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Power laws, polynomials and why big animals live longer

Power laws have lots of applications in biology. Watch Norman Wildberger introduce some of these connections in this video.
This week, we’re going to go beyond the linear, quadratic, and inverse relationships that we’ve been learning about to include more general kinds of relations that are still quite useful. We’ll start by talking about cubic relations. They’re all based on cubic functions and cubic curves, which have a very interesting geometry as well, studied by Isaac Newton, and Descartes, and also Fermat, and others. So we’ll have a look at more general power laws as well, where one quantity is proportional to another quantity raised to another power, perhaps even a fractional power, like x to the 1/2 or x to the 3/4. We’ll see that these kind of unusual power laws still have quite interesting applications to biology, and economics as well.
In fact, there’s a surprising relationship between the number of gas stations in a city and the population of that city. We’ll also see another very important historical example of that, in the inverse square law of Isaac Newton, which is at the heart of the understanding of planetary motion. Then it will turn out that these same kinds of power laws play an important role in modern biology, connecting sizes of animals with various things like lifespans, heartbeats, metabolic rates. We’ll even have a look at the fiddler crab’s claw and how its size is related to the crab.
And finally, we’ll have a look at some extra kinds of relations going beyond the kinds that we’ve been talking about, talking a little bit about log and exponential functions, and some new directions that are heading towards calculus.
This is an introductory video to our final Week 4 and the topics of power laws, polynomials and their applications to biology. Many of you have worked very hard in the first three weeks, really well done!
We are going to start off this week by looking at cubic relations, and more generally cubic functions and curves. Cubic relations connect to volume, cells and biology. They are a step up from quadratic relations of course, and while there are a lot of similarities there are some important differences too. One of these is that we don’t have the quadratic formula around to help us.
It turns out that cubic curves have a rich and interesting algebraic structure too, which is important in modern cryptography.
And what happens once we go beyond cubic polynomials? Then we get more general polynomial functions, whose general shape and properties follow the pattern of cubics. The simplest kinds of these are the pure powers \(\normalsize y=x^n\) where \(\normalsize n\) is a ‘natural’ (whole, non-negative) number.
In many applications however, more general power laws appear, of the form
\[\Large y=ax^k\]
where \(\normalsize k\) is a fractional number, such as \(\normalsize k=1/2\) or \(\normalsize k =3/4\). To understand those, we will want to review some fundamental index laws.
Such power laws occur frequently in biology, and we will be having a look at biological applications this week. And finally we will be venturing beyond polynomial laws, towards exponential functions and relations.
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Maths for Humans: Linear, Quadratic & Inverse Relations

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