Isometry Groups

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Kaleidoscope

We have looked at three different types of symmetry groups: rosette groups, border patterns, and wallpaper patterns. The next obvious questions are: “What is a symmetry group?”, and “What do we use these groups for?” The last question is maybe the easiest to answer. When we study animals or plants, we divide them into similar subgroups. Animals and plants allow a quite sophisticated classification. Similarly, when we study patterns we divide them up into smaller groups. In this instance we study rosette, border and wallpaper patterns. These patterns can be subdivided according to their symmetries. If we compare it to biology, it is like knowing the genus, and trying to determine the species.

Explorations

After reading the text below further questions can be found here:

Isometries

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.

In this course we focus on four isometries of the plane - also called rigid motions of the plane.

  1. Translation: An object is moved along a vector.
  2. Rotation: An object is rotated about a center of rotation.
  3. Reflection: An object is reflected over a mirror line.
  4. Glide-Reflection: This is a combination of a reflection and a translation. You reflect over a mirror line, then translate along a vector in the direction of the mirror line.

There are very nice illustrations of these four isometries in Visions of Symmetry. On page 34 you will find the ilusttrations Escher used to illustrate these concepts.


In the previous sections, we said that a figure has symmetry if it looks the same after being rotated or being reflected over a mirror line.

A very common question is: "What is the difference between a rotation (as an isometry) and rotational symmetry?" Or similarly: "What is the difference between a reflection (as an isometry) and refelectional symmetry?". The answer is that a rotation is an operation we do on an object. We can take the image of a bird and rotate it about a center of rotation. Rotational symmetry is a property of a pattern. We have rotational symmetry is we can take the pattern, rotate it through some angle and have the resulting pattern line up exactly with our original. Similarly a reflection is an operation we perform and reflectional symmetry is a property a pattern could have.

Why isn't there are glide-rotation, or a rotation-reflection? This is a good question. In fact, in three dimensions there is a glide-rotation, or helical symmetry which is the symmetry of a spiral staircase, a metal spring, DNA, and many viruses. Helical symmetry is a combination of a rotation around a line while at the same time translating along the line. However, neither glide-rotation nor rotation-reflection occur in flat images.

Definition of a Group

Introduction

In mathematics we study something called a group. You all actually know an example of a group already. The integers with addition, <math>(Z,+)</math>, is a mathematical group. What makes it a group? The following properties make <math>(Z,+) = </math>the integers with addition into a group:

  • We have a collection of numbers ...<math>{, -2, -1, 0, 1, 2,}</math>...
  • We have an operation: addition. This is a rule which tells us how to combine two numbers in our collection.
  • This group satisfies four different properties.
  1. Our collection is closed under addition. What this means is that if we add any two integers together we get another integer. As long as we stick to addition we can never fall outside our collection that we started with.
  2. 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.
  3. 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? Then just 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 –N, and note that if we combine the integer and its inverse we get the identity object: N + (-N) = 0. Also note that the inverse of 0 is itself: 0.
  4. Addition is associative. As long as we leave the order in place we can group the numbers how we want. For instance: <math>2 + 3 + 5 = (2 + 3) + 5 = 2 + (3 + 5)</math>


Another example of a group is <math>(Q,x)</math> the rational numbers (all numbers <math>p/q</math>, where p and <math>q</math> are relatively prime) with multiplication.

  • Our collections is Q = all numbers <math>p/q</math>, where p and <math>q</math> are relatively prime.
  • Our operation is multiplication.
  • Our properties all hold:
  1. Q is closed under multiplication: multiply two fractions, and you get another fraction.
  2. The identity in this group is 1. After all multiplying by 1 doesn’t change anything.
  3. The inverse of any fraction <math>p/q</math> is just <math>q/</math>p another fraction in our collection. And note that <math>p/q \times q/p = 1</math> (a number times its inverse is the identity!)
  4. Our operation is associative. <math>(a/b \times c/d) \times e/f = a/b \times (c/d \times e/f)

</math>

In general a group consists of three things:

  • A collection of objects
  • An operation
  • Properties that have to hold:
  1. The collection of objects is closed under the operation.
  2. There is an identity object in our collection.
  3. Every element of our collection has an inverse (that also belongs to our collection)
  4. The objects are associative.

Examples of collections that are not groups

It is useful to look at some collections with operations that are NOT groups.
For instance the collection of all odd numbers with additions fails rather spectacularly at being a group. First of all it is not closed under addition. If you add two odd numbers you get an even number. E.g. <math>3 + 5 = 8</math>. So the result of an operations will drop you outside your collection! 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. There are inverses and the addition is associative, but that doesn’t matter. All requirements have to be met in order for your collection to be a group.

The integers with multiplication is also not a group.
Here the problem is with the inverses. Under multiplication 1 has to be the identity and 1 ia part of the integers, but with respect to multiplication the inverse of for instance 2 is 1/2 and 1/2 is not a member of the integers. So, every element of the group has to have an inverse and that inverse has to belong to the group.

Subgroups

We looked at the collections of all odd numbers and we noticed that this was not a group. The even numbers with addition is a different story. We sometimes write this collection of numbers as <math>2Z = {\ldots , -4, -2, 0, 2, 4, 6, \ldots}</math>. Adding two even numbers gives an even number (closed under addition), the identity (0) is in there, the inverse of every number is part of the collection and addition is associative, so <math>(2Z,+)</math> is a group. Because the collection of even numbers is a sub-collection of the integers we say that <math>(2Z,+)</math> is a subgroup of <math>(Z,+)</math>.

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 group. So <math>{0}</math> 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 <math>(Z,+)</math> is a subgroup of <math>(Z,+)</math>. 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 that <math>(2Z,+)</math> is a subgroup of <math>(Z,+)</math> tells us that there are other groups embedded in the larger one. You should be able to convince yourself that <math>(3Z,+) = {\ldots , -9, -6, -3, 0, 3, 6, 9,\ldots}</math> is also a subgroup of <math>(Z,+)</math>.
Even more interesting is the fact that <math>(4Z,+) = {\ldots,-8, -4, 0, 4, 8, 12, \ldots}</math> is a subgroup of both <math>(Z,+)</math> and <math>(2Z,+)</math>.


Some Examples using Escher Art

Consider this design by Escher - made up of two lizards. Ignoring color we see that the symmetry group is a C2.

  • Our collection of isometries: {E (do nothing), R (rotate 180 degrees)}
  • The operation is a multiplication. R x R means rotate, then rotate again. And we do the isometries right to left!
  • We have our four properties:
  1. Any combination of these isometries gives us another isometry in our collection. For instance E x R = R, R x R = E, R x E = R, etc.
  2. E is the identity.
  3. The inverse of E is E, The inverse of R is R, because if you rotate 180 degrees, how would you get back to normal? You would just rotate through another 180 degrees.
  4. The operation is associative: (R x R) x R = R x (R x R). Note that (R x R) x R = E x R = R and R x (R x R) = R x E = R, so the two equations are equal to one another.

Finite Isometry Groups (Rosette Groups)

Isometries-rectangle.png

Here we see the three isometries that act on a rectangle. This means that the isometry group of the rectangle contains at least four elements: the identity (<math>E</math>), the reflection <math>M1</math>, the reflection <math>M2</math> and the rotation <math>R</math>.

What is the result of combining two operations (what is called multiplying them)? In symbols, what is <math>M2</math> x <math>M1</math>? First we have to decide if the composition of the two isometries <math>M2</math> x <math>M1</math> means that we first do <math>M2</math>, and then <math>M1</math> (perform the operations reading left to right) or if we first do <math>M1</math> and then do <math>M2</math> (read right to left). Different mathematicians use different conventions; here we will use the convention that we perform the operations in the order of natural reading. Recap: remember that we read of the isometries from right to left!

Composition-isometries.png

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 <math>M2</math> x <math>M1</math> = <math>R</math>. Similarly, it is easy to show that <math>M1</math> x <math>M1</math> = <math>E</math>, and that <math>M2</math> x <math>M2</math> = <math>E</math>. 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 <math>R</math> x <math>R</math> = <math>E</math>. Below is a multiplication table for this group. If you want to know what <math>M2</math> x <math>R</math> is equal to, then go over in the row for <math>M2</math> and down in the column for <math> R</math> and look at where the row and column intersect. In this example you find <math>M1</math> in that spot. This means that <math>M2</math> x <math>R</math> = <math>M1</math>. If you wish to find <math>R</math> x <math>M2</math>, then go to row <math>R</math> and column <math>M2</math>, and you find that <math>R</math> x <math>M2</math> = <math>M1</math>.

Multiplication table for the symmetry group of the rectangle, 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


This means that 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 = {<math>E</math>, <math>M1</math>, <math>M2</math>, <math>R</math>}

It is illuminating to quickly determine the symmetry groups for a parallelogram. Remember that a parallelogram only has 180 degrees rotation. It has no reflectional symmetry at all. This means that the only elements of the symmetry groups C2 are <math>E</math> (the identity) and <math>R</math> (the rotation).

Multiplication table for the symmetry group of the parallelogram, C2
E (identity) R (reflection)
E E R
R R E


Because <math>R</math> x <math>R</math> = <math>E</math> we have a very simple multiplication table. In this case we can rewrite the symmetry group C2 as follows: C2 = {<math>E</math>, <math>R</math>}

The Symmetry Group of the Square is D4

Symmetry-group-square.png

We have shown the square with all possible rotation and reflections. 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: <math>E</math>, <math>R</math>, <math>R^2</math>, <math>R^3</math>, <math>M1</math>, <math>M2</math>, <math>M3</math>, and <math>M4</math> (where <math>R^2</math> denotes twice the rotation <math>R</math>, i.e. a rotation by 180 degrees, and <math>R^3</math> is <math>R</math> 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 <math>M1</math> x <math>M1</math> = <math>M1^2</math> = <math>E</math>, <math>M2</math> x <math>M2</math> = <math>M2^2</math> = <math>E</math>, and similarly for <math>M3</math> and <math>M4</math>.
  • This shows that the inverse of <math>M1</math> (written <math>M1^{-1}</math>)--meaning, that which undoes it--is <math>M1</math> 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 <math>M2</math> x <math>M1</math> = <math>R^3</math>, and <math>M1</math> x <math>M2</math> = <math>R</math>. Note that the order in which you apply the two operations (<math>M1</math> and <math>M2</math>) 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 <math>M2</math> x <math>M1</math> worked out
Square-reflections.png
  • Lets look at the rotations. If you take <math>R</math> (90° rotation), apply another rotation <math>R</math>, then you get <math>R^2</math> (180° rotation). Another rotation gives you <math>R^3</math> (270° rotation), and one more rotation gets you back to <math>E</math>, the identity. The rotations form a subgroup: {1, <math>R</math>, <math>R^2</math>, <math>R^3</math>}. 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, <math>R</math>, <math>R^2</math>, <math>R^3</math>} will give another element from the group. For instance: <math>R</math> x <math>R^2</math> = <math>R^3</math>, <math>R^2</math> x <math>R^2</math> = <math>E</math>, <math>R^3</math> x <math>R^3</math> = <math>R^2</math>, 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

Symmetry Groups for the Border Patterns

Symmetry-group-border.png

The border pattern above has symmetry group MM, and the isometries here are reflection over a vertical mirror line (<math>MV</math>), reflection over a horizontal mirror line (<math>MH</math>), and translation in the horizontal direction (<math>T</math>). Translation is an interesting isometry here. If you apply <math>T</math> twice, <math>T^2</math>, you translate two circles to the right. . If you apply <math>T</math> three times, <math>T^3</math>, you translate three circles to the right. You can keep translating without ever coming back to the original spot. The inverse of <math>T</math>, <math>T^{-1}</math>, just translates to the left. Similarly, you can keep translating to the left without ever returning to your starting point. This implies that you have the following subgroup: { …, <math>T^{-3}</math>, <math>T^{-2}</math>, <math>T^{-1}</math>, <math>E</math>, <math>T</math>, <math>T^2</math>, <math>T^3</math>, …}. This is an isometry with all of its powers. For that reason we call it a cyclic group, but the size is infinite! This is called the infinite cyclic group. Note that <math>MV</math> x <math>MH</math> = <math>MH</math> x <math>MV</math>. Also note that <math>MV</math> x <math>MV</math> = <math>E</math>, and <math>MH</math> x <math>MH</math> = <math>E</math>. This means that there are at least two subgroups: {<math>E</math>, <math>MV</math>} and {<math>E</math>, <math>MH</math>}. Both of these are cyclic subgroups. The number of elements is two, hence they are both C2 groups.

There are also 2-fold rotations in this symmetry group; in fact, there is a separate group element for each rotation center. Also, because we can combine <math>T</math> (or any power of <math>T</math>) with <math>MH</math>, MM contains glide-reflections. In groups such as 12 and 1G, it would be crucial to name the rotations and glide-reflections, in describing the operation of the group.

Some questions can be found in the Symmetry Group for Border Pattern Exploration