Hope you had a good week so far. If you have read my previous post, you were probably expecting this post. While visiting Edinburgh Butterfly & Insect World I have spent some time watching the ants they have there. Looking at their organized trails and how they were carrying things around, I was sure there was some mathematics in there. So, when I came home afterwards I had to do some research on this.
I was not surprised to find out about 3 interesting examples of how ants use mathematical concepts. Or, maybe I should say, 3 interesting examples of ants’ behaviour that helped mathematicians and scientists understand mathematics and apply the concepts in different industries.
First example is on an old mathematical problem: estimating area of a shape. If the shape is regular then you can think of any of those area formulas you have learnt at school, but what it is not? What if it is uneven, with wholes and different stone sizes, not to mention extremely dark? It turns up that ants of the species Temnothorax albipennis can measure and compare the areas of two or three such irregular shapes to find their perfect home. These ants use a method, which can also be used to calculate/estimate pi. More specifically they use the geometry of their trails to estimate the area. The process they use seems quite easy to understand: they make some random trails (using pheromones) around the shape in their first visit, then they repeat the same process a second time or a third time depending of the size of the shape. Next time they revisit the place, all they have to do is “count” how many times the trails intersected each other. There is a mathematical formula which shows that the area is inverse proportional to the intersection frequency; in plain English: the smaller the place, the more times the trails will intersect. The formula was actually first explained by a French mathematician called Georges Louis Leclerc in 1777, when he was studying probability through geometry and found a way to estimate the value of pi.
Another research from December 2010 at University of Sydney published in the Journal of Experimental Biology talks about efficient paths. A couple of scientists tested something very strange and interesting at the same time. They wanted to see if Argentine ants (Linepithema humile) could solve a dynamic optimisation problem. To test this they converted the classic Towers of Hanoi maths puzzle into a maze. Towers of Hanoi is a very known mathematics puzzle which consists of three rods and some disks of different sizes that slide onto the rods. It starts with all the disks stack from big to small on one rod, making a shape that looks like a party hat (a cone). The objective is to move the entire stack to another rod following some specific rules. Everything sounds very interesting and my only problem is that I couldn’t find out more about how exactly they changed this puzzle and how the maze actually looks like. If there is anyone that has any idea, let me know.
Anyway, from entering the maze the ants had 32, 768 possible paths to the food source. If you ask me, this is a huge number and at first I thought this will be a fail for the ants, but in around an hour they have found the two shortest paths (the optimal solution). Then, they changed the maze a little and repeated the test. Again, it took the ants another hour to find the optimal solution.
This type of research is very important to our society. Just think at delivery drivers, telephone routes, internet cables and so on. Wouldn’t it be great if we could always find the most efficient path?
Another research of ants’ movement was a study by researcher from Spain and U.S.A. Again they were observing the Argentine ants, Linepithema humile, (they seem to be a favorite). This time they were studying their movement (foraging, exploring) through empty spaces. They have discovered that their paths were following a very interesting pattern. And who doesn’t love patterns?
María Vela Pérez, researcher explained: “To be more specific, they are a mixture of Gaussian and Pareto distributions, two probability functions which are commonly used in statistics, and that in this case dictate how much the ant ‘turns’ at each step and the direction it will travel in”. The Gaussian distribution, also called the normal distribution, is a common continuous probability distribution, which means that it has a cumulative distribution function that is continuous. On the other hand, the Pareto distribution is a power law probability distribution, which means that it uses the exponential function. Both these distributions are well known in statistics and they are used in many domains from natural sciences to social sciences.
Hope you enjoy this post. More nature related posts are coming soon. Let me know what you’ve been up to in the past month. Have a great day. You can find me on Facebook, Tumblr, Google+, Twitter and Instagram. I will try to post there as often as possible.
Lots of love and don’t forget that maths is everywhere! Enjoy!