CIRCLES

Vassily Kandinsky 'Squares with Concentric Rings'

For centuries people have been fascinated about circles. Many scientists, philosophers, mathematicians, artists, and religious people have been working with the symbol of the circle and giving it different meaning according to the applications of the circle in real life. And there is one great reason for that interest. Above us in the sky every day we can see the big and magical solar ring. People depend on the sun in so many ways. It gives us warm, light, it is always up there as a loyal friend. People from the whole human world have the solar ring as a very important symbol in their cultures. In mythology the sun is usually a symbol of the divine and superior God. It is full of mysterious power. The shape of the sun is a circular one. There has been a great respect and fear about the sun for centuries. Even people from ancient cultures were realising that the sun is something that they cannot control and has a behaviour they cannot predict, so they cannot ignore the giant circle. It is like a big ball that could bring either life or destruction. All the changes happening with that giant circle were affecting human lives in so many ways. The solar eclipses for examples were seen as a very bad sign which was a warning before God's punishment. So the sun represented the God Himself in many cultures.

The circle is a universal symbol with extensive meaning. It represents the totality, the wholeness, the original perfection, the Self, the infinite, the eternity, the timelessness, all cyclic movement. Being a symbol of God, the circle doesn’t have a circumference. Or it is not in the human power to find and to calculate the circumference of the circle as a symbol of God. Its circumference is everywhere, there are no limits. From a spiritual point of view God is the centre as well. The symbol of the circle also implies in the idea of movement, and symbolizes the cycle of time, the motion of everything that moves, the planets' journey around the sun (the circle of the zodiac), the great rhythm of the universe. The circle is also zero in our system of numbering, and symbolizes potential, or the embryo. It has a magical value.

A circle is a round shaped figure that has no corners or edges. In geometry, a circle can be defined as a closed, two-dimensional curved shape. We know today that the shape of the sun and the earth is not a flat, 2D shape.

Can a perfect circle exist in nature? It's not easy to say with certainty whether a perfect circle or a sphere, a circle's three-dimensional image, exist outside of mathematical abstraction. To the human eye, circles and spheres are abundant in nature and in our universe. They can occur naturally — in planets, stars, celestial bodies, tree rings, rain drops — or they can be man-made — such as traffic roundabouts, buttons, volleyballs, pizza and many others. But there is a difference and a nuance to what our eyes see as a circle and what Maths would tell us about their true shape. Some of these circles in nature are formed by natural growth, some by the action of the wind or waves on natural textures, some by animals making homes or escaping from homes, some are nature cause and effect and some are a byproduct unwanted by nature. We can see circles very often in nature, for example: beach stones, poppy seed heads, hazelnuts, acorn cups, rings in onion, small bore holes in coral, annular solar eclipse, rings in cedar wood, slice of bamboo, holes, lid of the eucalyptus seed pod, acorn cup surrounded by the bark formed around tree branch, slice through citrus, ripples in water, section through a seed head of cone flower, the pupil of the eye, crop circles, lotus flower seed pod, the arc of the rainbow as a semicircle and many other objects.

We see circles in so many flowers, seeds and plants and the food we eat. If we think about the cycles and about their shape in our understanding, it is a shape of a circle. For example we connect with the rhythm of a circle, the plant cycles, the 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', where every point on the circumference is exactly equidistant from the circle’s center.

Because of their symmetry, circles were seen as representations of the “divine” and “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 the recorded history. The invention of the wheel is a fundamental discovery of properties of a circle. The Greeks considered the Egyptians as the inventors of geometry. Later on, the shape would become a vital foundation for the wheel and other simple machines. A focus on circles is evident among structures built throughout history. The symbol of the circle is very important today - they are often used to symbolize harmony and unity. For example the Olympic symbol, it has five interlocking rings of different colours, which represent the five major continents of the world united together in a spirit of healthy competition.

Mathematics has a great respect and admiration for the circles. Mathematics is actually the area where the perfect circles exist. So the mathematical definition of the circle is ‘A circle, as a curve, is the set of points equidistant from a given point. That is, a circle is a curve consisting of all those points of a plane that lie at a fixed distance a from a particular point, called the center of a circle.’

Circles are also very efficient: they cover 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 a disc and gives us the wheel, perhaps the most famous of all inventions. The parts of the circle in mathematical point of view are an arc, a segment, a sector, a secant, a chord, a diameter, a radius, a tangent. Half of the circle is a semicircle. We can find the circumference and an area of a circle, and for that purpose we need to know the radius, or the diameter and the value of the number ‘pie’.

Once we draw some lines inside a circle, or outside the circle as a tangent for example, we create angles between them and the radius. For example, drawing two radii forms a triangle with two equal sides – an isosceles triangle. The relationship between the properties of the circle and the properties of the formed angles are an object of the Circle theorems. Circle theorems are used in geometric proofs and to calculate angles. There are eight Circle theorems.

Circles have great importance and significance in Visual Art as well. There is a very famous artist, called W. Kandinsky, whose paintings are full of colourful happy circles. For the artist the mysterious circle was always a thing of beauty and a joy forever. For them the mystery element combined with the aesthetic one, producing varied meanings in various semiotic conventions. Circles appear in paintings either as circles themselves or as their derivatives such as the sun, the moon and other stellar orbs, as wheels, sundials, clock faces and circular paths. In the field of architecture architects used circles, semi - circles and spirals in different places like arches, windows and others.

Here are some activities for you:

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

Activity 2: Can you try to experiment and to think of some patterns that include circles? Can you make a picture that uses only circles? Please use different colours. What is the message you are spreading to the world? If you have to paint a happy circle, what would it look like?

What does the circle symbolise in our life? Its shape is close to perfection with no beginning and end, like God Himself. The voice of the circle is in its metaphorical and abstract message. The circle could be a symbol of equality, chairs at a round table mean that there is no leader but a meeting of equal people. Is equality possible in society? There are different types of equalities. In society we all have equal human rights, this equality should be compulsory for a democratic society. We all are born with different abilities but in a fair competition we all should have equal chances and opportunities to show our differing abilities.

We very often associate with the closeness of the circle the idea of the society itself with its inner social circles or classes. Can we escape the social circle? It is very often a tempting idea to follow. If the circle represents human civilisation itself there is no point to escape it because we cannot live for longer out of society and carry on being humans. If we stay out of human society we will lose our human characteristics, what makes us humans, what makes us different from animals. What makes us different from animals is our longing about the divine, it is our secret desire and articulated intention to improve. The simplicity of the circle is so captivating. It is like endless human development in a moral and in a spiritual way. The better qualities we can nurture in ourselves, the better place for living our planet will be. Is perfection possible? Nothing is perfect but everything is in constant motion of improvement. If we say that we are perfect, that means that we are becoming something that doesn’t change at all. We can get used to perfection and even a perfect beauty can make us bored because it will take from us the whisper that we deserve something even better. There is always a way up and high to our own idea of God as something perfect. The circle is the symbol of divine energy. This is something that should remind us that there is always a chance for us to improve. Tomorrow’s image should be better than it was today and yesterday. This is the way the energy is flowing - forward but not backwards. It is so important for our existence to develop higher consciousness and responsible awareness about life around us. This will give us the vision of how to take care of everything around us. Nothing is perfect. Neither are human relationships. But if we try with love, care and intelligence to reach the others and if they do the same, there is a great chance that we can make our common home Earth a peaceful place full of joy and mutual respect for each other. This is what the divine circle is about.

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