Writing about symmetry is both a great challenge and a great joy. It is a challenge because, like the topic of infinity, it is all around us, yet it is so profound and deep that there is a great risk that, by trying to say something, we may fail to say very much at all. I am willing to take this risk, mainly because I am not trying to achieve the impossible—to say everything. No, that would require much more research and much more knowledge than I have at the moment.
I will simply try to lift the curtain on the topic and make connections, on the one hand, with the mathematical language of symmetry and how it reflects real life in order to develop practical tools for describing and working with symmetry. On the other hand, as we have mentioned many times, mathematical language developed from people's practical needs to solve practical problems, so we also need to make connections with other areas.
Symmetry is one of the most fundamental ways in which nature structures itself. It is one of the laws of nature that we humans simply imitate in our everyday activities. We are naturally attracted to symmetry.
When you hear the words symmetry and symmetrical, what is the first association that comes to your mind? Is it something beautiful? And what about asymmetrical? Are your associations less positive? If so, don't worry—you are not alone.
The idea and the principles of symmetry lie at the very foundation of how we see, perceive and understand the world. Symmetry is what holds different parts of the world together and makes them more pleasing to the eye. Everything is in harmonious proportion when it is symmetrical. Nature itself is governed by the principle of symmetry, which creates balance rather than chaos. In many ways, symmetry represents order—a beautiful, peaceful and well-ordered world. It helps us make sense of the world around us.
There are many things we could say about symmetry, and they would all be true. We can say that symmetry is a transformation that leaves an object unchanged. Symmetry is also a property of an object, such as a butterfly or a geometric shape. Its importance in our lives is enormous. It brings into our lives the ideas of beauty, harmony, balance, grace and elegance.
When we look at ourselves in a mirror, we see the same person reflected back at us. Our face in the mirror (the image) is symmetrical to us (the object). The mirror acts as the mirror line that separates the two identical parts. One side reflects the other and creates harmonious balance. Something is asymmetrical when the two sides of the mirror line are not the same.
In Ancient Greece, people were fascinated by symmetry and beauty. They believed that everything should be symmetrical—buildings, philosophy and even the principles of life. Philosophy reflected this broader cultural outlook. It was extremely important to mathematicians because they were trying to develop more precise mathematical tools that could express philosophical ideas. It is generally believed that the concept of symmetry originated in the work of Pythagoras, the great mathematician and philosopher.
Symmetry surrounds us. Look at your own body. Look at the shapes on your computer screen. Look at the buildings in your street. Look at your cat or your dog. Symmetry has been defined in many different ways: as "proportion", "perfect or harmonious proportions", or "a structure that allows an object to be divided into parts of equal shape and size". When we think about symmetry, we usually think of some combination of all these definitions. That is because symmetry, whether in biology, architecture, art or geometry, reflects all of these ideas.
Symmetry in nature is one of the things that brings beauty into our world. Everybody admires the symmetrical beauty of butterflies, snowflakes and leaves, for example. Many things in nature come in pairs, with the two sides mirroring each other. Look at the human body—we have left and right sides that are almost identical. Why did nature create our bodies in this way? Is it because symmetry is important for survival and evolution? If one side stops functioning, we still have the other. Is symmetry a clue to a deeper order in the universe?
Humans follow nature's principles of symmetry and create gardens in a symmetrical way so that every part is in proportion to the others. Symmetry is so pleasing to the eye. There is also great symmetry in the repetition of the seasons, at least in some parts of the world. But nature is not entirely symmetrical. There are many asymmetrical forms as well.
Symmetrical forms can also be found in the inanimate world. The planets, with slight variations caused by chance, exhibit radial symmetry. Snowflakes provide another beautiful example. Every snowflake has sixfold symmetry around an axis that runs through its centre. This symmetry results from the way water molecules arrange themselves as ice forms. It reminds us that symmetry is part of the structure of the world around us. By understanding symmetry, we can make predictions. The language of nature is remarkably consistent. Symmetry lies at the very heart of how we understand the universe and may even be a clue to its deeper order.
Symmetry is reflected in the language of mathematics because mathematics seeks practical explanations for how symmetry works. This is especially clear when we study transformations of shapes. Mathematics translates the beauty and wonders of nature into a more rational language. The most common types of symmetry in mathematics are reflection symmetry and rotational symmetry. Shapes also have lines of symmetry.
In mathematics there are also lines of symmetry, and different shapes have different numbers of lines of symmetry (reflectional and rotational). Let us remind ourselves of a few examples. An equilateral triangle has three lines of symmetry, an isosceles triangle has one line of symmetry, a scalene triangle has no lines of symmetry, a square has four lines of symmetry, a rectangle has two lines of symmetry, a parallelogram has no lines of symmetry, a circle has infinitely many lines of symmetry, and a rhombus has two lines of symmetry. For regular polygons, the number of lines of symmetry is equal to the number of sides. For example, a regular pentagon has five lines of symmetry.
The principles of symmetry are very often used in philosophy. Since the time of Ancient Greece, philosophers have tried to understand how nature builds its structure according to the principles of symmetry. They have also applied these principles to moral and aesthetic concepts by arranging them in pairs that mirror one another in opposite ways. They speak 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 that reflect one another through contrast.
In natural human languages, there are synonyms and antonyms related to the word symmetry. Antonyms, in particular, help us understand the properties of a word more clearly by showing us its opposite.
The definitions of symmetry found in different dictionaries include several meanings that relate to the real-life uses and forms of the concept. They include:
- The correspondence of the form and arrangement of elements or parts on opposite sides of a dividing line or plane, or around a centre or an axis, for example, the symmetry of a butterfly's wings.
- A relationship in which there is correspondence or similarity between entities or parts, for example, the symmetry of a play that opens and closes with a speech by the same female character.
- Beauty resulting from pleasing proportions or a harmonious arrangement, as in the quotation: "Here were the ringlets, framing a face of exquisite symmetry." (Clive Barker)
- In physics, invariance under transformation. For example, a system that remains unchanged under rotation possesses rotational symmetry.
These definitions show that symmetry implies regularity in the form and arrangement of corresponding parts. For example, we may speak about the perfect symmetry of matched columns. Balance, on the other hand, implies equilibrium between different parts, often used to create emphasis—for example, a balance between humour and seriousness. Proportion refers to the proper relationship among parts, as in the sentence: "His long arms were not in proportion to his body." Harmony suggests a pleasing, consistent and orderly combination of parts, for example, the harmony of colour.
Symmetry is of enormous importance in the arts, music, dance, ballet and architecture. We naturally associate these forms of human creativity with symmetry because it lies at the heart of their beauty. For them, symmetry is an aesthetic category. They are beautiful to watch because they express some of the finest human emotions.
For example, Leonardo da Vinci's painting The Last Supper (15th century) is an excellent illustration of symmetry in art. The figure of Jesus stands at the centre, with an equal number of people and objects arranged on either side. Leonardo da Vinci is one of the greatest representatives of the Italian Renaissance, a movement deeply influenced by the ideals of humanism, proportion and symmetry. Leonardo himself is perhaps the finest example of the Renaissance man. His achievements covered an extraordinary range of fields, including invention, painting, sculpture, architecture, science, music, mathematics, engineering, literature, anatomy, geology, astronomy, botany, history, cartography and writing. He is well known for saying, "Learning never exhausts the mind."
In architecture, symmetrical and proportional lines form the foundation that holds an entire structure together. Symmetry finds its way into architecture at every scale, from the overall appearance of buildings such as Gothic cathedrals and the White House, through the layout of individual floor plans, down to the design of architectural details such as tile mosaics.
In dance, and especially in classical ballet, the principles of symmetry are equally important. Ballet is a beautiful form of dance in which much of the visual impression comes from symmetrical lines created by the arms, legs and body, all working together to express deep emotions. Many positions of the arms, legs and neck are highly symmetrical when they move in the same direction. Even the movements themselves are full of harmony, beauty, elegance, grace and balance.
Symmetry can express both calmness and intense passion in ballet; it all depends on the music. It is predictable, familiar and often fascinating to watch because it conveys harmony. Many forms of dance are based on this principle. However, symmetry cannot be fully appreciated without asymmetry nearby. Many modern choreographers have developed the use of asymmetry, although it has always existed naturally in movement. Spatial patterns, the number of dancers and body positions have increasingly become asymmetric. Today, we often see a balance between symmetry and asymmetry. Asymmetry creates interesting patterns and possibilities that symmetry alone cannot provide. It is less predictable, more surprising and often more expressive. It reflects nature and our roots, giving movement greater freedom. Yet too much asymmetry can make a dance appear random and lose its artistic impact. A balance between symmetry and asymmetry is needed so that the performance does not simply resemble the randomness of everyday life.
Symmetry can also be found in literature. A simple example is the palindrome, in which a word or a short text reads the same forwards and backwards. Stories may also have symmetrical structures, and even the pages of a book can mirror one another by presenting the same text in different languages.
Here are two activities for you.
Activity 1: Be detectives! Make a list of as many examples of symmetry around you as you can. They may include shapes, plants, paintings or other objects. How many lines of symmetry does each one have?
Activity 2: Translate the word symmetry into your own language. It probably sounds quite similar, doesn't it? Then make a list of five idioms, sayings or proverbs in your language that include the word symmetry. Translate them into English. What happens to the word symmetry? Does it remain the same, or is it replaced by another word? If you cannot find examples using symmetry, try looking for expressions that include the word mirror.
 |
| A painting by Jason Galles |
A mirror gives us the illusion of a double life—the illusion that we can do things better the next time and avoid our previous mistakes. It is like a new beginning, a new opportunity to improve ourselves if we do not like what we see. It is like a door to another dimension, a new world full of exciting opportunities for growth.
Our own reflection in the mirror is a quiet and honest image of who we are and who we want to become. Change does not always come easily. Most often, it is a painful decision born from what we see reflected back at us. Yet change is also a source of new energy and a wonderful challenge. It allows us to keep growing as a courageous response to the constantly changing world around us.
So, our own image is in our own hands!
(E. S. Lyubenova, LoveMaths Story for My Students)
No comments:
Post a Comment