Transformations and Symmetry

Symmetry can be seen everywhere in nature – but it even underlies completely invisible laws of nature. The mathematical properties of symmetry explain why that is the case.

Introduction

Many geometric concepts, like lines and points, were “invented” by mathematicians. Symmetry, on the other hand, is everywhere around us. Almost all plants, animals, and even we humans are symmetric.

Over time, we’ve imitated nature’s symmetry in art, architecture, technology and design. Symmetric shapes and patterns just seems to look more beautiful than non-symmetric ones.

But symmetry is much more important that simply looking beautiful. It lies at the very foundations of our universe, and can even explain the most fundamental laws of physics.

While symmetry is a very intuitive concept, describing it mathematically is more difficult than you might think. To start with, we have to learn about transformations.

Transformations

A transformation is a specific set of rules that convert one geometric figure into another one. Here are a few examples:

The result of a transformation is called the image. The image of a figure A is usually denoted by A (pronounced as “A prime”).

Initially, we will just think about transformations that don’t change the original figure’s size and shape. Imagine that it is made out of a solid material like wood or metal: we can move, turn and flip it, but we can’t stretch or otherwise deform it. These transformations are called rigid transformations.

Which of these transformations are rigid?

For rigid transformations, the image is always the original. There are three different types of rigid transformations:

A transformation that simply moves a shape is called a translation.

A transformation that flips a shape over is called a reflection.

A transformation that spins a shape is called a rotation.

We can also combine multiple types of transformation to create more complex ones – for example, a translation followed by a rotation.

But first, let’s have a look at each of these types of transformations in more detail.

Translations

A translation is a transformation that moves every point of a figure by the same distance in the same direction.

In the coordinate plane, we can specify a translation by how far the shape is moved along the x-axis and the y-axis. For example, a transformation by (3, 5) moves a shape by 3 along the x-axis and by 5 along the y-axis.

Translated by (, )

Translated by (, )

Translated by (, )

Now it’s your turn – translate the following shapes as shown:

Translate by (3, 1)

Translate by (–4 –2)

Translate by (5, –1)

Reflections

A reflection is a transformation that “flips” or “mirrors” a shape across a line. This line is called the line of reflection.

Draw the line of reflection in each of these examples:

Now it’s your turn – draw the reflection of each of these shapes:

Notice that if a point lies on the line of reflection, its image is the original point.

In all of the examples above, the line of reflection was horizontal or vertical, which made it easy to draw the reflections. If that is not the case, the construction becomes more complicated:

To reflect this shape across the line of reflection, we have to reflect every vertex individually and then connect them again.

Let’s pick one of the vertices and draw the line through this vertex that is perpendicular to the line of reflection.

Now we can measure the distance from the vertex to the line of the reflection, and make the point that has the same distance on the other side. (We can either use a ruler or a compass to do this.)

We can do the same for all the other vertices of our shape.

Now we just have to connect the reflected vertices in the correct order, and we’ve found the reflection!

Rotations

A rotation is a transformation that “turns” a shape by a certain angle around a fixed point. That point is called the center of rotation. Rotations can be clockwise or counterclockwise.

Try to rotate the shapes below around the red center of rotation:

Rotate by 90° clockwise.

Rotate by 180°.

Rotate by 90° anti-clockwise.

It is more difficult to draw rotations that are not exactly 90° or 180°. Let's try to rotate this shape by ${10*ang}° around the center of rotation.

Like for reflections, we have to rotate every point in a shape individually.

We start by picking one of the vertices and drawing a line to the center of rotation.

Using a protractor, we can measure an angle of ${ang*10}° around the center of rotation. Let’s draw a second line at that angle.

Using a compass or ruler, we can find a point on this line that has the same distance from the center of rotation as the original point.

Now we have to repeat these steps for all other vertices of our shape.

And finally, like before, we can connect the individual vertices to get the rotated image of our original shape.

Transformations are an important concept in many parts of mathematics, not just geometry. For example, you can transform functions by shifting or rotating their graphs. Other transformations don’t even have a visual representation at all. You’ll learn more about these transformations in future chapters, but for now let’s move on to symmetry.

Symmetry

Symmetry is everywhere around us, and an intuitive concept: different parts of an object look the same in some way. But using transformations, we can give a much more precise, mathematical definition of what symmetry really means:

An object is symmetric if it looks the same, even after applying a certain transformation.

We can reflect this butterfly, and it looks the same afterwards. We say that it has reflectional symmetry.

We can rotate this flower, and it looks the same afterwards. We say that it has rotational symmetry.

Reflectional Symmetry

A shape has reflectional symmetry if it looks the same after being reflected. The line of reflection is called the axis of symmetry, and it splits the shape into two halves. Some figures can also have more than one axis of symmetry.

Draw all axes of symmetry in these six images and shapes:

This shape has axes of symmetry.

A square has axes of symmetry.

This shape has axes of symmetry.

Many letters in the alphabet have reflectional symmetry. Select all the ones that do:

A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z

Here are some more shapes. Complete them so that they have reflectional symmetry:

Shapes, letters and images can have reflectional symmetry, but so can entire numbers, words and sentences!

For example “25352” and “ANNA” both read the same from back to front. Numbers or words like this are called Palindromes. Can you think of any other palindromes?

If we ignore spaces and punctuation, these letters also have reflectional symmetry. Can you come up with your own?

Never odd or even.
A for a jar of tuna.
Yo, banana !

But Palindromes are not just fun, they actually have practical importance. A few years ago, scientists discovered that parts of our DNA are palindromic. This makes that more resilient to mutations or damage – because there is a second backup copy of every piece.

Rotational Symmetry

A shape has rotational symmetry if it looks the same after being rotated (by less than 360°). The center of rotation is usually just the middle of the shape.

The order of symmetry is the number of distinct orientations in which the shape looks the same. You can also think about it as the number of times we can rotate the shape, before we get back to the start. For example, this snowflake has order .

The angle of each rotation is 360°order. In the snowflake, this is 360°6 = °.

1 2 3 4 5 6 60°

Find the order and the angle of rotation, for each of these shapes:

Order , angle °

Order , angle °

Order , angle °

Now complete these shapes, so that they have rotational symmetry:

Order 4

Order 2

Order 4

Symmetry Groups

Some shapes have more than one symmetry – let’s have a look at the square as a simple example.

You have already shown above that a square has axes of reflection.

It also has rotational symmetry by °, ° and °.

And finally, we can think about “doing nothing” as another special kind of symmetry – because the result is (obviously) the same as before. This is sometimes called the identity.

In total, we have found different “symmetries of the square”.

Now we can actually start doing some arithmetic with these symmetries. For example, we can add two symmetries to get new ones:

+=
+=

Whenever you add two symmetries of a square, you get a new one. Here is a “symmetry calculator” where you can try it yourself:

+
=
×
+
+
+
+
+
+
+
+

Spend some time playing around with the symmetry calculator, and try to find any patterns. Can you complete these observations?

  • Adding two rotations will always give (or the identity).
  • Adding two reflections will always give (or the identity).
  • Adding the same two symmetries in the opposite order result.
  • Adding the identity .

You might have realised already that adding symmetries is actually very similar to adding integers:

  1. Adding two symmetries/integers always gives another symmetry/integer:
    +=
    12+7=19
    Continue
  2. Adding symmetries/integers is associative:
    ++=++
    4+2+5=4+2+5
    Continue
  3. Every symmetry/integer has an inverse, another symmetry/integer which, when added, gives the identity:
    +=
    4+–4=0
    Continue

In mathematics, any collection that has these properties is called a group. Some groups (like the symmetries of a square) only have a finite number of elements. Others (like the integers) are infinite.

In this example, we started with the eight symmetries of the square. In fact, every geometric shape has its own symmetry group. They all have different elements, but they always satisfy the three rules above.

Groups appear everywhere in mathematics. The elements can be numbers or symmetries, but also polynomials, permutations, matrices, functions … anything that obeys the three rules. The key idea of group theory is that we are not interested in the individual elements, just in how they interact with each other.

For example, the symmetry groups of different molecules can help scientists predict and explain the properties of the corresponding materials.

Groups can also be used to analyse the winning strategy in board games, the behaviour of viruses in medicine, different harmonies in music, and many other concepts…

The properties of the CCl4 molecule (left) and the Adenovirus (right) are determined by their symmetries.

Wallpaper Groups

We have now seen two different kinds of symmetry, that correspond to two different transformations: rotations and reflections. But there is also a symmetry for the third kind of transformation: .

Translational symmetry does not work for isolated objects like flowers or butterflies, but it does for regular patterns that extend into every direction:

Hexagonal honyecomb

Ceramic wall tiling

In addition to reflectional, rotational and translational symmetry, there even is a fourth kind: glide reflections. This is a combination of a reflection and a translation in the same direction as the axis of reflection.

A pattern can have more than one type of symmetry. And just like for squares, we can find the symmetry group of a pattern, which contains all its different symmetries.

These groups don’t tell you much about how the pattern looks like (e.g. its colours and shapes), just how it is repeated. Multiple different patterns can have the same symmetry group – as long are arranged and repeated in the same way.

These two patterns have the same symmetries, even though they look very different. But symmetries are not about colours, or superficial shapes.

These two patterns also have the same symmetries – even though they look more similar to the corresponding patterns on the left, than to each other.

It turns out that, while there are infinitely many possible patterns, they all have one of just 17 different symmetry groups. These are called the Wallpaper Groups. Every wallpaper group is defined by a combination of translations, rotations, reflections and glide reflections. Can you see the centers of rotation and the axes of reflection in these examples?

Type P1
Just translations

Type P2
Rotations of order 2

Type P3
Four rotations of order 2 (180°)

Type P4
Rotations of order 3 (120°)

Type P6
Rotations of order 6 (60°), order 3, and order 2

Type PM
Parallel axes of reflection

Type PMM
Perpendicular reflections and rotations of order 2

Type P4M
Rotations of order 2 and order 4 (90°), reflections, and glide reflections

Type P6M
Rotations of order 2 and order 6, reflections, and glide reflections

Type P3M1
Rotations of order 3, reflections in three directions, and glide reflections

Type P31M
Rotations of order 3, reflections in three directions, and glide reflections

Type P4G
Rotations of order 2 and order 4, reflections, and glide reflections

Type CMM
Perpendicular reflections and rotations of order 2

Type PMG
Reflections, glide reflections, and rotations of order 2

Type PG
Just parallel glide reflections

Type CM
Reflections and glide reflections

Type PGG
Perpendicular glide reflections, and rotations of order 2

Unfortunately there is no simple reason why there are 17 of these groups. Proving it requires much more advanced mathematics…

Instead, you can try drawing your own repeated patterns for each of the 17 wallpaper groups:

Examples of other students’ drawings

The wallpaper groups were all about flat, two-dimensional patterns. We can do something similar for three-dimensional patterns: these are called crystallographic groups, and there are 219 of them!

In addition to translations, reflections, rotations, and glide reflections, these groups include symmetries like glide planes and screw axes (think about the motion when unscrewing a bottle).

Boron-Nitride has its molecules arranged in this crystal lattice, which has a 3-dimensional symmetry group.

Symmetry in Physics

So far, all the symmetries we looked at were visual in some sense: visible shapes, images or patterns. In fact, symmetry can be a much wider concept: immunity to change.

For example, if you like apple juice just as much as you like orange juice, then your preference is “symmetric” under the transformation that swaps apples and oranges.

In 1915, the German mathematician Emmy Noether observed that something similar is true for the laws of nature.

For example, our experience tells us that the laws of Physics are the same everywhere in the universe. It doesn’t matter if you conduct an experiment in London, or in New York, or on Mars – the laws of Physics should always be the same. In a way, they have .

Similarly, it shouldn’t matter if we conduct an experiment while facing North, or South, or East or West: the laws of nature have .

And finally, it shouldn’t matter if we conduct an experiment today, or tomorrow, or in a year. The laws of nature are “time-symmetric”.

F =GMmr2 F =GMmr2

These “symmetries” might initially seem quite meaningless, but they can actually tell us a lot about our universe. Emmy Noether managed to prove that every symmetry corresponds to a certain physical quantity that is conserved.

For example, time-symmetry implies that Energy must be conserved in our universe: you can convert energy from one type to another (e.g. light, or heat or electricity), but you can never create or destroy energy. The total amount of energy in the universe will always stay constant.

CERN is the world’s largest particle accellerator. Scientists smash together fundamental particles at enourmous speeds, to learn more about their properties. Can you see the person at the bottom, for size comparison?

The paths taken by particle fragments after a collision

It turns ou that, just by knowing about symmetry, physicists can derive most laws of nature that govern out universe – without ever having to do an experiment or observation.

Symmetry can even predict the existence of fundamental particles. One example is the famous Higgs Boson: it was predicted in the 1960s by theoretical physicists, but not observed in the real world until 2012.

Similarity

So far, we have just looked at transformations. Now let’s think about one that is not: a dilation changes a shape’s size by making it larger or smaller.

All dilations have a center and a scale factor. The center is the point of reference for the dilation and scale factor tells us how much the figure stretches or shrinks.

If the scale factor is between 0 and 1, the image is than the original. If the scale factor is larger than 1, the image is than the original.

Scale factor: ${s}

COMING SOON – More on Dilations and Similarity

Triangles are not just useful for measuring distances. In the next chapter we will learn a lot more about triangles and their properties.