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## 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 …

## 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 …

## 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 …

## Beyond power and polynomial relations

In this video we discuss other kinds of relations going beyond the elementary polynomial and power laws, culminating with a formula of Cayley on the number of labelled trees on …

## 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 …

## Scaling laws in biology

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

## Metabolic rate and Kleiber’s law

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

## Pareto and the distribution of wealth

More than a hundred years ago, the Italian economist Vilfredo Pareto (1848- 1923) studied income and wealth distributions. And he noted a curious fact: that certain power laws seemed to …

## Lifespans and heartbeats

In this video we explain the mathematical power laws that relate animal mass to heart rate and lifespan. Allometry In biology, the application of power laws is part of allometry, …

## Lifespans of animals

What is the relationship between the size of an animal and how long it lives? This interesting topic raises lots of questions, many quite pertinent to our modern lives. It …

## Starbucks in your city

Sometimes in real life we have data that does not fit into our preconceived patterns or formulas. There are so many questions we can ask about the modern world: it …

## The power of exercise

It is commonly accepted that regular exercise adds years onto life, but what exactly is the power of exercise? And is this power a power law? In this step, we …

## How many gas stations has your city?

A power law relation has the form $$\normalsize{y=ax^k}$$ for some number $$\normalsize{k}$$, not necessarily an integer. In this video we introduce this more general kind of relation, explain how the …

## Gas stations and population

The larger a city is, the more gas stations it will have. On average, there ought to be some kind of quantitative relationship between the number of gas stations $$\normalsize{G}$$, …

## Cubic polynomials and their roots

Just as for quadratic functions, knowing the zeroes of a cubic makes graphing it much simpler. Typically a cubic function will have three zeroes or one zero, at least approximately, …