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Unit 6

The Beauty of Symmetry

6.2 Types of Symmetry Leading to Groups

FOUNDATIONS OF BEAUTY

Symmetry is perhaps most familiar as an artistic or aesthetic concept. Designs are said to be symmetric if they exhibit specific kinds of balance, repetition, and/or harmony. In mathematics, symmetry is more akin to something like "constancy," or how something can be manipulated without changing its form. In other words, the mathematical notion of symmetry relates to "objects" that appear unchanged when certain transformations are applied.

Flatworm, Ant, Human Note that reflecting any of these forms about the axis leaves the design unchanged.

Think of the form of a butterfly; its right and left halves mirror each other. If you knew what the right half of a butterfly looked like, you could construct the left half by reflecting the right half over a line that bisects the butterfly.

Butterflies exhibit a type of symmetry called "bilateral symmetry" or a "mirror symmetry," (either half of the butterfly is the mirror image of the other) one that is very common among living things. Perhaps most familiar to us is our own bilateral symmetry, the symmetry of our left and right arms and hands, or our left and right legs and feet, or the approximate symmetry of our bodies if bisected vertically into left and right halves. In general, bilateral symmetry is present whenever an object or design can be broken down into two parts, one of which is the reflection of the other. Given any motif, one can generate a design with bilateral symmetry by choosing a line and reflecting the motif over it. Conversely, if a motif already possesses bilateral symmetry, it can be reflected over a line and we would notice no difference between the original and the reflected versions. This action, reflection, leaves the original design apparently unchanged, or invariant.

Bilateral symmetry is quite common in nature, but it is by no means the only form of visual symmetry that we see in the world around us. Another common form is rotational symmetry, such as that seen in sea stars and daisies.

BEAUTIFUL STARFISH Item 1576/Corel Corporation, BEAUTIFUL STARFISH (2008). Courtesy of Corel Corporation.
OXEYEDAISY Item 1253/NPS Photo by William S. Keller, OXEYEDAISY (CHRYSTANTEMUM LEUCANTHEMUM) (1962). Courtesy of National Park Service.


Recall that to be symmetric an object must appear unchanged after some action has been taken on it. An object that exhibits rotational symmetry will appear unchanged if it is rotated through some angle. A circle can be rotated any amount and still look like a circle, but most objects can be rotated only by some specific amount, depending on the exact design. For example, an ideal sea star, having five arms, is not symmetric under all rotations, but only those equivalent to 1/5 of a full rotation, or 72° 360/5.

rotating Rotating an ideal sea star 72° leaves itís appearance unchanged


A daisy, on the other hand, is rotationally symmetric under smaller rotational increments. Let’s say it has 30 petals, all of which are the same in appearance— no such daisy exists in the real world, of course—this is an ideal mathematical daisy. The flower will be symmetric under a rotation of 12° 360/30or any multiple thereof.

rotating The ideal daisy is symmetric under rotations of 12°


You might have observed that the sea star and the daisy are not limited to rotational symmetry. Depending on how you choose an axis of reflection, they can each display bilateral (reflection) symmetries as well. Notice, however, that only certain dividing lines can serve as axes of reflection.

reflection

This brings us to an important point: an object may have more than one type of symmetry. The specific symmetries that an object exhibits help to characterize its shape. Remember, the motions associated with symmetries always leave the object invariant. This means that combinations of these motions, which are how mathematicians tend to think of symmetries, will also leave the original object invariant. Let’s explore this idea a bit further by looking at the symmetries of an equilateral triangle.

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THE EQUILATERAL TRIANGLE

reflection

Notice that there are three lines over which the triangle can be reflected and maintain its original appearance.

rotationConsidering rotations, we find that there are only three that will return the triangle to its original appearance. We can rotate it through 1/3 of a complete revolution (120°), 2/3 of a complete revolution (240°), or one full revolution (360°).

As we have seen, an equilateral triangle has three distinct reflections and three rotations under which it remains invariant. Furthermore, since all of these symmetries leave the triangle invariant, the combination of any two of them creates a third symmetry. For example, a rotation through 120°, followed by a reflection over a vertical line passing through its top vertex, leaves the triangle in the same position it was in at the start. Let’s look at all possible combinations of symmetries in an equilateral triangle a little more closely. To do this, it will be helpful to label each vertex so that we can keep track of what we have done.

reflection

If we do nothing to the triangle, this is called the identity transformation, I.

reflection

This symmetry is simply a rotation of 120° counterclockwise; let’s call it R1.

reflection

A rotation of 240° degrees counterclockwise is another symmetry of the equilateral triangle; let’s call it R2.

reflection

This diagram represents a reflection over the vertical axis (notice how vertices A and B have switched sides); let’s call it L.

reflection

This symmetry is a reflection over the line extending from B through the midpoint of AC; let’s call this motion M.

reflection

In this diagram the triangle has been reflected over the line extending from C through the midpoint of AB; let’s call this action N.

Now that we have identified all the possible motions that leave the triangle invariant, we can organize their combinations in a chart. In these combined movements, the motions in the left column of the chart are done first, then the motions across the top row. For instance, the rotation R1, followed by the Identity, I, yields the same result as simply performing R1 by itself. Performing the reflection M, followed by N, gives us the same result as simply performing the rotation R2.

chart of all possible combinations

Notice that as we complete this chart of all possible combinations of two motions, every result is one of our original symmetries. This is an indication that we have found some sort of underlying relational structure. Mathematicians call sets of objects that express this type of structure a group.

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STAY WITH THE GROUP

A group is just a collection of objects (i.e., elements in a set) that obey a few rules when combined or composed by an operation that we often call"multiplication." This may seem like a vague, even unhelpful, description, but it is precisely this generality that gives the study of groups, or group theory, its power. It is also amazing and a bit mysterious that out of just a few simple rules we can create mathematical structures of great beauty and intricacy.

The symmetries of an equilateral triangle form a group. Remember, these symmetries are all the rigid motions that leave the triangle invariant. One of the powers of group theory is that it allows us to perform operations that are"sort of" arithmetic with things that are not numbers. Notice that the operation we used in the triangle example above was simply the notion of "followed by." This is going to be completely analogous to the idea of combining two integers by addition and getting another integer, or multiplying together two nonzero fractions and getting another fraction!

Group theorists study objects that don’t have to be numbers as well as operations that don’t have to be the standard arithmetical operations. Now we can be a little more precise about what we mean by a group and how groups function. For example, we would like to be able to use the members of a group to do arithmetic and even to solve simple equations, such as 3x = 5. To solve this equation, we need the operation of multiplication, and we need the number 3 to have an inverse. An inverse is simply a group member that, when combined with another group member under the group operation, gives the Identity.

In the case of 3x = 5, the inverse is 1/3 , which when combined with 3 under the operation of multiplication, gives 1, the multiplicative identity.

This scenario has pointed out the first two rules of a group. First, the group must have an element that serves as the Identity. The characteristic feature of the Identity is that when it is combined with any other member under the group operation, it leaves that member unchanged.

Second, each member or element of the group must have an inverse. When a member is combined with its inverse under the group operation, the result is the Identity.

In addition to these two basic rules of group theory, there are two more concepts that characterize groups. The third property, or requirement, of a group is that it is closed under the group operation. This means that whenever two group members are combined under the group operation, the result is another member of the group. We saw this as we looked at all possible combinations of symmetries of the equilateral triangle above. No matter which symmetry was"followed by" which, the result was always another symmetry. For simplification as we go forward in our exploration of groups, we might as well use the term"multiplication" to express the operation of "followed by."

The fourth and final requirement of a group is that it is associative. In other words, if we take a list of three or more group members and combine them two at a time, it doesn’t matter which end of the list we start with. Arithmetic with numbers is governed by the associative property, so if we want to do arithmetic with members of a group, we need them to be associative as well.

A group is a set of objects that conforms to the above four rules. It is worth noting that although groups obey the associative property, the commutative property generally does not apply; that is, the order in which we combine motions usually matters. For example, in the table above for the equilateral triangle symmetries notice that the rotation R1 followed by the reflection L gives the reflection M as a result, whereas L followed by R1 gives the reflection N as a result.

As a side note, specialized types of groups that do conform to the commutative property are called Abelian groups. For our purposes the current discussion will focus solely on more-general, non-commutative groups.

In examining the equilateral triangle, we saw that its symmetries formed a group. Another example of a group would be the set of integers under the operation of addition. If you add any two integers together, you get another integer, this demonstrates that this set is closed. There is an identity element, zero, that you can add to any integer without changing its value. Every integer also has an inverse. For instance, if you take positive 3 and add to it negative 3, you get the Identity, zero. (Zero, just in case you were wondering, serves as its own inverse, which is perfectly acceptable!) Finally, we know intuitively that adding more than 2 numbers gives the same result no matter how we choose to group them. For example:

(3+2) + 6 = 3 + (2 +6)

This demonstrates the associativity of the group of integers.

Group theory is very useful in that it finds commonalities among disparate things through the power of abstraction. We will explore this idea in more depth soon, but first let’s return to the concept we introduced at the beginning of this section. With all of this focus on rules and axioms, it’s easy to forget that we are chiefly concerned with understanding and characterizing symmetry in a mathematical fashion. Now that we have introduced the basic requirements of groups, we can start to characterize a wide variety of designs using groups. In the next section, we will focus on one- and two-dimensional patterns and the groups that describe them.

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Next: 6.3 Frieze and Wallpaper Groups


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