## Archive for May, 2014

### The High-Power Hypar

Maths frequently shows us surprising and illuminating connections between physical systems that are not obviously related: the analysis of one system often turns out to be ideally suited for describing another. To illustrate this, we will show how a surface in three dimensional space — the hyperbolic paraboloid, or hypar — pops up in unexpected ways and in many different contexts.

Warszawa Ochota railway station, a hypar structure
[Image Wikimedia Commons].

### The Chaos Game

The term “Chaos Game” was coined by Michael Barnsley [1], who developed this ingenious technique for generating mathematical objects called fractals. We have discussed a particular fractal set on this blog: see Cantor’s Ternary Set.

The Chaos Game is a simple algorithm that identifies one point in the plane at each stage. The sets of points that ultimately emerge from the procedure are remarkable for their intricate structure. The relationship between the algorithm and fractal sets is not at all obvious, as there is no evident connection between them. This element of surprise is of one of the delights of mathematics.

### The Future of Society: Prosperity or Collapse?

The article in this week’s That’s Maths column in the Irish Times ( TM045 ) is about a mathematical model to simulate the future of society.

Our extravagant lifestyle is draining the Earth’s natural resources. Population is climbing and climate change looms ever larger. Is the collapse of society imminent?

The historical precedents are ominous. Many civilizations have ended abruptly with drastic population reductions and centuries of oblivion. The fall of the Roman, Han, Mayan and Gupta Empires show that advanced and sophisticated civilisations can be fragile and impermanent. There are many causes, but over-consumption of resources and inequalities within society are primary.

### Predator-Prey Models

Next week’s post will be about a model of the future of civilization! It is based on the classical predator-prey model, which is reviewed here.

Solution for X (blue) and Y (red) for 30 time units. X(0)=0.5, Y(0)=0.2 and k=0.5.