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.
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.
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.
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.
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?
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.
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?
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.
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.
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.
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?
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.
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.
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.
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.
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
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 \(r_{360}\text{?}\)
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.
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.
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 \(r_0\text{.}\)
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.
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.
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 \(\Delta ABC\text{?}\)
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.
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.
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.
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.
\(r_{120}\circ m_p\text{,}\) by first rotating the triangle in home position 120 degrees counterclockwise and then reflecting the resulting triangle across line \(p\text{,}\) and
\(r_{120}\circ m_v\text{,}\) by first rotating the triangle in home position 120 degrees counterclockwise and then reflecting the resulting triangle across line \(v\text{.}\)
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.
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.
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.
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.
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 \(g(y)=\frac{y}{5}\) ?
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.
