# Lesson Ten

This lesson looks at:

Patterns and number sequences

The Fibonacci sequence

Using Python to calculate number sequences

The Golden Ratio

## Patterns in nature

A sequence is a set of numbers that have an order and follow a pattern. Patterns appear in nature, from waves on sand dunes to the shapes of leaves. Take a look at the pictures below. Can you spot any patterns?

## Number Sequences

The simplest number sequence is:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

The rule to make the next number in the sequence is to add 1 to the last number.

Find the rules for the number sequences below:

2, 4, 6, 8, 10, 12, 14, 16, 18, 20

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024

Take a look at this sequence. Can you work out how to get the next number?

1, 1, 2, 3, 5, 8, 13, 21, 34, 55

That final one is a little trickier. It is called the Fibonacci sequence. It was discovered by a mathematician called Leonardo Fibonacci in the year 1202. This number sequence is interesting as it appears quite often in plants. Take a look at the flowers below. How many petals can you count on each flower?

Did you notice anything about the numbers of petals you counted? The Fibonacci sequence appears quite often in nature in things like the number of petals a flower has. We can use computers and programming to calculate sequences. Can you think of any reasons when this might be useful?

## Programming in Python

Follow the instructions and write a program using the Python programming language that can calculate sequences.

## The Golden Ratio

We can use Fibonacci numbers to create special rectangles called Golden Rectangles. If we start with two 1 x 1 squares next to each other, and draw a third square next to them that is 2 x 2, we will have created a Golden Rectangle. If we keep on going, drawing bigger and bigger squares, we will end up with a rectangle that has the same proportions as the first. For all of these rectangles, if we divide the longer side by the shorter side, we get a number called the Golden Ratio, which is approximately 1.61.

If we draw an arch going corner to corner in each of the squares, we can create a shape called a Golden Spiral.

Like the Fibonacci numbers, the golden spiral often appears in nature! Take a look at a sunflower head. Are the seeds all lined up in rows? Or do they follow a different pattern?

The seeds in a sunflower grow in a spiral – a Golden Spiral!

Try drawing your own Fibonacci spirals on paper or by using Python and Trinket!