
3
4
5
2
2
4
3
3
4
4
5
5
20
(d)
Like the multiplication table above, we can create a table for the composition of symmetry isomorphisms on an equilateral triangle using the following steps as a guide. Each entry you add to Table 4.6.16 will always be one of the six symmetries of the triangle; namely, \(r_0\text{,}\)\(r_{120}\text{,}\)\(r_{240}\text{,}\)\(m_n\text{,}\)\(m_p\text{,}\) or \(m_v\text{.}\) When you use Figure 4.6.12 to compute compositions, be sure to reset the triangle to home position before performing the first of the two actions and then perform the second action without resetting the triangle. Compare the final result to the vertex labels you recorded in Task 4.6.4.b and then use the symbol, \(r_0\text{,}\)\(r_{120}\text{,}\)\(r_{240}\text{,}\)\(m_n\text{,}\)\(m_p\text{,}\) or \(m_v\) that corresponds to the output of the composition.
Table4.6.16.Composition Table
\(\circ\)
\(r_0\)
\(r_{120}\)
\(r_{240}\)
\(m_n\)
\(m_p\)
\(m_v\)
\(r_0\)
\(r_0\)
\(r_{120}\)
\(r_{240}\)
\(m_n\)
\(m_p\)
\(m_v\)
\(r_{120}\)
\(r_{120}\)
\(r_{240}\)
\(r_{240}\)
\(m_n\)
\(m_n\)
\(m_p\)
\(m_p\)
\(m_v\)
\(m_v\)
(i)
The row and column headed by \(r_0\) have already been filled in. Explain why this row and column is the same as the header row and header column. In particular, what is the action of \(r_0\) on the vertices of triangle \(\Delta ABC\text{?}\)
Hint.
Do you notice the similarity between the action of \(r_0\) here and the way the number 1 acts in multiplication?
(ii)
Determine the compositions of two rotations, namely \(r_{120}\circ r_{120}\text{,}\)\(r_{120}\circ r_{240}\text{,}\)\(r_{240}\circ r_{120}\text{,}\) and \(r_{240}\circ r_{240}\text{.}\) Enter these results into the table, using the appropriate \(r_x\) or \(m_x\) shorthand in the appropriate locations.
Hint.
The column will be headed by the first action performed and the row will be headed by the second action. As an example, the result of \(r_{240}\) (rotating by 240 degrees) followed by \(r_{120}\) (rotating by 120 degrees) would be entered into the cell below 240 and to the right of 120.
(iii)
The nine cells in the upper left should now be filled. Starting with the triangle in the home position of Figure 4.6.12, perform \(m_n\) followed by \(r_{120}\text{.}\) 2
This operation is the same as \(r_{120}\circ m_n\text{,}\) the composition of \(r_{120}\) and \(m_n\) since the composition of two functions \(f\) and \(g\) is defined by mathematicians to be \(f\circ g(x)=f(g(x))\text{.}\) See Definition 4.4.1. For this activity, we will refer to the first and second action throughout to maintain clarity.
. Record the symbol in Task 4.6.4.b that corresponds to the resulting image in the cell in the \(m_n\)-column and the \(r_{120}\)-row.
Hint.
\(m_n\) followed by \(r_120\) results in \(m_p\text{.}\) In symbols, \(r_120\circ m_n=m_p\text{.}\)
(iv)
Is \(m_n\circ r_{120}\) the same as \(r_{120}\circ m_n\text{?}\) To determine \(m_n\circ r_{120}\text{,}\) start at the home position and first perform \(r_{120}\) followed by \(m_n\text{.}\) Write the symmetry you just found in the cell of the \(r_{120}\)-column and \(m_n\)-row. Compare the two results.
Hint.
The result depends on the order in which you perform the isometries!
(v)
To complete the \(r_{120}\)-column, compute the two compositions:
(A)
\(r_{120}\circ m_p\text{,}\) by first rotating the triangle in home position 120 degrees counterclockwise and then reflecting the resulting triangle across line \(p\text{,}\) and
(B)
\(r_{120}\circ m_v\text{,}\) by first rotating the triangle in home position 120 degrees counterclockwise and then reflecting the resulting triangle across line \(v\text{.}\)
(C)
The \(r_{120}\)-column should now be completely filled. Each of the symmetries, \(r_0\text{,}\)\(r_{120}\text{,}\)\(r_{240}\text{,}\)\(m_n\text{,}\)\(m_p\text{,}\) and \(m_v\) will appear exactly once in this colunn.
(D)
The \(r_{120}\)-column should now be completely filled. Each of the symmetries, \(r_0\text{,}\)\(r_{120}\text{,}\)\(r_{240}\text{,}\)\(m_n\text{,}\)\(m_p\text{,}\) and \(m_v\) will appear exactly once in this colunn.
(E)
Similarly, we will complete the \(m_n\)-column by reflecting the home triangle across the line \(n\) before performing the symmetry corresponding to the row heading. Check again to make sure that no symmetry appears twice in the same column.
(F)
Complete the last two columns where the first transformation is reflection across the lines \(p\) and \(v\text{,}\) respectively.
Hint.
Ignoring the headers, each of the six isometries should appear exactly once in each row and each column.
(e)Observations and Summary.
Refer to your completed version of Table 4.6.16, as you answer the following:
(i)
Of the six symmetry transformations, \(r_0\text{,}\)\(r_{120}\text{,}\)\(r_{240}\text{,}\)\(m_n\text{,}\)\(m_p\text{,}\) and \(m_v\text{,}\) which transformation behaves like the number 1 does in multiplication? We call this transformation the identity transformation.
(ii)
Which transformation “undoes” \(R_{120}\text{?}\) In other words, which transformation can you perform after \(R_{120}\) so that this two-step action gives the same result as the identity transformation? Does the order of these two transformations matter?
Algebra connection: How is this relationship like composing the ‘multiply by five’ function \(f(x)=5x\) and the ‘divide by 5’ function \(g(y)=\frac{y}{5}\) ?
(iii)
Which transformation “undoes” \(m_n\) ? Does \(m_n\) undo that transformation?
Algebra connection: Is there a real number \(k\) such that the multiplication by \(k\) function, \(h(x)=kx\text{,}\) undoes itself?
(iv)
Algebra connection: The operation multiplication on the set of real numbers satisfies the commutative property, \(a\cdot b=b\cdot a\) for all real numbers, \(a\) and \(b\text{.}\) Does composition of symmetry transformations satisfy the commutative property? Support your answer.
Exercises4.6.4Exercises
Building Our Toolbox
1.
add
Skills and Recall
2.
Add some. Identify symmetry of polyominos or letters. Pentagon with reflective but no rotational symmetry.
Extending the Concepts
3.
Add exploring connections between \(n\)-fold symmetry and number of reflecting lines.
