The Mathematics of Shell Construction

and other patterns

GROWING UP IN MORECAMBE 2008 GROWING UP IN MORECAMBE 2008

The Mathematics of Shell Construction

and other patterns

Contents

3 Fibonnaci Numbers12 Simple Rules and Complex Patterns14 The Game of Life16 Chaos

Fibonacci NumbersThe Fibonacci series of numbers is a sequence that starts with 0 and 1 and then increases by adding together the previous two numbers in the sequence in order to get the next:

0 + 1 = 11 + 1 = 21 + 2 = 32 + 3 = 53 + 5 = 8

So the sequence goes:

0 1 1 2 3 5 8 13 21 … etc

This repeating pattern of numbers can potentially go on increasing forever.

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The Fibonacci series of numbers is thought to be important because it is a pattern that appears regularly in the natural world. The number of petals on a flower is often a Fibonacci number – flowers often have 3, 5 or 8 petals, but rarely have only 4.

Plants like Sneezewort and some trees grow new shoots and branches in a pattern that approximates the Fibonacci sequence - during each stage of development the number of shoots that these plants are growing will usually be a Fibonacci number and the number of leaves will also be a Fibonacci number.

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You can make a Fibonacci series of rectangles by placing together squares as follows:

Start with the first two numbers in the Fibonacci series: 0 and 1. Draw a square with sides measuring 0 units next to a square with sides measuring 1 unit. A square with sides measuring 0 units cannot exist so you are left with a single square with a length and a width measuring 1 unit.

The next number in the Fibonacci series is also 1. Place another square with sides measuring 1 unit next to your first square.

This is the first rectangle. It has a width of 1 unit and a length of 2 units. Its length is equal to its width, plus the width of the previous square in the sequence.

1 + 1 = 2

Length = 1Width = 1

Length = 2Width = 1

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Two is the next number in the Fibonacci series of numbers, so now place a square with sides measuring two units on the end like this:

Length = 3 Width = 2

This is a Fibonacci rectangle. Its width is equal to the length of the previous rectangle in the sequence. This was 2. Its length is equal to its own width, plus the width of the previous rectangle in the sequence. The width of the previous rectangle in the sequence was 1. So in this case the length is equal to 1 + 2.

1 + 2 = 3

Three is the next number in the Fibonacci series of numbers, so now add a square on the end of this rectangle with sides that measure 3 units.

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Length = 5 Width = 3

Again, the width of the rectangle is equal to the length of the previous rectangle in the series.

The length of this rectangle is equal to its own width, plus the width of the previous rectangle in the series. Both of these widths are Fibonacci numbers, and the length of the rectangle is the next Fibonacci number in the series.

Like the Fibonacci numbers themselves you can continue this pattern forever, each time adding another square with sides that are the same length as the next Fibonacci number in the series.

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The squares follow each other round, moving outwards from the centre. If a quarter circle is drawn inside each square, starting and finishing at the corners where the square meets the next square, then a spiral is made.

This spiral pattern is the Fibonacci Spiral.

Like the Fibonacci series, it can continue getting bigger forever. However, it cannot continue getting smaller forever because it has a fixed start point – in the square with sides that measure 1 unit.

This means it is not a true mathematical spiral, since a true mathematical spiral can continue getting bigger and smaller forever. It is however very close to a true mathematical spiral called the Golden Spiral, which fits into a similar repeating pattern of rectangles known as the Golden Rectangle.

The Golden Spiral is one example of a particular kind of spiral, called a “growth spiral”*. This is because the line of the spiral grows faster the further away it is drawn from the central point where the spiral begins – the space inside the curve of the spiral therefore gets bigger further out from the centre.

This is different to other types of spiral, such as the Archimedean spiral. The space inside the curve of an Archimedean spiral remains the same size no matter how far out the spiral is drawn from the centre.

The Golden Spiral and the Fibonacci numbers are * This type of spiral can also be called a logarithmic spiral, an equiangular spiral, the spira mirabilis (marvellous spiral) and the Bernoulli spiral.

Archimedean Spiral Growth Spiral

Both of these spirals grow, but in a different way.

The kind of spiral pattern called the growth spiral is regularly found in nature, particularly in the shells of sea creatures and snails. An example that is often used is the Nautilus shell. While the average spiral of a Nautilus shell isn’t exactly the Golden Spiral, it does tend to display a growth spiral that has a similar pattern.

Growth Spiral Nautilus Spiral Live Nautilus

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The shape of the growth spiral is seen in the Nautilus because as the animal that lives inside the spiral shell grows, it will need a bigger space inside its shell to live in. The animal is constantly building its shell outwards from the centre, so the spiral of the shell will therefore be built bigger as the animal grows and moves through the structure of the shell.

Many other species of shell also grow in patterns that are close to a growth spiral, but look very different from either the Golden Spiral or the Nautilus.

Mathematical growth spirals all follow the same rule, but can look very different depending on the numbers to which the rule is applied. In a similar way, natural forms like shells that resemble growth spirals can look very different from each other depending on the size, shape and behaviour of the animals that build them. Even within species of shell, there are always individual variations.

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Simple rules and complex patterns

The Fibonacci sequence of numbers follows a very simple rule, but the forms, such as the spiral, that can be produced by repeating the rule are more complex.

Structures that are found in nature, such as the shells of sea creatures, are complex shapes, but they display repeating patterns that can often be described mathematically using simple rules. By looking at the mathematical patterns that certain natural forms and structures are similar to, we can try to understand better how they work and why they are the shapes they are.

There are lots of complex patterns that appear in the natural world that may be modelled in some way using fairly simple and regular mathematical rules – the structures and forms of plants and shells are just two examples. Animal behaviour, the human body and the weather are all systems that are complex and difficult to understand, but they can be explored using simpler mathematical models as a tool for learning about how they work.

There are many more examples. Think of the way birds fly together in large groups, or the patterns that waves form in the sea.

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Above: Birds in flight at Morecambe Bay; Patterns in the sand left by waves, Morecambe Bay

The Game of LifeThe Game of Life, invented in 1970 by mathematician John Conway, provides a good demonstration of how very simple rules can lead to complex patterns. It has been used as a tool to think about how other complex patterns and systems might work.

The Game of Life is played on a grid of squares. Each square is called a cell and a cell can be “alive” or “dead”. This is usually displayed by colouring the live cells and not the dead cells.

Live cell Dead cell

Each cell has eight neighbours – these are the cells directly above and below, to the left and right and in each diagonal.

A cell surrounded by its neighbours.

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For each “round” or “generation” of the game, the following rules apply:

1) Any live cell that has less than 2 living neighbours dies of loneliness.2) Any live cell that has more than 3 living neighbours dies from overcrowding.3) Any live cell that has 2 or 3 living neighbours stays alive.4) Any dead cell that has exactly 3 living neighbours comes to life.

In each generation, the rules must be applied at the same time to every cell in the grid. Once this has happened the next generation can take place, and so on.

You can play the Game of Life by hand on graph paper or by using a wooden board and counters to get the idea of how the rules work. You can also play the game using a computer program that works out each new generation a lot faster than a person can.

When you use a computer to play the Game of Life it is easy to see that some very complex patterns are created by applying the rules to different combinations of living and dead cells.However if you are playing the game, you can start by choosing which cells in the grid you want to be alive or dead. Experiment with different combinations and see what patterns are created when you start to apply the rules of the Game of Life.

All of the materials to play the Game of Life will be available at the Shell-Lantern Building Workshops.

ChaosWhile the Game of Life always uses the same basic rules, you will see that very different complex patterns are created depending on the cells that are “living” or “dead” when the game is started.

The specific pattern of living and dead cells at the start of the game determines which patterns will be formed when the rules of the game are applied, but a single cell being set to “living” instead of “dead” might make a very large difference in the pattern that appears.

The Game of Life is sensitive to initial conditions – this means that a very small change in the layout at the beginning can produce a very large difference in the way that a pattern develops. It is difficult for a person to predict how a small change will affect the pattern, and so even though we know they are governed by simple rules, patterns in the Game of Life can appear random.

The appearance of random behaviour in non-random systems of patterns where small changes can have large effects is called chaos.

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Chaos is found in lots of complex systems. The weather, the patterns of waves in the sea, and the patterns of birds in flight are all systems of behaviour that are affected by chaos. Small changes at the start of the patterns can have large effects that are difficult to predict.

Mathematical tools such as the Game of Life are used to try and understand how other complex systems might work. They won’t exactly have the same rules or work in exactly the same way, but they might be similar.

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Use this booklet together with ‘Building the Shell Lanterns’ to find out more about the mathematical relationships used in constructing the shell lanterns. There will be opportunities to explore all the mathematical relationships explained here at our workshops.

GROWING UP IN MORECAMBE 2008 GROWING UP IN MORECAMBE 2008

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2008