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Mathematics Illuminated

The Beauty of Symmetry Online Textbook

The mathematical study of symmetry is the rigorous study of the commonalities between objects or situations.

1. Introduction

“The Theory of Groups is a branch of mathematics in which one does something to something and then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing…”

-James R. Newman

Why do we find butterflies appealing? What is it about a snowflake that can hold our attention? Why do we find some designs beautiful and others unattractive? Such subjective observations can hardly be thought to be within the realm of the objective explanatory power of mathematics, and yet the concept of “symmetry,” an idea that underlies what humans consider to be beautiful, drives right to the heart of mathematical thinking.

EMBROIDERED SUSANI

Item 2875 / Uzbek, EMBROIDERED SUSANI (nineteenth-twentieth century). Courtesy of Kathleen Cohen.



FIERY ORNAMENTAL PATTERN

Item 2925 /Tony Link Design, FIERY ORNAMENTAL PATTERN (2007). Courtesy of iStockphoto.com/Tony Link Design.

CUPOLA

Item 2926 /Adam Mandoki, CUPOLA (2007). Courtesy of iStockphoto.com/Adam Mandoki.



equilateral triangleThe mathematical study of symmetry is the rigorous study of the commonalities between objects or situations. For example, a plain equilateral triangle with no distinguishing markings to differentiate one side or angle from another has six geometric symmetries. These result from actions, or transformations, such as flipping or rotating the triangle, that leave it looking the same as when we started. Furthermore, we can compose symmetries to make other symmetries.

Interestingly, the ways in which we can move an equilateral triangle and have it appear as it did when we started have much in common with the ways in which we can rearrange a stack of three objects, such as pancakes. pancakesThis similarity is not accidental; rather, it is an indication of some deeper concept in action, one that transcends both equilateral triangles and stacks of pancakes. Mathematicians call objects that adhere to this deeper concept a group.

As we will see, the study of groups is one way mathematicians extend the notion of arithmetic to objects that are “beyond numbers.” Group theory unites the spatial thinking of geometry with the symbolic realm of algebra. It is beautiful in its generalization and abstraction, providing deep insights into a wide array of phenomena. In addition to its theoretical power and usefulness, the study of groups also has many applications to the real world. The techniques of group theory can be used to study the solvability of certain kinds of equations, explain the existence of elementary physical particles, verify the fact that physical laws are the same everywhere on the planet, or determine when a deck of cards is sufficiently shuffled. These applications are, of course, in addition to the fascinating use of group concepts and techniques in the traditional art forms of various cultures.

Apart from its applications, the study of symmetry and groups reveals deep and surprising connections between different areas of mathematics itself. Group theory is part of a larger discipline known as abstract algebra. Abstract algebra takes the skills and techniques learned in high school algebra—related to how the system of numbers works in a general sense through the use of variables— and extends these tools into the realm of geometry and beyond. Abstract algebra shows the way in which incredibly interesting and complex structures can be created simply by putting a few rules in place as to how a small group of symbols can be moved around and related to one another. In this symbolic and abstract setting we find a unity among things that, on the surface, may seem quite different. Just as the concept of bilateral symmetry unites the forms of butterflies, airplanes, and humans, group theory and abstract algebra show a relationship between the fundamental structures of logic and mathematical reality.

3D SCHEME

Item 3078 /Andrey Prokhorov, 3D SCHEME (2007). Courtesy of iStockphoto.com/Andrey Prokhorov.

In this unit we will start by examining different types of visual symmetry, and we will see how the concept of groups enables us to classify different design motifs in both one and two dimensions. From there we will examine the symmetries inherent in permutations, such as the ways in which one can stack pancakes or shuffle cards. With these concepts in hand, we will catch a glimpse of one of the crown jewels of abstract algebra, Galois theory. We will then shift gears to study the role that symmetry plays in our understanding of the physical universe.

2. Types of Symmetry Leading to Groups

FOUNDATIONS OF BEAUTY

  • Symmetry is a basic notion in the visual arts.The Ishango bone represents a level of early mathematics more sophisticated than simple counting.
  • Rotation, reflection, and translation are the most common types of visual symmetry.

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, HumanNote 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 it’s 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.

THE EQUILATERAL TRIANGLE

  • The symmetries of the equilateral triangle can be thought of as the transformations (i.e., motions or actions) that leave the triangle invariant— looking the same as before the motion.
  • Combinations of symmetries are also symmetries.

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.

STAY WITH THE GROUP

  • A group is a collection of objects that obey a strict set of rules when acted upon by an operation.

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.

3. Frieze and Wallpaper Groups

GROUPS IN VISUAL FORM

  • The symmetries of an infinite sine wave for a group.

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.

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?

moving sine wave

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.

sine wave reflection

If we reflect the sine curve over the vertical line x = pi over 2, however, the original look of the design is retained. So, this design remains invariant when reflected over vertical lines defined by x = pi over 2 plus (n)(pi). 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.

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.

translation

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.

leaving design invarient

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:

G*(T*(R*S))

G*(T*S)

G*S

S

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.

FRIEZES

  • Friezes are patterns along a line, which are commonly used in art.
  • All friezes fall into one of seven general types, each of which is a mathematical group.
ALEXANDER SARCOPHAGUS

Item 2876/Greek Hellenistic, ALEXANDER SARCOPHAGUS. DETAIL: DECORATIVE FRIEZES. (fourth century BCE). Courtesy of Kathleen Cohen.

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).

frieze symmetries

wallpaper patternMoreover, 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.

wallpaper group

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!

wallpaper group

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.

4. Card Shuffling

PERMUTATIONS

  • The symmetries of a geometric object can be expressed as permutations of elements.
  • Permutations can form a group, as symmetries do.

Let’s return to the equilateral triangle group that we studied earlier. We have been saying all along that the power of group theory lies in its capacity to reveal deep connections between seemingly unrelated things. We can catch a small glimpse of this by re-examining the motions that make up the equilateral triangle group.

Remember, to keep track of the reflections and rotations that make up the equilateral triangle group, we labeled the vertices A, B, and C. We can express all of the symmetries shown earlier by using just these labels. As a convention, we’ll express each configuration of the triangle as some sequence of A, B, and C, with the first letter of each sequence corresponding to the vertex at the top of the triangle, the second letter representing the right vertex, and the third letter representing the left vertex. Basically, this is just a clockwise sequence, starting from the top. Applying this convention yields the following sequences, shown corresponding to the motions that produced them.

motion sequence

Each of these motions has the effect of permuting the labels of the vertices. For example, a reflection over a vertical line through the top vertex of the equilateral triangle keeps A in the same (first) position in the label sequence, while switching the positions of vertices B and C. This motion takes ABC and gives us ACB.

Perhaps we don’t really even need an actual triangle to have a group. In fact, let’s forget about the triangle for a moment and simply look at the ways in which we can arrange three objects. Just for fun, let’s think of this as the ways in which we can stack three different-flavored pancakes, one apple (A), one banana (B), and one chocolate (C). We can use combinatorics to figure out how many different arrangements there are. Because these are ordered permutations of three objects, we know there will be 3! (3 × 2 × 1) or six arrangements. They are:

ABC
ACB
BAC
BCA
CAB
CBA

If we look at these sequences in terms of how such permutations are created, instead of simply enumerating the specific arrangements, we will see an interesting result. For the sake of simplicity, let’s say that each permutation action starts on ABC. Here are our actions:

permutations

Let’s see if this set of permutations forms a group under the operation of “followed by.” It’s clear that we have an Identity motion (P6). The easiest way to check for closure, inverses, and associativity will be to form a table, as we did with the equilateral triangle group. Note that as we perform these combined movements, the motions represented across the top row of the table are done first, followed by the motions in the left column.

For instance, P5 switches A with C and C with A while leaving B untouched. Doing this twice in succession yields the same result as doing nothing, so P5 serves as its own inverse. The same can be said for P1 and P2. What happens when we perform the P3 motion twice in succession? P3 replaces A with B, B with C, and C with A. Starting from ABC, P3 creates BCA, so applying this transformation to BCA yields CAB, the same result obtained from performing P4 once. In other words, P3 followed by P3is equivalent to P4, which demonstrates that P3 is not its own inverse. However, a little bit further examination reveals that P4 undoes the changes that P3 creates, and, thus, serves as its inverse.

permutations

As we fill in this table, we find that every combination of two permutations gives another permutation, thereby confirming that the set of permutations of three objects is closed. Furthermore, we can see that each permutation has an inverse. Finally, by using the reasoning of the previous examples, namely that we should be able to combine motions in any way we choose without affecting the result, we can convince ourselves that the elements are associative under the operation “followed by.”

Having confirmed that the four requirements are met, we see that the set of permutations of three objects indeed forms a group. In other words, the possible ways in which we can stack three distinguishable pancakes forms a group. It’s interesting to note that the number of elements in this group is the same as the number of elements in the group of symmetries of the equilateral triangle—six. Furthermore, if we examine the table we obtained for the triangle with the table we obtained for the permutations, we find that they have identical structure. This suggests to us that these two situations are both manifestations of the same fundamental structure. In other words, what do equilateral triangles have to do with stacks of three pancakes? They both have the same group structure!

This is, of course, no coincidence and is merely an example of a much broader observation, first documented by Arthur Cayley, an influential British mathematician working in the late 1800s. Cayley proved that every finite group of symmetries can be represented by a group of permutations. When two groups, such as a group of symmetries and a group of permutations, have the same structure, we say that they are isomorphic to each other.

Now, to be clear, this does not mean that the group of symmetries of a square, for instance, will have the same structure as the set of all permutations of four objects. This is patently obvious from the fact that a square has eight symmetries, whereas there are twenty-four possible permutations of four objects.

permutations

Cayley’s Theorem does assure us, however, that there is some group of permutations that contains a subgroup (that is, a subset of the group that does itself satisfy all the group axioms) that corresponds with the eight symmetries of a square. In this case, the group of twenty-four permutations of four objects suffices as the broader group that contains the corresponding subgroup. This connection between geometric symmetries and permutations illustrates but one example of the connecting power of group theory.

THE PERFECT SHUFFLE

  • The symmetries of permutations are evident in card shuffling.
  • Eight perfect “riffle” shuffles will return a standard deck of cards to the order in which it started.

Let’s consider the connection between permutations and group theory in a little bit more detail. In the example above, we considered permutations of three objects, modeled by a stack of pancakes. Increase the number of pancakes to 52 and we can (and probably should!) shift our thinking from food to cards. We may not normally consider a deck of cards to have symmetry, but if we view symmetries as permutations, and permutations as shuffles, we see yet another fundamental connection made possible by group theory.

As it turns out, we can use group theory to determine when a deck of cards has been sufficiently shuffled. Let’s take a step back, though—generally, when we think of a group, we’re thinking of some kind of symmetry, and to have a symmetry, something must remain invariant. So, what remains invariant when a deck of cards is shuffled? Isn’t the point of shuffling to mess everything up? When we shuffle a deck of cards, what remains invariant is the deck itself. Although it will probably end up in a different order than when we started, it remains a deck of cards.

perfect shuffle

While there are only six permutations of three pancakes, you might well imagine that there are a staggering number of permutations of a deck of 52 cards. A good shuffle would make each of the billions of possible deck orderings equally likely. Is there a technique that can accomplish this?

If you were able to perform a perfect shuffle, one in which the deck is cut into two stacks of 26 cards which are then riffled together one card at a time alternately from each stack, the result would be anything but random. The cards would be interleaved in some mathematically predictable way. The easiest way to analyze this is to assume that the deck you start with is in ascending order (2345678910JQKA) for each suit. After one perfect shuffle, this sequence will be messed up, but in a very particular way. To get a better idea of what happens, let’s simplify our example to just the lowest ten cards of a suit, with the ace designated as “1.”

After a perfect cut, we have two stacks, 1 2 3 4 5 and 6 7 8 9 10. Riffling these together perfectly, starting with the left stack, gives 1 6 2 7 3 8 4 9 5 10. If we were to perform a second perfect riffle shuffle, the cut would give us:

1 6 2 7 3 and 8 4 9 5 10

. . . and riffling these together would give us: 1 8 6 4 2 9 7 5 3 10.

Cutting again gives us:

1 8 6 4 2 and 9 7 5 3 10

. . . which gives us 1 9 8 7 6 5 4 3 2 10 when riffled perfectly back together. Note that all of the sequences of ten cards that we have seen so far are quite predictable provided we know the starting order and perform perfect riffle shuffles every time.

Cutting once again gives us 1 9 8 7 6 and 5 4 3 2 10, which, combined, yield the sequence 1 5 9 4 8 3 7 2 6 10.

In yet another perfect shuffle: 1 5 9 4 8 and 3 7 2 6 10 come back together in the sequence 1 3 5 7 9 2 4 6 8 10.

Okay, just once more. When riffled perfectly, this last action produces the sequence 1 2 3 4 5 6 7 8 9 10, which was our original ordering! We have found that six perfect riffle shuffles return our ten-card deck to its original ordering. In addition to our previous observation that perfect riffle shuffles do not randomize a deck, we now see that if you perform six of them, the deck actually isn’t shuffled at all! You might think that a full deck of 52 cards would take many more shuffles than this to achieve the same result, but actually it only takes eight!

This means that, if we want to shuffle a deck of cards in a way that is mathematically unpredictable—in other words, truly random—then we must be imperfect shufflers. We should make a certain amount of errors, but not too many, in our shuffling process. Each of these errors will propagate through other shuffles until the deck is truly randomized. The number of shuffles required to randomize a full deck of 52 cards, depending on our measure of randomness (and that’s a whole other story!), is then either six or seven.

The unexpected connection between symmetries, permutations, and card shuffling is just a small sampling of the broad explanatory power of group theory. Symmetry and group theory not only tell us about patterns and shuffles, they can also be used to tackle some of the more abstract challenges that arise in mathematics itself. One of the classic examples of this is the story of Evariste Galois and the remarkable connections he made in an attempt to solve a problem that had vexed some of the greatest mathematical thinkers for centuries.

5. Galois Theory

EQUATIONAL SYMETRY

  • The study of permutation groups is related to the roots of algebraic equations.

The study of groups is part of the larger discipline called abstract algebra. We have seen how group structure represents an abstraction of two seemingly different situations: an equilateral triangle and a stack of three pancakes. When we think of algebra, we normally think of typical algebraic problems such as”solve 3x +5 = 7″. We use these types of algebraic statements to make general observations about how our number system works. To do this, we use variables to represent unknown numbers, thus freeing us from the constraints of specific numerical values so that we can see commonalities in different types of mathematical expressions and equations. For instance, we know that equations of the form y = mx + b all have something in common:, they give us straight lines that are completely characterized by their slope and y-intercept.

Certain equations have symmetry in the form of invariance under permutation. In other words, we can tell something about an equation in two variables by seeing what happens when we swap the variables. For example, the equation y = –x + 5 has the following graph:

y = -x + 5

If we permute x and y (i.e., swap their positions), we get x = – y + 5, which has the following graph:

x = -y + 5

The two graphs are the same! This is because the equation y = –x + 5 is invariant under permutation of x and y. By contrast, the equation y = x2 + 1 has the following graph:

y = x squared + 1

The graph of its permutation, x = y2 + 1, looks like this:

x = y squared + 1

This altered look shows that y = x2 + 1 is not invariant under permutation of x and y. This notion of permuting parts of equations will play a role as we attempt to answer a question similar to “what makes an equation solvable?”

GALOIS

  • Many mathematicians, including Evariste Galois, tackled the question of whether or not a polynomial has a general solution by radicals.

If we combine the notions of abstraction seen in groups with the arithmetical power of algebra, we get a system of thought that brings the explanatory power of mathematics to things that are not numbers, such as the motions of symmetries and permutations. In short, abstract algebra is the study of how collections of objects behave under various operations. Its power lies in its generality.

Surprisingly, things such as symmetries and permutations can be used to understand problems from the realm of what we normally consider algebra. One of the first people to realize the vital link between symmetry and algebra was a 19th century French mathematician named Evariste Galois.

Galois is a legendary figure, known for both his extraordinary insight and his dramatic life. A revolutionary, Galois was as much obsessed with politics as with mathematics, to the point that he was expelled from school. He spent time in prison, led protests, and all the while continued to do groundbreaking mathematics. Shot fatally in a duel at the age of 20, the young mathematician is said to have written down all of his mathematical knowledge in a letter the night before his demise. Whether or not this tale is true, the creative insight that Galois brought to mathematics is difficult to overstate.

Galois made an astonishing breakthrough while attempting to resolve one of the great questions of his age. Mathematicians had for centuries been trying to find general formulas that could give the roots to any polynomial using only the coefficients in the polynomial. Recall that a polynomial is a simple function that is a linear combination of powers of the input. For example, a quadratic equation is a second-degree polynomial such as p(x) = 3x2 + 7x – 2. The “roots” of p(x) are precisely the variable inputs that result in p(x) being 0. In this case, the famous quadratic formula tells you what the roots are in terms of the coefficients. Given a general quadratic equation:

p(x) = ax2 + bx + c

. . . the roots are:

quadratic equation and quadratic equation

Similar formulas exist for third-degree polynomials (cubics) and fourth-degree polynomials (quartics). For some time it had been a question of whether or not a fifth degree polynomial, or “quintic,” was solvable, in the sense of having a similarly simple expression for the roots in terms of coefficients, square roots, cube roots, etc. and simple arithmetic. After years of work, mathematicians were able to find solutions only for specific cases, the simplest of these being quintic equations of the form ax5 + b = 0, for which the solution would be the fifth root of b/a. Solutions that hold only for specific cases, however, are a far cry from the complete mastery that a general solution implies.

PERMUTING ROOTS

  • The general quintic equation cannot be solved by a formula.
  • Galois theory can determine the solvability by formula of an equation by examining the ways that permutation groups of its roots behave.

In 1824, the Norwegian algebraist Niels Henrik Abel published his “impossibility” theorem in which he proved that there is no general solution by radicals—no nice formula, in other words—for polynomials of degree five or higher. Earlier, in 1799, the Italian mathematician Paolo Ruffini had published a similar finding, though his proof was somewhat flawed. The fact that no general solution exists for quintic or higher-degree polynomials is now known as the “Abel-Ruffini Theorem” or, alternatively, “Abel’s Impossibility Theorem.” The methods that Galois used to reach a similar conclusion were more general, opening new doors to mathematical exploration. Where Abel and Ruffini showed simply that quintic and higher-degree polynomials could have no general solution by radicals, Galois showed why. Furthermore, his contribution was general enough to explain why polynomials of degree four and lower have general solutions and why those solutions take the form that they do. His ideas form the basis for what we now call Galois theory, the basis of modern group theory.

Galois’ epiphany was to consider the symmetries of the roots of a polynomial. He discovered a set of conditions in terms of these symmetries that would determine if that polynomial had a solution by radicals. To do this, he worked backwards. Instead of starting with a polynomial and trying to find its roots, he started with a set of roots and looked to find the polynomial that they would form.

For example, we can use the roots -1, -2, -3, -5, and -7 to construct a polynomial and find its coefficients. To do this, we can recognize that these roots correspond to the following binomial factors:

(x+1)(x+2)(x+3)(x+5)(x+7)

When multiplied together, these factors produce the quintic polynomial expression:

x5 + 18x4 + 118x3 + 348x2 + 457x + 210

Galois studied the conditions under which the coefficients and roots were related in such a way as to permit a solution by radicals. To do this, he started with a set of roots and combined them with the rational numbers to serve as basic building blocks for creating other, somewhat arbitrary, numbers using multiplication and addition. Using these blocks, he identified a series of key equations relating the roots and examined how those equations behaved when the roots were permuted—i.e., shuffled in a way similar to the equations and graphs we saw in the previous section.

Galois discovered that the ability to solve a polynomial built out of a given set of roots depends on the invariance under permutation of the roots of those key equations, created from the roots. In essence, he saw that certain symmetries in a polynomial’s roots determined whether or not that polynomial has a nice solution.

Galois not only resolved one of the great challenges of his day, he discovered an important connection between symmetry, permutation groups, and solvability of equations. Galois’ discovery is perhaps one of the more unexpected results in mathematics and shows yet again how group theory provides a way to see past the superficial in order to find underlying connections. It should come as no surprise then that group theory has played an important role in understanding not only the artistic, gambling, and mathematical worlds, but also many underlying connections inherent in the physical world.

6. Physics

PLEASE BE DISCRETE

  • Discrete symmetries are responsible for physical invariance.
    CPT-symmetry is a fundamental prediction of the standard model of particle physics and is due to the interaction of three kinds of discrete symmetries.

The idea of symmetry plays an important role in physics. It primarily manifests as the concept of invariance, the idea that certain quantities or properties do not change under certain actions. A famous invariant is energy; the energy of a closed system does not change. This is more commonly understood to mean that energy is neither created nor destroyed. Conservation laws abound in physics and have an important connection to symmetry. To get an idea of the ubiquitous role of symmetry in physics, we have to consider both discrete and continuous symmetries.

Up until this point, we have really only considered discrete symmetries, such as those of the equilateral triangle. An object with discrete symmetries has motions that leave it invariant that cannot be smoothly turned into one another. In other words, with our triangle, both 120° and 240° rotations leave it invariant, but all of the rotations between 120° and 240° are not symmetries, so we cannot smoothly change one symmetry into another. Continuous symmetry, on the other hand, can be seen in the rotation of a circle; every rotation can be smoothly turned into every other rotation while the circle remains invariant. Both discrete and continuous symmetries are important in physics. Let’s first look at the discrete type.

An important discrete symmetry in physics is actually made up of three situations that are thought to remain invariant. The first of these is the conjecture that the universe would behave the same if every particle were interchanged with its anti-particle. For instance, if we were made of antiprotons and positrons instead of protons and electrons, we would not be able to observe any difference. This swapping is called a charge-conjugation transformation, and like permuting pancakes, can be thought of as a motion that leaves the initial system invariant. It is referred to as C-symmetry.

The second idea is that the universe would behave the same if left and right were interchanged. This is known as a parity inversion and is basically what we observe in a mirror. The idea that the mirror universe behaves no differently than our own is known as P-symmetry. Unfortunately, both Cand P-symmetries do not always hold. There are certain situations in which inverting charge, orientation, or both, leads to a different outcome than if the inversions had not occurred.

The third discrete symmetry is that of time reversal. Now, over long scales, this idea is nonsense; of course the future is distinctly different from the past. However, if one considers two billiard balls colliding (ignoring friction), the incoming speeds and angles of the balls are the same as the outgoing, so if this collision were run backwards in time, like rewinding a videotape, we would not be able to tell. This is called T-symmetry, and it is less general than both C- and P-symmetries. In fact, T-symmetry is, by itself, not really true, but something fascinating happens when all three symmetries, C, P, and T, are considered together.

Each of the C-, P-, and T-symmetries acting alone, or even any two of them acting as a pair, do not leave a physical system invariant. However, these “broken” symmetries tend to cancel each other out when all three are taken together in what is known as CPT-symmetry. Basically, what this means is that if every particle were swapped out with its anti-particle, and all coordinates were inverted, and time was run backwards, the universe would behave no differently than it does now.

CPT-symmetry is a fundamental prediction of the Standard Model of Particle Physics. The standard model predicts what kind of particles should and do exist, as well as their properties such as mass, charge, and spin. This model has been remarkably accurate in its ability to explain the interactions of all particles observed so far, not only for protons, neutrons, and electrons, but also more fundamental particles such as quarks and neutrinos.

CONTINUITY AND CONSERVATION

  • Noether’s theorem implies that every conserved physical quantity is based on a continuous symmetry.

Let’s now turn our attention to continuous symmetries. Recall that something possessing continuous symmetries can remain invariant while one symmetry is turned into another. Space itself, or more precisely, spacetime (the combination of both space and time) possesses such continuous symmetries. For instance, if two billiard balls collide in one location, I expect the result would be no different than if they had collided a foot to the left, or a centimeter to the left, or a micron to the left. The collision remains invariant under translations of any magnitude in spacetime. This is a continuous symmetry.

A remarkable theorem, proved at the beginning of the 20th century by Emmy Noether, has the physical consequence that for every continuous symmetry in spacetime, a quantity is conserved. Noether was a German mathematician and theoretical physicist who made fundamental contributions in both physics and algebra and greatly expanded the role of women mathematicians in Germany. In 1933, despite her accomplishments, she was forced to flee Nazi Germany because she was Jewish. Once safely in the United States, she taught at Bryn Mawr and also lectured at Princeton’s Institute for Advanced Studies until her death in 1935.

According to Noether’s theorem, every quantity that we consider to be conserved corresponds to an underlying symmetry of spacetime. For instance, a common notion in physics is that momentum is conserved, such as in the collision of two billiard balls. This conserved quantity stems from the fact that spacetime has continuous spatial translational symmetry, or put in other terms, “every location is just as good as every other location.”

Conservation of energy, the law that helps roller coasters to function, stems from the idea that if we conduct an experiment at one particular time and then conduct the same experiment under the same conditions ten minutes later, we should not expect to find a different result. Conducting an experiment ten minutes later is what we call a translation in time. It’s basically like when we imagined picking up our sine wave frieze motif and shifting it to the right, except that now we are shifting an event forward in time. Noether’s theorem implies that this time-translational invariance is what gives rise to the law of conservation of energy.

Finally, the fact that spacetime is invariant under continuous rotations gives rise to another important conserved physical quantity. If we ignore things such as stars, planets, people, and dust and simply focus on perfectly featureless spacetime, we would find that every direction is just as good as (i.e., equivalent to) every other direction. This is rotational invariance and it, like the space and time translations, is both continuous and gives rise to its own conserved quantity, angular momentum in this case. Conservation of angular momentum, for example, is why a figure skater can spin faster if she pulls in her arms.

Noether’s theorem is yet another example of how symmetries show profound connections among seemingly disparate ideas. It is this mysterious power to bring some rhyme and reason to our world that compels mathematicians to study the structure of symmetries and groups. There are undoubtedly many surprising connections left to uncover via the power of group theory.

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