CIRCLES

Vassily Kandinsky 'Squares with Concentric Rings'

For centuries, people have been fascinated by circles. Many scientists, philosophers, mathematicians, artists and religious thinkers have worked with the symbol of the circle, giving it different meanings according to its applications in real life.

There is one great reason for this interest. Above us in the sky, every day, we can see the great and magical solar circle. People depend on the Sun in so many ways. It gives us warmth and light, and it is always up there like a loyal friend. People throughout the world have regarded the solar circle as a very important symbol in their cultures.

In mythology, the Sun is usually a symbol of the divine and of a supreme God. It is full of mysterious power. The shape of the Sun is circular. For centuries, people have felt both great respect and fear towards it. Even people in ancient cultures realised that the Sun was something they could not control and that its behaviour could not always be predicted, so they could not ignore the giant circle. It was like a great ball that could bring either life or destruction.

All the changes happening to that giant circle affected human lives in many ways. Solar eclipses, for example, were seen as very bad signs, warning people of God’s punishment. Therefore, in many cultures, the Sun represented God Himself.

The circle is a universal symbol with many meanings. It represents totality, wholeness, original perfection, the Self, infinity, eternity, timelessness and all cyclical movement. As a symbol of God, the circle has no visible boundary. Or perhaps it is simply beyond human power to find and calculate the circumference of the circle when it is understood as a symbol of God. Its circumference is everywhere; there are no limits. From a spiritual point of view, God is also the centre.

The symbol of the circle also implies the idea of movement. It symbolises the cycle of time, the motion of everything that moves, the planets’ journeys around the Sun, the circle of the zodiac and the great rhythm of the universe. The circle is also represented by zero in our number system and symbolises potential or the embryo. It has magical value. 

A circle is a round shape that has no corners or edges. In geometry, a circle can be defined as a closed, two-dimensional curved shape. Today, we know that the Sun and the Earth are not flat, two-dimensional shapes.

Can a perfect circle exist in nature? It is not easy to say with certainty whether a perfect circle - or a sphere, the three-dimensional equivalent of a circle - exists outside mathematical abstraction. To the human eye, circles and spheres are abundant in nature and throughout our universe.

They can occur naturally in planets, stars, celestial bodies, tree rings and raindrops, or they can be human-made, such as traffic roundabouts, buttons, volleyballs and pizzas. However, there is a difference between what our eyes perceive as a circle and what mathematics tells us about its true shape.

Some circles in nature are formed through natural growth. Others are created by the action of wind or waves upon natural materials. Some are made by animals building or escaping from their homes. Others are the result of natural processes, while some are unwanted by-products of nature.

We can see circles very often in the natural world. Examples include beach stones, poppy seed heads, hazelnuts, acorn cups, onion rings, small holes in coral, annular solar eclipses, rings in cedar wood, slices of bamboo, holes, the lids of eucalyptus seed pods, acorn cups surrounded by bark formed around tree branches, slices of citrus fruit, ripples in water, cross-sections of coneflower seed heads, the pupil of the eye, crop circles, lotus flower seed pods and the arc of a rainbow appearing as a semicircle.

We also see circles in many flowers, seeds, plants and foods. When we think about cycles and imagine their shape, we often imagine a circle. We connect the rhythm of the circle with plant cycles and life cycles, like the never-ending pattern of life.

The circle has been adopted in every aspect of human experience, from money and art to architecture and religion.

The world is full of circles, but almost nothing in the physical world passes the “perfect circle test”, in which every point on the circumference is exactly the same distance from the centre of the circle.

Because of their symmetry, circles were seen as representations of the divine and of natural balance in ancient Greece. Greek mathematicians were fascinated by the geometry of circles and explored their properties for centuries. The study of the circle goes back beyond recorded history.

The invention of the wheel was a fundamental discovery connected with the properties of the circle. The Greeks considered the Egyptians to be the inventors of geometry. Later, the circular shape became the vital foundation of the wheel and other simple machines.

A fascination with circles is evident in structures built throughout history. The symbol of the circle is still very important today, and circles are often used to symbolise harmony and unity. The Olympic symbol, for example, consists of five interlocking rings of different colours, representing the major regions of the world united in a spirit of healthy competition.

Mathematics has great respect and admiration for circles. Mathematics is, in fact, the area in which perfect circles exist. The mathematical definition of a circle is:

“A circle, as a curve, is the set of points that are equidistant from a given point. In other words, a circle is a curve consisting of all the points on a plane that lie at a fixed distance from a particular point, called the centre of the circle.”

Circles are also highly efficient. They enclose the maximum possible area for a given perimeter, or have the minimum possible perimeter for a given area. They are useful too. A filled-in circle is called a disc, and it gives us the wheel, perhaps the most famous of all inventions.

From a mathematical point of view, the parts associated with a circle include an arc, a segment, a sector, a secant, a chord, a diameter, a radius and a tangent. Half of a circle is called a semicircle. We can calculate the circumference and area of a circle, and for this purpose we need to know its radius or diameter and the value of the number pi.

Once we draw lines inside a circle, or outside it as tangents, for example, we create angles between them and the radii. Drawing two radii, for instance, can form a triangle with two equal sides - an isosceles triangle.

The relationships between the properties of circles and the properties of the angles formed within them are studied through circle theorems. Circle theorems are used in geometrical proofs and to calculate angles. There are eight main circle theorems.

Circles are also of great importance and significance in visual art. There is a very famous artist called Wassily Kandinsky whose paintings are full of colourful, joyful circles. For the artist, the mysterious circle has always been a thing of beauty and a joy forever. The element of mystery combines with the aesthetic element, producing different meanings within various artistic and semiotic conventions.

Circles appear in paintings either as circles themselves or through related images such as the Sun, the Moon and other celestial bodies. They also appear as wheels, sundials, clock faces and circular paths.

In architecture, circles, semicircles and spirals have been used in many forms, including arches, windows and decorative structures.

Here are some activities for you:

Activity 1: Try to find circular shapes around you. Which objects or things in nature have a circular shape? Can you try to calculate their area and circumference?

Activity 2: Can you experiment and think of some patterns that include circles? Can you create a picture using only circles? Please use different colours. What message would you like to share with the world? If you had to paint a happy circle, what would it look like?

What does the circle symbolise in our lives? Its shape is close to perfection, with no beginning and no end, like God Himself. The voice of the circle lies in its metaphorical and abstract message.

The circle can also be a symbol of equality. Chairs placed around a round table suggest that there is no single leader, but rather a meeting of equal people.

Is equality possible in society? There are different types of equality. In society, we all have equal human rights, and this equality should be an essential part of every democratic society. We are all born with different abilities, but in a fair competition, we should all have equal chances and opportunities to demonstrate those different abilities.

We very often associate the closed shape of the circle with the idea of society itself, with its inner social circles or classes. Can we escape the social circle? It is often a tempting idea.

However, if the circle represents human civilisation itself, there is no point in trying to escape it, because we cannot live outside society for long and continue to be fully human. If we remain outside human society, we may lose the characteristics that make us human and distinguish us from animals.

What makes us different from animals is our longing for the divine. It is our secret desire and our clearly expressed intention to improve.

The simplicity of the circle is captivating. It is like endless human development in a moral and spiritual sense. The more positive qualities we nurture within ourselves, the better a place our planet will become.

Is perfection possible? Nothing is perfect, but everything is in a constant process of improvement. If we say that we are perfect, it suggests that we have become something that no longer changes.

We can become accustomed even to perfection, and perfect beauty itself may eventually make us feel bored because it takes away the whisper that tells us we deserve and can achieve something even better.

There is always a way upwards, towards our own idea of God as something perfect. The circle is a symbol of divine energy. It should remind us that there is always an opportunity to improve.

Tomorrow’s image of ourselves should be better than today’s and yesterday’s. This is the way energy flows - forwards, not backwards.

It is essential for our existence that we develop a higher consciousness and a responsible awareness of the life around us. This will give us a vision of how to care for everything around us.

Nothing is perfect. Human relationships are not perfect either. But if we try, with love, care and intelligence, to reach other people, and if they do the same, there is a great chance that we can make our shared home, the Earth, a peaceful place full of joy and mutual respect.

This is what the divine circle is about.

(E. S. Lyubenova; LoveMaths Story for My Students)