PATTERNS

The idea of patterns has accompanied human life since the early years of human civilisation. It is an abstract and very powerful idea which holds human life in order and gives it good structure. Everything we do in life is subject to rules, models and repetitions. Since we were born, this has been the only way to discover the world around us, by following certain patterns.

In every area of our work or personal life, we obey some models, examples and rules. This is the only way to learn new things by investigating and identifying the forces that cause the patterns. We learn by repeating knowledge in a certain way, following certain patterns.

A pattern is a way in which something is done or organised, or in which something happens. We can talk about weather patterns, behaviour patterns, shape patterns, colour patterns and many other types of patterns depending on the nature of the activities we are involved in.

The idea of patterns is an abstract philosophical one with great applications in every aspect of human life. Whatever we do, we instinctively organise ourselves in a way that repeats itself again and again throughout space in an identical but not random way. Especially if the patterns are a good model and show progress and development, we tend to keep them for the next time as well.

A lack of everyday patterns that we could follow is a signal to our brain of chaos and can cause mental health problems. We will feel lost and frustrated, as if something is not right. We naturally like patterns and recognise them; it is part of our human intelligence. We cannot imagine our lives separated from patterns because the ability to organise ourselves in a specific way and to maintain this way is what makes us human as well. Pattern recognition is what allows us to do everything from recognising individuals and people's emotional states to solving jigsaw puzzles and sensing when a storm is about to come.

In our greatest teacher, nature, patterns are visible regularities of form found everywhere. They create the structure of plants, for example. Tree branches grow in a certain pattern. When we make patterns, it happens as a result of our plan to put the elements into a certain place. In nature, somehow natural forces agree to create patterns that look beautiful.

These patterns recur in different contexts and can sometimes be modelled mathematically. Many Greek philosophers studied patterns in order to explain the order in nature. Among them were Plato, Pythagoras and Empedocles. Pythagoras (c. 570–c. 495 BC), for example, explained patterns in nature, such as the harmonies of music, as arising from numbers.

It was a British scientist called Alan Turing who concluded that every living organism, from the simplest cells to the most detailed plants, must be created by a mathematical formula which produces this art in nature. Some invisible mathematical processes create the patterns in every living organism. The Greek philosopher Pythagoras had suggested this more than 2,000 years ago when attempting to interpret the entire physical world in terms of numbers.

The modern understanding of visible patterns developed gradually over time. Mathematics, physics and chemistry give us explanations about patterns at different levels. Mathematics, for example, seeks to discover and explain abstract patterns or regularities of all kinds.

There are a few types of natural patterns, and they include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Patterns can be seen everywhere in nature, from the leaves on a tree to the microscopic structure of those leaves. Shells and rocks have patterns, animals and flowers have patterns, and the human body follows a pattern and includes an infinite number of patterns within it.

Symmetry is widespread in living things. Animals mainly have bilateral, or mirror, symmetry, as do the leaves of plants and some flowers. Plants often have radial or rotational symmetry, as do many flowers and some groups of animals such as sea anemones. Fivefold symmetry is found in the group that includes starfish and sea lilies.

Among non-living things, snowflakes have striking sixfold symmetry. Crystals, in general, have a variety of symmetries. Rotational symmetry is found at different scales among non-living things. Symmetry has a variety of causes. Radial symmetry suits organisms like sea anemones whose adults do not move: food and threats may arrive from any direction. But animals that move in one direction necessarily have upper and lower sides, a head and a tail, and therefore a left and a right side.

In trees, the patterns can be seen when a parent branch splits into two or more child branches. The surface areas of the child branches add up to that of the parent branch. An equivalent is that if a parent branch splits into two child branches, then the cross-sectional diameters of the parent and the two child branches form a right-angled triangle. This allows trees to better withstand high winds.

Spirals are another type of pattern. They are common in plants and in some animals. A growth spiral can be seen as a special case of self-similarity. Plant spirals can be seen in phyllotaxis, the arrangement of leaves on a stem, and in the arrangement of other parts, as in composite flower heads and seed heads like the sunflower, or fruit structures like the pineapple and snake fruit, as well as in the pattern of scales in pine cones, where multiple spirals run both clockwise and anticlockwise.

Each inner part of the spiral resembles the larger spiral. These arrangements have explanations at different levels – mathematics, physics, chemistry and biology – each individually correct, but all necessary together. Branches, spirals and waves are all examples of fractals. Fractals are curves or geometric figures, each part of which has the same statistical character as the whole. Fractals are useful in modelling structures (such as eroded coastlines or snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth, fluid turbulence and galaxy formation.

For example, the shape of the smallest branchlet on the fern frond repeats the shapes of the larger branches. This is described as 'self-similar', meaning the shapes are similar to each other even if they are of different sizes. This repetition of self-similar shapes is one of the features of fractals and can also be seen in spirals, waves and other patterns. Wave-like patterns can be found on land, in the sky and in water as well.

Fractal geometry is the geometry of the natural world. In the natural world, objects aren't squares, boxes, pyramids or cubes. There are few straight lines. Instead, we find curves, wrinkles and broken lines.

Patterns have so many uses in the natural world. Whether they are related to survival or structural integrity, patterns have developed over time to become more and more useful due to evolution, which is similar to an algorithm.

The fact that we can observe mathematical patterns in nature is proof enough that things such as the Fibonacci Sequence actually work and are functional as axioms in mathematics. The Fibonacci sequence itself is found frequently in nature. It is a natural pattern of growth: 0, 1, 1, 2, 3, 5, etc. (every new number is the sum of the two previous numbers).

While many natural patterns occur, the Fibonacci sequence appears very often in nature, whether it is connected with the Golden Ratio or exists independently of it. In nature, there is an incredible richness of patterns. This shows how intelligent life is and makes us appreciate it even more.

Patterns are present in every human activity. We can see them in music and art as well. The patterns are just so striking, beautiful and remarkable. Music is made up of patterns. We sing songs in which words and melodies are repeated. Patterns are the vocabulary of the language of music. Rhythm patterns, for example, are the main rhythmic elements in a piece of music. Tonal patterns provide a shorthand harmonic outline of a song.

Some musicians make a parallel between learning a human language and learning music. According to them: Letters = Music Notes; Words = Tonal or Rhythm Patterns; Sentences = Groups of Patterns; Paragraphs = Phrases; Chapters = Songs.

Art uses geometrical shapes and includes them in different ideas and arrangements to create abstract designs or original paintings. In art, a pattern is a repetition of specific visual elements. They are aesthetically pleasing because they embody a sense of harmony.

Every culture has its own set of folk patterns that appear on textiles, architecture, manuscripts, masks and other objects. Even if the patterns are simple, the paintings are still eye-catching and compelling because of the bright colours and the diversity of simple and complex patterns. Patterns can come in many forms. Humans also try to replicate nature with man-made patterns.

Mathematics itself is all about patterns, rules, models and formulas. Patterns are at the heart of maths. Mathematicians are like painters and poets, but the patterns they create are more permanent because they are made with ideas. A mathematician is a maker of patterns.

There are patterns with colours, shapes and sizes. But numbers can have interesting patterns as well.

Are patterns and sequences the same thing? There are two different types of patterns:

1. Repeating patterns: repetitions of symbols, shapes, colours, numbers, etc., that recur in a specific way.
2. Increasing (growing) and decreasing (shrinking) patterns: an ordered set of shapes or numbers that are arranged according to a rule. Typically, the term sequence is used to describe this type of pattern, as opposed to a repeating pattern.

A repeating pattern is the shortest sequence that repeats. Sequences are more linear, unlike repeating patterns. They have a tendency to increase or decrease in specific ways. That's why they are also referred to as increasing and decreasing patterns. The way in which the individual parts of sequences, called 'terms', are ordered is governed by a rule.

We need to be able to identify this rule in order to extend the numerical sequence. This can happen by examining each given term and identifying what has happened between it and the next term, i.e. whether the numbers increase or decrease, and by how much.

For example, take the simple numerical sequence: 2, 4, 6, 8, 10. The observation is that the numbers increase by 2. This is their 'common difference', and the rule is '+2'. If we need to predict the 6th term, we simply add 2, so the next term is 12.

Odd and even numbers are examples of increasing patterns/sequences, as the difference between each term is +2.

If we have to summarise, a sequence is a string of organised objects following certain criteria, which can be ordered (increasing or decreasing) and established by a pattern. A pattern, on the other hand, is a form or model proposed for imitation. In the case of sequences, their patterns are models that serve to construct them.

There are a few types of numerical sequences:

1. Arithmetic Sequences are made by adding the same value each time. For example: 1, 4, 7, 10, 13, 16, … This sequence has a difference of 3 between each number. The pattern is continued by adding 3 to the last number each time. The value added each time is called the "common difference". The common difference can also be negative: 21, 19, 17, 15, … Here the common difference is −2.
2. A Geometric Sequence is made by multiplying by the same value each time. For example: 1, 3, 9, 27, 81, … This sequence has a factor of 3 between each number. The pattern is continued by multiplying by 3 each time. The value by which we multiply each time is called the "common ratio". Another example of a geometric sequence is: 1, 2, 4, 8, 16, 32, 64 … This sequence starts at 1 and has a common ratio of 2. The pattern is continued by multiplying by 2 each time.
3. The Triangular Number Sequence is generated from a pattern of dots that form a triangle. By adding another row of dots and counting all the dots, we can find the next number in the sequence: 1 dot, 3 dots, 6 dots, 10 dots, 15 dots…
4. The Fibonacci Sequence is found by adding the two previous numbers together. For example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, … The number 2 is found by adding the two numbers before it (1 + 1). The same rule applies to all the other numbers.

Some other special sequences are the Square Numbers Sequence and the Cube Numbers Sequence.

There are patterns in other parts of mathematics – Algebra, Geometry and Statistics as well. Learning maths is about learning patterns.

Patterns are present in our personal and spiritual lives in a very powerful way. Through patterns, we can learn to predict the future, discover new things and better understand the world around us. They make the world more predictable.

In our behaviour, emotions and thoughts, we follow some repetitive patterns. They give our lives routines, which can reduce anxiety. Routines give structure to our day. Patterns in behaviour are part of our civilisational belonging. We all know what is right and what is wrong when it comes to how to behave in society.

Patterns in our emotions are somehow predictable as well. We know what joy, happiness, sadness, kindness, guilt, shame, respect and rudeness are. And we are aware of when to feel these emotions. However, very often, since our childhood, we have been told the wrong things about how to show and when to experience some of these emotions.

Too much guilt and shame can paralyse your entire inner world for a long time. Too much reserve and keeping your emotions only to yourself can be damaging to your mental health; it can turn your heart into ice. So, when it comes to our emotional patterns, it is healthy from time to time to change the pattern. We do not want to be caught in a repetitive cycle of the same mistakes.

If we want to free ourselves from an unwanted behavioural or emotional pattern, we need to analyse the situation and the outcomes it leads to. We need to be able to see and recognise a certain pattern as a wrong one. Meditation, for example, is a good technique to give us some healthy distance from the situation we are in and from our thoughts and emotions.

Once we see the pattern, we can start thinking about changing it. We can start looking for motivation and inspiration to change some patterns for the better. Motivation is the drive behind the reasons we do things. Life outside changes so quickly that inside we cannot keep following the same old pattern that is damaging to us. Change allows us to develop and to grow.

We all have role models whom we admire. They could be people who have an impressive career, and by mirroring our lives in theirs, we could mirror their ambition as well. This could boost our professional life. Our role models could also be people who inspire our spiritual growth. Often they are people who have achieved some level of spirituality, and we find their kindness, modesty and service to others uplifting. We try to follow their example of becoming better people and spreading light and love.

The patterns in our emotional reactions and spiritual life are quite powerful and rather stubborn. Gradually they become like a good old habit that is sentimental and attached to the past but doesn't like the idea of us becoming new. Change likes honesty. This is the only way to break the old patterns and become freer and more spontaneous. Because happiness doesn't like chains.

Here are some activities for you:

Activity 1: Can you look around yourself and try to make a list of all the patterns you can see? What are the shapes that repeat themselves? What colours are they? What's the idea behind these patterns?

Activity 2: Can you make a design for a curtain using some shapes as a pattern? Which shapes would you use and which colours? Make it beautiful. What story do you want to tell?

Activity 3: Can you say which of your behavioural and emotional patterns you like and think make you happy and successful?

Activity 4: Can you describe the people whom you consider your role models in life? Why are they so special to you? What is it that makes them different from others?

(E. S. Lyubenova, LoveMaths Story for my Students)