Teacher professional development and classroom resources across the curriculum
Teacher professional development and classroom resources across the curriculum
In the preceding section, we examined two specific types of visual symmetry, reflection and rotation. We saw that objects, such as a sea star, a daisy, or an equilateral triangle, can possess one or both of these symmetries. It might be tempting to think that these two kinds of motion are the only symmetries that an object can possess. To see whether or not this hunch is correct, we need to revisit our understanding of what a symmetry is.
We often think of symmetry as a descriptor or quality of an object. We use the term "symmetric" as an adjective to describe beautiful, balanced objects such as the ones we studied in the preceding section. We've now begun to think of symmetry in terms of the motions that leave an object appearing the same as before the motion took place. Once we have freed our thinking from the idea of symmetry as a property of an object and shifted to considering symmetry to be a motion, we open ourselves up to more options as to what we consider a symmetry to be.
For instance, consider a sine wave.
Looking at this design, and assuming that it continues its pattern forever both to the left and to the right, what motions could we apply to it that would leave it invariant? In other words, imagine that we can pick this design up off of the page, move it in some way, and replace it on the page. In what ways can we move it so that when we replace it, we can't tell that we did anything?
If we reflect the image over the y-axis, we will not end up with the same design. The same is true if we reflect it over a horizontal line such as the x-axis.
If we reflect the sine curve over the vertical line , however, the original look of the design is retained. So, this design remains invariant when reflected over vertical lines defined by . The nπ term of this expression represents the fact that the sine curve goes to infinity and, thus, has an infinite number of possible vertical reflection axes.
Now let's consider rotation. It is not hard to conceive that any rotation of less than 180° will not leave the sine wave invariant. If, however, we rotate the entire design exactly 180° about the origin, or about any point on the x-axis with an x-value of the form 0 + n π , we get a result that coincides with the original configuration.
We have established that the sine wave has both rotational symmetry and reflection symmetry under certain conditions. What if we simply shifted the curve horizontally along the x-axis? Would this motion leave the design invariant?
A little thought will reveal that as long as we move it in increments of 2 π , which is the period of a standard sine wave, the design will be unchanged. This type of motion is called a translation, and it, along with reflection and rotation, is a symmetry of the sine wave. We must consider shifts of different magnitudes to be separate motions, however. Consequently, because we can shift the sine curve by any integer multiple of 2 π that we choose, the number of translations that leave this design invariant is infinite.
What if we shifted the sine wave horizontally by a value of π , and then reflected it over the x-axis? Neither of these actions alone is a symmetry, because neither action alone leaves the design invariant. Together, however, these two motions actually do leave the design invariant. This type of symmetry is called a glide reflection, and because it involves a translation, it also has infinite varieties. Note that translations and glide reflections can be symmetries only of designs that are infinite in extent.
So, we've seen that our relatively innocent-looking sine wave has many symmetries: reflection over certain vertical lines, rotation by 180°, translation by multiples of 2 π , and glide reflections—and let's not forget the Identity! As before, with our analysis of the equilateral triangle group, we can check to see if these transformations of the sine wave form a group under the operation of"followed by."
First, we have an Identity, but does each motion have an inverse? A bit of examination should convince you that both the vertical reflections and the 180° rotations can be undone by themselves. The translations and glide reflections can be undone by shifting the curve in the opposite direction of the initial shift. If the first motion was to translate the curve to the right by 2 π, the inverse is to translate it left by 2 π. This shows that the symmetries of a sine wave have inverses.
With the first two requirements of a group, identity and inverse, confirmed, let's turn our focus to closure. Since the sine wave remains invariant under an infinite number of translations and glide reflections, we cannot simply construct a table and check to see that every combination of symmetries gives us another symmetry. We should, however, be able to convince ourselves fairly easily that, as long as we are careful about how we perform our reflections, rotations, translations, and glide reflections, the result will still be a symmetry of the design.
For example, say that S represents the sine wave. G, T, and R represent symmetries of S—motions that leave S invariant. We'll use * to represent the operation of applying G, T, or R to S.
G*S means "apply motion G to sine wave S." Because G is a symmetry, we expect that G*S will result in an unchanged S, which is the same result obtained with the Identity motion. Therefore, G*S = S, as does R*S, and T*S.
We can compose G, T, and R together to demonstrate both closure (that any combination of group elements results in an element within the group) and associativity (that multiple motions can be grouped however we choose without affecting the result).
Let's consider G*T*R*S. We can do this as follows:
Notice that we end up with an unchanged S, a symmetry equivalent to the Identity. This shows that combinations of G, T, and R are still symmetries, demonstrating closure. Hopefully, it's obvious that we would have reached the same result regardless of the order of the G, T, and R motions in the original expression, so the associativity requirement is confirmed.
We've now seen that the symmetries of this infinite sine wave do indeed form a group of the type known as frieze groups. A frieze is simply a repetitive design on a linear strip.
The number of designs possible with a frieze is limited only by the imagination of the artist creating it. However, every frieze will have at least one of the aforementioned symmetries or a symmetry of reflection over a horizontal line (a symmetry the sine wave does not have).
Moreover, every frieze, if sufficiently "stripped" of its ornamental elements, falls into one of the seven symmetry groups described here:
Friezes extend in only one dimension, but if we consider patterns that extend in two dimensions, patterns that cover the Euclidean plane for instance, we find a similar result. These planar patterns, constructed of the same basic motions as the linear patterns (except that rotations of less than 180° are possible), also form symmetry groups. Every one of these socalled "wallpaper patterns" can be classified as one of seventeen planar groups.
This is an example of a wallpaper group consisting of reflections over two axes and rotations of 180°. The following diagram is a simplified representation:
The blue lines represent the axes over which this design can be reflected, and the pink diamonds show the centers of rotation. Friezes and wallpaper patterns are not the only geometric designs that can be classified into groups. Similar results hold for three-dimensional patterns, such as those found in crystals, and even for patterns in higher dimensions!
We've now seen how symmetry in patterns can be captured mathematically in the concepts of friezes and wallpaper groups. Let's now turn our attention away from design toward an area that may at first seem wholly unrelated—permutations. Through this exploration we'll see the power of abstraction (and in this case, the abstract concept of a group) to tie together ideas and situations that seem to have little in common.
Next: 6.4 Card Shuffling