Isometry Groups

Kaleidoscope

Symmetries as Groups

The classification of the symmetry groups of plane figures is complete. Each figure is either a rosette, frieze, or wallpaper pattern, and then within these broad classes there are the cyclic and dihedral rosette symmetry groups, the seven frieze groups, and the seventeen wallpaper groups.

So far, the term symmetry group has meant the collection of all possible symmetries of a figure, or more concretely, the pattern made when all the symmetries are marked. The process began with a symmetric figure, and the figure was the focus of the investigations.

In this section, the focus is on the symmetry group itself, and so the material is more abstract than in the previous sections. A figure has symmetry when it looks the same after some sort of motion, for example after sliding or turning or flipping. Two figures which look different but have the same symmetry group can be moved around by the same set of motions, so these motions, called isometries, are the key to understanding symmetry groups.

This section begins with a discussion of isometries, then introduces the mathematical concept of a group, then puts the two together to solidify the concept of a symmetry group.

Explorations

• Composition Exploration
• Regular Triangle Symmetry Group Exploration
• Symmetry Group for Border Pattern Exploration
• More on Isometries Exploration

Isometry

A rigid motion is a motion that does not distort shape. Picking something up and moving it around is a rigid motion, but stretching or warping it is not. Because a rigid motion does not change size or shape, it is also called an isometry, from the Greek iso (meaning equal) and metry (meaning measure or distance).

We are particularly interested in isometries of the plane. The plane is a mathematical abstraction of a piece of paper — a big flat sheet. However, there are some critical differences between the plane and a real sheet of paper:

1. The plane has no thickness. Paper is thick (and not very flat either, if you look with a microscope [1]).
2. The plane does not have "sides". Paper has two sides, because it is thick. After flipping a sheet of paper, you see the other side. Flipping over the plane shows the same points as before (although mirror imaged). If it helps, imagine that any image printed on the plane soaks through to the other side.
3. The plane is infinite. Paper has edges, which makes it far easier to carry around.

To perform a rigid motion of the plane, you slide it around and/or flip it over, just like you could do to a sheet of paper. Since the plane has no edges we must illustrate isometries of the plane by showing what happens to images painted on the plane. But you should keep in mind that the whole plane is moving around — images, background, and the whole infinite sheet.

Translation	Rotation
Reflection	Glide Reflection

Any rigid motion of the plane must be one of four types, which correspond to the four types of symmetry in previous sections:

Translation

The plane is moved in one direction (the translation vector).

Rotation

The plane is rotated around a point (the center of rotation).

Reflection

The plane is reflected across a line (the reflection axis).

Glide-Reflection

A combination of a reflection and a translation, reflecting across a line while translating along that same line.

This list of four types is a complete classification of isometries of the plane. Though one might perform a rigid motion by turning, flipping, sliding and otherwise tossing around a piece of paper, the end result of all that movement must be exactly one of these four types. Proving of this is beyond the scope of this book.

Still, with the appearance of glide-reflection, it's easy to wonder why there is no glide-rotation. Using a bit of geometry, it's possible to prove this fairly remarkable fact: a combination of translation with rotation is the same thing as a rotation around a different center. Try this yourself using a piece of paper.

Stairs with helical symmetry. Treasure Island, California.

In three dimensions there is a glide-rotation, also called helical symmetry. Helical symmetry is the symmetry of a spiral staircase, a metal spring, DNA, and many viruses. The reason glide-rotations are possible in three dimensions (space) is that rotations in space happen around a line (the rotation axis), rather than around a point. A helical motion is a combination of a rotation with a translation along the same axis. In the photograph of stairs, the axis is the central support pole for the staircase.

The key point of this section is the distinction between a symmetry and an isometry. An isometry is an operation performed on an object. You can take the image of a butterfly and rotate it by 90°. On the other hand, symmetry is a property of an object: the butterfly has bilateral mirror symmetry because its left and right halves are identical.

Butterfly with symmetry	Butterfly rotated 90°

Abstract Groups

Symmetry groups are one special case of a mathematical concept called a group. Groups are fundamental objects in mathematics, showing up in almost every area of study. And though abstract, the notion of a group has a broad range of applications in other sciences as well.

A group needs two things:

• A collection of objects
• An operation

The operation takes two of the objects and produces another object in the collection, subject to the following rules:

1. There must be an identity object.
2. Every element of our collection has an inverse.
3. The operation is associative.

Example: The Integers

The integers (...,-3, -2, -1, 0, 1, 2, 3, 4,...), also called $\mathbb {Z}$ , form a group with addition as the operation. What makes $(\mathbb {Z} ,+)$ a group?

• The collection of objects is the numbers: ..., -3, -2, -1, 0, 1, 2, 3, 4, ...
• The operation is addition. Addition is the rule which tells us how to combine two numbers and produce another number.

Addition of numbers satisfies the three required properties.

1. There’s an identity object. In this example this is the number 0. You can add 0 to any number and nothing changes. We know that for any number $n$, $n+0=n$. In other examples this identity object can play quite an important role. It is some object that does not change or affect the others.
2. Every integer has an additive inverse. This is a difficult way of saying that we can undo any addition if we want to. For example: If you just added 3 and you want to undo this, you add –3. We say that –3 is the additive inverse of 3. It “undoes” the action of the number 3. Similarly the inverse of 5 is –5, the inverse of –7 is 7, and so forth. In general the inverse of the integer $n$ would be the integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle –n} . Note that if we combine a number and its inverse we get the identity object: $n+(-n)=0$. Also note that the inverse of 0 is itself: 0.
3. Addition is associative. As long as we leave the order in place we can group the numbers how we want. For instance: $2+3+5=(2+3)+5=2+(3+5)$

Example: The Positive Rational Numbers

Another example of a group is $(\mathbb {Q} ^{+},\times )$. The symbol $\mathbb {Q}$ means the collection of all positive rational numbers. Recall that a rational number is a fraction, and can be written as $p/q$. The little '+' means that only positive rational numbers are included (numbers bigger than zero). The operation $'\times '$ is multiplication. Let's see why $(\mathbb {Q} ^{+},\times )$ forms a group:

• The collections is $\mathbb {Q} ^{+}$, all positive rational numbers.
• The operation is multiplication.

Multiplication satisfies the following properties:

1. There is an identity object. Here, the identity object is 1, because multiplying by 1 doesn’t change anything.
2. The inverse of any fraction ${\frac {p}{q}}$ is just ${\frac {q}{p}}$. Note that ${\frac {p}{q}}\times {\frac {q}{p}}=1$ (a number times its inverse is the identity!)
3. Multiplication is associative: $({\frac {a}{b}}\times {\frac {c}{d}})\times {\frac {e}{f}}={\frac {a}{b}}\times ({\frac {c}{d}}\times {\frac {e}{f}})$

One important point here is that we did not include 0 in our collection of numbers. This is because zero has no inverse. Multiplication by zero cannot be undone, or alternatively, you cannot divide by zero.

Non-Examples

It is useful to look at some collections with operations that are NOT groups.

The collection of all odd numbers with addition

This fails rather spectacularly at being a group. First of all if you add two odd numbers you get an even number, which is not even in our collection of objects! Another issue is that the identity is not in the collection. The identity would have to be 0, because it is the only number you can add without changing the numerical value. But 0 is not an odd number. On the other hand, there are inverses (since the negative of an odd number is still odd) and the addition is associative, but that doesn’t matter. All requirements have to be met in order for a collection to be a group.

The integers with multiplication

Here the problem is with the inverses. With respect to multiplication, the inverse of 2 is 1/2, but 1/2 is not a member of the integers. Every object is required to have an inverse which is also an object in the group. Using mulitiplication as the operation, the integers do not form a group.

Subgroups

We looked at the collections of all odd numbers and we noticed that this was not a group. The even numbers (... , -4, -2, 0, 2, 4, 6,...) are different. We write the collection of even numbers as $2\mathbb {Z}$.

Adding two even numbers gives another even number, the identity (0) is in there, the inverse of every number is part of the collection and addition is associative, so $(2\mathbb {Z} ,+)$ is a group. Because the collection of even numbers is a sub-collection of the integers we say that $(2\mathbb {Z} ,+)$ is a subgroup of $(\mathbb {Z} ,+)$.

A subgroup is a subset of the original group which on its own also satisfies all the requirements to be a group.

The collection of just the identity element will always be a subgroup. So ${0}$ is a subgroup of the integers. This subgroup is not terribly interesting, but it does always exist.

The other extreme is also true. The entire group is technically a subgroup of itself. So $(\mathbb {Z} ,+)$ is a subgroup of $(\mathbb {Z} ,+)$. This also does not give us much information.

Scientists are often interested in the subgroups that are neither trivial nor the whole group. The fact we found one subgroup of $\mathbb {Z}$ suggests that there may be other groups embedded in $(\mathbb {Z} ,+)$. You should be able to convince yourself that the multiples of three (...,-9, -6, -3, 0, 3, 6, 9,...), written $(3\mathbb {Z} ,+)$, is also a subgroup of $(\mathbb {Z} ,+)$.

Even more interesting is the fact that $(4\mathbb {Z} ,+)$ = ...,-8, -4, 0, 4, 8, 12,... is a subgroup of both $(\mathbb {Z} ,+)$ and $(2\mathbb {Z} ,+)$.

Symmetry Groups

Plane Filling Motif with Reptiles

The power of the abstract concept of a group is that it applies to far more than numbers and numerical operations such as addition and multiplication. In particular, the symmetry groups of figures are also abstract groups, meaning that they consist of objects and an operation satisfying the rules for a group.

To begin, consider the design by Escher - made up of two lizards - shown to the right. The design has C2 symmetry (ignoring color). Each symmetry of the figure corresponds to an isometry that preserves the figure. One symmetry, the 180° rotation, will be called R. The second object in the group of symmetries is the identity object, called E, which is the isometry 'do nothing', a rigid motion that involves no motion at all. These two isometries, E and R, are the objects in the symmetry group.

The operation that combines two isometries to give another isometry is called composition. It means to perform one motion followed by the other. In this text, the composition operation is written as 'x', a multiplication symbol. So, R x R means rotate, then rotate again.

When writing compositions, the order is important. This is difficult to get used to, especially since you are accustomed to switching the order of numbers when performing multiplication of numbers. In this book, isometries are performed from right to left, so that E x R means first rotate, then do no move, while R x E means to do nothing, then rotate. A little silly in this example, maybe, but an important distinction later.

So, in summary, the symmetry group C2 has two objects, E and R, and the operation composition, written x.

The first thing to notice is that all the compositions result in another object in our collection. For instance E x R = R, R x R = E, R x E = R, and E x E = E. The operation is concisely summarized by a 'multiplication' table for the objects in the group.

Multiplication table for the symmetry group C2

	E (identity)	R (reflection)
E	E	R
R	R	E

Next, we check the three properties that the operation should have:

1. There is an identity object, E. Clearly, 'do nothing' has no effect on other motions.
2. There are inverses: The inverse of E is clearly E. What is the inverse of R? If you rotate by 180°, how would you return the figure to its starting position? The answer is to rotate 180° again. This means that the inverse of R, written $R^{-1}$, is R again. In formal notation, $R=R^{-1}$, or R x R = E.
3. The operation is associative. This is a bit painstaking to fully check. It is easy to see that any expression involving E's simplifys quickly. So, we only check that (R x R) x R = R x (R x R). On the left, (R x R) x R = E x R = R and on the right, R x (R x R) = R x E = R, so the two expressions are equal.

These three properties mean that C2 is a group with two objects and the operation of composition. The collection of symmetries of any pattern, including rosette, frieze, and wallpaper patterns, also form groups in this way.

Example: The symmetry group of a rectangle

A rectangle has D2 symmetry, and the figure below shows it's three symmetries:

There are three corresponding isometries of the plane that move the rectangle back onto itself. The isometry group of the rectangle contains four elements: the reflection $M1$, the reflection $M2$, the rotation $R$, and finally the identity $E$, the isometry which does nothing.

The operation, as it was for C2, is composition, written 'x'. Consider $M2$ x $M1$. First, recall that composition is read from right to left, so that $M2$ x $M1$ means to perform $M1$ followed by $M2$.

Note that $M1$ reflects the rectangle over the vertical mirror line, and thereby switches B and C, and switches A and D. Note that the resulting labeled rectangle is the same we would get if the rectangle had been rotated through 180 degrees. This implies that $M2$ x $M1$ = $R$. Similarly, it is easy to show that $M1$ x $M1$ = $E$, and that $M2$ x $M2$ = $E$. Reflecting twice over the same mirror line gets us back to our original labeled rectangle. This means that reflecting over the same mirror line twice is the same as doing nothing. In exactly the same way $R$ x $R$ = $E$. Below is a multiplication table for this group. If you want to know the result of $M2$ x $R$, go over in the row for $M2$ and down in the column for $R$ and look at where the row and column intersect. In this example you find $M1$ in that spot. This means that $M2$ x $R$ = $M1$. If you wish to find $R$ x $M2$, then go to row $R$ and column $M2$, and you find that $R$ x $M2$ = $M1$.

Multiplication table for the symmetry group D2

	E	M1	M2	R
E	E	M1	M2	R
M1	M1	E	R	M2
M2	M2	R	E	M1
R	R	M2	M1	E

The symmetry group D2 (the rosette symmetry group for the rectangle) is a group consisting of four elements, and the multiplication table is as above. We would say that D2 = {$E$, $M1$, $M2$, $R$}

Example: The symmetry group of a square

The symmetry group of the square is D4. All possible rotation and reflection symmetries of the square are shown below:

The labels on the square represent the labels after the isometry has been applied to the square.

• Note that there are eight basic elements in this group: $E$, $R$, $R^{2}$, $R^{3}$, $M1$, $M2$, $M3$, and $M4$ (where $R^{2}$ denotes twice the rotation $R$, i.e. a rotation by 180 degrees, and $R^{3}$ is $R$ performed three times; this same notation applies below).
• If we apply any of the reflections twice, we get back to our original square. This means that $M1$ x $M1$ = $M1^{2}$ = $E$, $M2$ x $M2$ = $M2^{2}$ = $E$, and similarly for $M3$ and $M4$.
• This shows that the inverse of $M1$ (written $M1^{-1}$)--meaning, that which undoes it--is $M1$ itself. A similar fact is true for the other reflections: The inverse of any reflection is that reflection. If you have applied a reflection, and you want to return to the original position, then you just apply the same reflection again.
• You can check that $M2$ x $M1$ = $R^{3}$, and $M1$ x $M2$ = $R$. Note that the order in which you apply the two operations ($M1$ and $M2$) matters. We say that the isometries are non-commutative. This is very different from how regular numbers behave: 2x3 = 3x2, and it doesn’t matter in what order you multiply a string of numbers; but with isometries, the order in which you apply them makes a big difference.
• Below you see $M2$ x $M1$ worked out

• Lets look at the rotations. If you take $R$ (90° rotation), apply another rotation $R$, then you get $R^{2}$ (180° rotation). Another rotation gives you $R^{3}$ (270° rotation), and one more rotation gets you back to $E$, the identity. The rotations form a subgroup: {1, $R$, $R^{2}$, $R^{3}$}. Not every collection of elements will be a subgroup. If you take any two elements from the subgroup and multiply them you should get another element from the subgroup. In this case you can check that any two elements from the subgroup {1, $R$, $R^{2}$, $R^{3}$} will give another element from the group. For instance: $R$ x $R^{2}$ = $R^{3}$, $R^{2}$ x $R^{2}$ = $E$, $R^{3}$ x $R^{3}$ = $R^{2}$, and so on. Note that a group with only rotational symmetries is a cyclic group. This implies that there is a C4 group sitting inside D4!

Test your understanding by completing the Regular Triangle Symmetry Group Exploration

Example: The symmetry group of a frieze pattern

The frieze pattern below has symmetry group pmm2:

Three symmetries of this pattern are marked on the figure. They correspond to isometries which are:

• $MV$ - reflection across a vertical mirror line.
• $MH$ - reflection across the horizontal mirror line.
• $T$ - horizontal translation one unit to the right.

These are not all the symmetries of the pattern, and the symmetry group contains many more isometries than just these three. However, every isometry in the group can be written in terms of these three.

Translation is an interesting isometry here. If you apply $T$ twice, $T^{2}$, you translate two circles to the right. . If you apply $T$ three times, $T^{3}$, you translate three circles to the right. You can keep translating without ever coming back to the original spot. The inverse of $T$, $T^{-1}$, just translates to the left. Similarly, you can keep translating to the left without ever returning to your starting point.

Composing $T$ repeatedly gives infinitely many translations, { …, $T^{-3}$, $T^{-2}$, $T^{-1}$, $E$, $T$, $T^{2}$, $T^{3}$, …}. This list is the 'translation subgroup of pmm2. This subgroup consists of the isometry $T$ along with all of its powers, so it is called a cyclic group, even though it has infinitely many elements. This group is called the infinte cyclic group.

Notice that $MV$ x $MH$ = $MH$ x $MV$. Also, $MV$ x $MV$ = $E$, and $MH$ x $MH$ = $E$. We have found two subgroups of pmm2: {$E$, $MV$} and {$E$, $MH$}. Both of these are cyclic subgroups. The number of elements is two, hence they are examples of C2 groups as subgroups of pmm2..

There are also 2-fold rotations in this symmetry group; in fact, there is a one group element for each rotation center. Also, because we can combine $T$ (or any power of $T$) with $MH$, the group contains glide-reflection isometries. When classifying symmetries of patterns, we were careful to only identify glide reflections which did not decompose into reflection and translation symmetries also present in the pattern. This simplified the task of identifying symmetry type.