Starting with maths: Patterns and formulas

This free course is available to start right now. Review the full course description and key learning outcomes and create an account and enrol if you want a free statement of participation.

Free course

# 9 Patterns in nature and elsewhere

If you take a cauliflower and break off one of the florets, the floret appears to be just the same shape as the original cauliflower but on a much smaller scale.

Figure 30 Romanescu cauliflower

This idea of self-similarity occurs throughout nature – the frond of a fern looks like the whole fern, a branch of a tree splits into further branches which look like trees themselves, and so on.

Self-similarity now forms part of an area of mathematical research known as fractal geometry. Beautiful patterns can be generated by fractal formulas, such as the Barnsley fern and the fractal below. Fractal geometry also has many practical applications in the physical sciences, medical research, economics and computing, particularly in compressing images.

Figure 31

You can generate part of a fractal yourself, by using Pascal's triangle that you constructed earlier in the course. If you shade in (or cover) all the odd numbers in the triangle, a pattern starts to emerge as shown below.

Figure 32

This is part of a fractal known as Sierpinski's triangle (Waclaw Sierpinski (1882–1969) was a Polish mathematician). It can also be generated by starting with a triangle, splitting it into four smaller triangles and removing the middle one. Then split each of the remaining triangles into four smaller triangles and remove the middle one. Carrying on in this way results in something like the sequence below.

Figure 33 Constructing Sierpinki's triangle
Y162_2

### Take your learning further

Making the decision to study can be a big step, which is why you'll want a trusted University. The Open University has 50 years’ experience delivering flexible learning and 170,000 students are studying with us right now. Take a look at all Open University courses.

If you are new to university level study, find out more about the types of qualifications we offer, including our entry level Access courses and Certificates.

Not ready for University study then browse over 900 free courses on OpenLearn and sign up to our newsletter to hear about new free courses as they are released.

Every year, thousands of students decide to study with The Open University. With over 120 qualifications, we’ve got the right course for you.

Request an Open University prospectus