SYMMETRY

Writing about symmetry is a great challenge and a joy at the same time. It is a challenge because it is like the topic of infinity - it is all around us, but it is so profound and deep that there is a great risk that by trying to say something we will not manage to say even a little. I will take this risk mainly because I am not aiming to achieve the impossible thing - to say everything. No, I will need to do a lot of research for that and I will need more knowledge that I don’t have for the moment. I will try to lift the cover of the topic and to make connections, on one hand, with its Mathematical language and how it reflects real-life in order to develop tools for describing and working with symmetry in a more practical way. On the other hand, the mathematical language that we mentioned many times, developed from the people’s practical needs to deal with practical issues so we need to make a connection with other areas.

Symmetry is one of the most fundamental and important ways that nature structures itself. It is one of the laws of nature which we, humans, just copy in our everyday activities. We are naturally attracted to symmetry. When we say the word ‘symmetry’ and ‘symmetrical’ what is the first association that comes to your mind? Is it about something beautiful? What about ‘asymmetrical’? Are your associations rather not very positive? It is not only you, don’t worry. The idea and the principles of symmetry lies in the fundamentals of how we see, perceive and understand the world. Symmetry is what holds different parts of the world together and makes it more pleasing for the eyes. Everything is in a harmonious proportion when it is symmetrical. Nature itself is governed by the idea of symmetry that creates balance not chaos. In many ways symmetry represents order, a beautiful peaceful ordered world. It helps us to make sense of the world around us.

There are many things we could say about symmetry and they all will be true. We can say that a symmetry is a transformation that leaves the object unchanged. A symmetry is also a property of an object (butterfly, shapes). Its importance in our life is huge. It brings into our lives the ideas of beauty, harmony, balance, grace, elegance. When we look at ourselves in the mirror we see the same person, we mirror ourselves. And our face in the mirror (the image) is symmetrical to us (the object). The mirror is the mirror line that separates the two identical parts. One side mirrors the other, and creates harmonious balance. Asymmetrical is when the sides from both sides of the mirror line are not the same.

In Ancient Greece people were obsessed with symmetry and beauty. They thought that everything should be symmetrical - buildings, philosophy, principles of living. The ideas of philosophy followed this common cultural mood. Philosophy was very important for every mathematician because they were trying to find more precise tools (mathematical ones) which would respond to the philosophical ideas. It is considered that the idea of symmetry came from the works of Pythagoras, that great mathematician and philosopher.

Symmetry surrounds ourselves. Look down at your body. Look at the shapes on the computer screen. Look at the buildings on your street. Look at your cat or dog. Symmetry is variously defined as "proportion," "perfect, or harmonious proportions," and "a structure that allows an object to be divided into parts of an equal shape and size." When we think of symmetry, we think of some combination of all these definitions. That's because symmetry, whether in biology, architecture, art, or geometry reflects all of those definitions.

Symmetry in nature is something that brings beauty in our world. Everybody admires the symmetrical beauty of butterflies, snowflakes and leaves for example. Many things in nature come in a pair in which the two sides mirror each other. Look at the human body - we have left and right sides which are almost identical. Why did nature create our body in that way? Is it because it is important for evolution to survive - if the one stops working we still have the other side. Is symmetry a clue for a deeper order of the universe? Humans follow nature’s principles of symmetry and create gardens in a symmetrical way so every part is in the same proportion as the others. Symmetry is something so pleasing for the eyes to watch and to admire. There is a great symmetry in repetition of the seasons as well, at least in some parts in the world. But in nature not everything is symmetrical, there are many asymmetrical parts as well.

Symmetrical forms can be found in the inanimate world as well. The planets, with slight variation due to chance, exhibit radial symmetry. Snowflakes also provide an example of radial symmetry. All snowflakes show a hexagonal symmetry around an axis that runs parallel to their face. The fact that all snowflakes have this sort of symmetry is due to the way water molecules arrange themselves when ice forms. It's a reminder that symmetry is part of the structure of the world around us. Knowing the symmetry we can make predictions. The language of nature is so persistent. Symmetry is in the very heart how we understand the universe and maybe is a clue to the deeper order of the universe.

Symmetry is present in mathematical language in a way that mathematics tries to find practical explanations of how the symmetry happens. Especially when it comes to some transformations of shapes we can clearly see how mathematics tries to translate into a more rational way the natures beauty and wonders. The most common types of symmetry in mathematics are reflection symmetry and rotational symmetry. The shapes have as well lines of symmetries.

Mathematicians took the idea of symmetry from philosophy and science and generalised it completely. The concept of symmetry is mainly developed in geometry but could be applied in other parts in mathematics. Symmetry of the object in the space is about when moving the object and the object after the transformation stays the same, so the image is identical to the object. Rotation doesn't change the object, we can rotate an equilateral triangle, for example, and it doesn't matter how many degrees it will rotate. It stays exactly the same as it was before the rotation to take place. Reflection mirrors the same object but doesn’t change it, like when we see ourselves in the mirror - our image is symmetrical to us, because we are the same thing. Translation or movement of the object up, down, left or right doesn't have a power to change the properties of the object, it doesn’t make it smaller or bigger, it is just closer or a little bit far away from the image. So the object and the image are symmetrical.
In mathematics there are as well lines of symmetry and different shapes have different lines of symmetry (reflectional and rotational). Let’s remind ourselves some of them: the equilateral triangle has three lines of symmetry, the isosceles triangle has 1 line of symmetry, the scalene triangle has no lines of rotational symmetry, the square has four lines of symmetry, the rectangle has two lines of symmetry, the parallelogram has no lines of symmetry, the circle has infinite lines of symmetry, the rhombus has two lines of symmetry. For the regular polygons the number of lines of symmetry is equal to the number of sides, for example the regular pentagon has five lines of symmetry.

A painting by Jason Galles

The principles of symmetry are used very often in philosophy. Since the times of Ancient Greece philosophers are trying to understand how nature is building its structure according to the principles of symmetry and the philosophers are doing the same but with some moral and aesthetic concepts putting them in pairs which mirror each other in an opposite way. They talk about left and right, good and evil, night and day, love and hate, male and female, life and death. In these pairs we see concepts which mirror each other in an opposite way.

In natural human languages there are synonyms and antonyms for the word ‘symmetry’. The antonyms help us to see from a different perspective more clearly the words’ properties.

The definition of the symmetry in different language dictionaries includes a few meanings, which relates to the real life uses and forms of the concept of symmetry.
And they are:

1. ‘The correspondence of the form and arrangement of elements or parts on opposite sides of a dividing line or plane or about a center or an axis: the symmetry of a butterfly's wings.’
   2. ‘A relationship in which there is correspondence or similarity between entities or parts: the symmetry of the play, which opens and ends with a speech by a female character.’
   3. Beauty as a result of pleasing proportions or harmonious arrangement: "Here were the ringlets, framing a face of exquisite symmetry" (Clive Barker).
   4. Physics Invariance under transformation. For example, a system that is invariant under rotation has rotational symmetry.

In this definition symmetry implies a regularity in form and arrangement of corresponding parts, for example: the perfect symmetry of pairs of matched columns. Balance implies equilibrium of dissimilar parts, often as a means of emphasis, for example: a balance of humor and seriousness. Proportion implies a proper relation among parts, for example: His long arms were not in proportion to his body. Harmony suggests a consistent, pleasing, or orderly combination of parts, for example: harmony of colour.

Symmetry has huge importance for different kinds of arts, music, dance, ballet and architecture. We associate these parts of human creative expressions with symmetry, this is their essence to be deeply symmetrical. Symmetry for them is an aesthetic category. They are beautiful to watch. They express the finest human emotions.
For example the painting ‘The last supper’ (from Leonardo da Vinci, 15th century) is a good example of how the symmetry is applied in art. The figure of Jesus in the middle and from his both sides there are an equal number of people and things on the table. Leonardo da Vinci is the greatest example of Italian Renaissance which followed the ideas of humanism, proportion and symmetry. And Leonardo da Vinci himself represents the Renaissance man in the best way, he achieved so much in so many fields - including invention, painting, sculpture, architecture, science, music, mathematics, engineering, literature, anatomy, geology, astronomy, botany, writing, history, and cartography. He is known to have said, "Learning never exhausts the mind."

In architecture the symmetrical and proportional linea are the basics that holds the whole structure. Symmetry finds its ways into architecture at every scale, from the overall external views of buildings such as Gothic cathedrals and The White House, through the layout of the individual floor plans, and down to the design of individual building elements such as tile mosaics.

In dance and especially in classical ballet the lines of symmetry are so important. Ballet is a very beautiful type of dance in which we receive the impressions from the symmetrical lines, the arms and legs together expressing deeper emotions. Many positions of arms, legs and the neck if they go into the same direction are highly symmetrical. And even the movements they are doing are full of harmony, beauty, elegance, grace and balance. Symmetry can express calm or very passionate emotions in ballet; it depends on the music. Symmetry is predictable, familiar and fun to watch sometimes. It presents harmony. Many dances are based on this principle. But symmetry cannot be highlighted without asymmetry close by. Many modern works developed the use of asymmetry though it already existed naturally with movement. The spatial pattern, count of dancers and the sets or body positions started being overwhelmingly asymmetric. Today we see more of a balance of symmetry and asymmetry. Asymmetry presents some interesting patterns and possibilities that symmetry doesn’t. It is unpredictable, interesting and odd. It represents nature and roots. It gives movement more possibility. Too much asymmetry is not going to keep a dance interesting though. There needs to be a balance as to not suggest randomness of everyday life.

Symmetry can be found in various forms in literature, a simple example being the palindrome where a brief text reads the same forwards or backwards. Stories may have a symmetrical structure as well and the pages can mirror themselves presenting the text in different languages.

Here are two activities for you:

Activity 1: Be detectives! Try to make a list of all examples of symmetry around you. It could be shapes, plants, paintings or other things. How many lines of symmetry do they have?

Activity 2: Translate the word ‘symmetry’ in your own language. It sounds pretty much the same, doesn’t it? Make a list with 5 idioms (phrases, sayings, proverbs) in your language which include the word symmetry. Translate them into English. What happens with the words symmetry - does it stay or there is another word to replace it? If you cannot find some with the word ‘symmetry’ try to look with the word ‘mirror’.

In everyday life symmetry language refers to a sense of harmonious and beautiful proportion and balance. In our life we look at the mirror every day. What do we see there? Do we see an image of ourselves in a way people see us or in a way we see ourselves? Is it possible to see both images? Is it actually important to see both images or we need to be true to ourselves and ignore the other image? Mirror gives us the illusion of double life. The illusion that we can make things right next time without mistakes. The mirror is like a new beginning, a new chance for us to improve ourselves if we don’t like what we see. It is like a door to a different dimension, to a new world full of exciting opportunities for us to improve.
Our own image in the mirror is like a quiet honest reflection of what we are and what we want to be. Change doesn’t always come in an easy way. Most of the time it is a painful decision as a result of what we see in the mirror. But change is as well a new energy and a great challenge for ourselves to keep growing as a brave response to the changing world around us.

So, our own image is in our own hands!

(E. S. Lyubenova; LoveMaths Story for my students)