## Cubic curves I: historical aspects

In this video, we discuss cubic curves and their history, going back to Isaac Newton. The basic cubic function The basic cubic function is of the form (normalsize y = …

## Power laws, polynomials and why big animals live longer

This is an introductory video to Week 3 and the topics of power laws, polynomials and their applications to biology. Many of you have worked very hard in the first …

## Benford’s law

In this video we describe Benford’s Law: a curious observation about the unequal distribution of first digits in random numbers. It was first enunciated by Simon Newcomb (also of Newcomb’s …

## Zipf’s law

In this video we discuss Zipf’s law: a curious relation that connects distributions of words and populations of cities to inverse relations. It was first noticed by George Kingsley Zipf, …

## Rational functions and Mobius transformations

In this video we discuss rational functions and their special cases called Mobius transformations. In our earlier course Maths for Humans: Linear and Quadratic Relations, we saw that by translating …

## Inverse relations from travel, gases and electricity

In this video we give an overview of different examples of inverse relations associated with physics: velocity, Boyle’s law and Ohm’s law. Two quantities have an inverse relation if their …

## Hyperbolas: the coolest conic sections

The hyperbola is a rather special conic section. It is the curve that we get when we slice a cone with a plane that meets both the top and bottom …

## Inverse relations, hyperbolas and Zipf’s Law

Here we introduce our first interesting topics: Inverse relations, hyperbolas and Zipf’s law, giving an overview to what is ahead. So this video covers both Weeks 1 and 2 in …

## Trees and Cayley’s formula

Of course the world is a complicated place, and there are many kinds of relationships that do not fit smoothly into the kinds we have discussed here. In fact once …

## Steiner’s regions of space problem

Jakob Steiner was a prominent Swiss geometer who made many remarkable contributions to mathematics. In 1826 he asked: what is the maximum number of regions you could divide the plane …

## Gravitation and Newton’s inverse square law

Isaac Newton was a giant of modern science, and one of the key architects of the modern world. One of his major insights was that the force of gravitational attraction …

## Kepler’s Third Law: the law of harmonies

In 1609 Kepler formulated his first two laws to summarize the remarkable astronomical observations his mentor Tycho Brahe had made over many years. Ten years later he published his third …

## Allometry and the Fiddler crab’s claw

The study of how things scale in biology, usually with respect to body size, is called allometry. The term was introduced by Huxley & Tessier in 1936 in their study …

## Metabolic rate and Kleiber’s law

Kleiber’s law was proposed by Max Kleiber, a Swiss chemist, in the 1930s. The idea is to try to quantify the relation between how big an animal is, and what …

## Scaling laws in biology

It has been observed that the heart rate and metabolic rate per cell is less for large animals, and greater for small animals. All well and good, but having some …