Sections
Tasks
- How many pegs would you need to get an accurate measurement of the exhibit coastline?
- Imagine that you have an endless amount of string and pegs. What measurement would you get for the coastline now?
- Is it possible for a finite area to have an infinite perimeter?
Scroll down to the maths section below to find out the answers!
Maths
Explain the paradox:
The coastline of the UK has many peninsulas, inlets and caves which make finding its true length extremely difficult.
We can approximate the coastline by using straight line segments, known as rulers, around the UK and then we can count how many we used to find the estimated distance.
𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐶𝑜𝑎𝑠𝑡𝑙𝑖𝑛𝑒 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑟𝑢𝑙𝑒𝑟 × 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑢𝑙𝑒𝑟𝑠 𝑢𝑠𝑒𝑑
We can find a more accurate measurement by using a smaller length ruler. The length of the coastline will increase as we are able to more accurately model the complex shape of the coastline.
When we choose very small length rulers such as 1m, the length of the coastline becomes very large, and approaches infinity as we continue to shorten the length of our ruler to say 1cm or 1mm.
No matter how much you zoom into the coastline of the UK there will a shape sufficiently small enough which cannot be fully approximated by our straight-line rulers.
So, what is the length of the coastline?
Ans: is that it depends on the length of your ruler
Task Answers
How many pegs would you need to get an accurate measurement of the exhibit coastline?
As many as possible.
Imagine that you have an endless amount of string and pegs. What measurement would you get for the coastline now?
An infinite length.
Is it possible for a finite area to have an infinite perimeter?
Yes, there are shapes known as Fractals which have this property!
Fractals
What is a Fractal:
A fractal is an object which has a “self-similarity” property.
Fractals are present in nature for example in Romanesco Cauliflowers (shown in the image) to even trees, rivers and snowflakes.
The Cauliflower can be seen to have similar patterns which seem to exist even on an infinitely small level.
The most famous Fractal structure is the Mandelbrot set.
This is created using a recursive equation with imaginary numbers as inputs. 𝑍𝑛+1=𝑍𝑛2+𝑐
If you continue to zoom into this set there will always be a repeated pattern or structure.
How are fractals related to the Coastline Paradox?
The UK Coastline exhibits a fractal property, which is the reason why it is difficult find the exact length.
History
Who discovered the paradox?
In 1951 Lewis Richardson discovered that the border between Portugal and Spain was measured differently.
The Portuguese believed the distance was 987km but the Spanish had a distance of 1214km. Richardson discovered that this was due to the ruler length chosen of estimating the border.
The Spanish used a smaller measurement which resulted in a more accurate estimate of the border.
People
Lewis Fry Richardson 1881-1953
L F Richardson was a British mathematician most famous for his work in physics, meteorology, and psychology.
He is noted for his research on fractals and the modified Richardson iteration which is a method of solving a system of linear equations.
Benoit Mandelbrot 1924 – 2010
B Mandelbrot was a Polish born, French mathematician most famous for his Mandelbrot set and contributions to the field of Fractal Geometry.
Applications
Financial Markets
Fractal Market Hypothesis is the theory which supports the idea that stock prices move in fractal like way. Analysts can then use this to predict further price movements by using historic data.
Medicine
Fractals can be used in medicine to analyse and treat cancer cells. Blood cells grow and develop in a fractal manner, in a repeated and self-similar way, whereas cancer cells are the opposite and grow in an abnormal way which makes it easier to detect.
Maths at Home
Play around with this online Fractal generator to create your very own fractal structure.
The website lets you colour, trail and even animate your fractal is it develops over time.
The software is from the sciencevsmagic page mentioned below:
https://sciencevsmagic.net/fractal/#0060,0090,1,1,0,0,1