A square and a circle, as I imagine you probably know, look completely different – and you would certainly be right to say that they are different. Mathematically, however, a circle and a square share many structural properties and it may come as a surprise to some that in some cases mathematicians may go so far as to not even bother to distinguish between the two – in other words, the circle and the square can be considered one and the same thing. This is where topology comes in…

Topology is a very abstract area of geometry that simplifies many problems that would be very difficult if not impossible to solve. A formal introduction to topology requires knowledge of some basic set-theory and a knowledge of some analysis and metric spaces is usually helpful. I want to give a simple and informal introduction to this fascinating area of mathematics.

Given a set, $X$, a topological space is a couple formed from the set $X$ and a collection $\tau$ of subsets (called a topology) of $X$ that satisfy

  • $\emptyset \in \tau$ and $X \in \tau$
  • $\tau$ is closed under finite intersections
  • $\tau$ is closed under arbitrary unions

Anything (yes anything) that satisfies the above definition is a topological space. The set $X$ may be a region of the $x$-$y$ plane, a $3$-dimensional region of space, or even an abstract collection of objects – the level of abstractness of topological spaces can be demonstrated in the following example of a topological space

Let $X=\{Amy, Boris, Charlie\}$.

One possible topology, $\tau$, for $X$ could be $\{\emptyset, \{Amy\}, \{Amy, Boris\}, X\}$.

What has all this got to do with the square and the circle? Well two topological spaces can be considered the same (topologically indistinguishable) if a homeomorphism exists between the two. Homeomorphism is mathematical jargon but in simple terms it is a way of “moulding one space into the shape of another space.” Homeomorphisms can be informally visualised by imagining a circular piece of plasticine. It wouldn’t be too difficult to re-shape the plasticine into the shape of a square and vice-versa – this way of deforming the plasticine obeys the rules of a homeomorphism and therefore the circle and the square are topologically equivalent.

A circle and a square

A graph of a circle and a square showing topological equivalence

One possible function that takes the square onto the circle is $f(x,y)=\dfrac{(x,y)}{\sqrt{x^{2}+y^{2}}}$

The word topology doesn’t usually manage to make its way out of university maths departments, yet topology does have practical uses. If you have seen a London underground map before then you have seen a practical application of topology – if you have not seen an underground map before then you can see one here. In reality the tube tracks do not run in perfect straight lines but by representing the tube tracks as straight lines the whole thing is much easier to understand – the map of the London underground is topologically identical to the real thing. In other words, even though they are two different things, they have identical structures.

There are many other examples of topologically equivalent pairs of objects such as

  • the interval $(0,1)$ and the real-line $\mathbb{R}$
  • a sphere and a cube
  • $\mathbb{R}^{2}$ and the punctured sphere ($S^{2}$ with a point removed)

There are also many examples of pairs of objects that have significant structural differences and are not topologically equivalent such as

  • A sphere and a torus – these are not homeomorphic because the torus has a hole but the sphere does not; both are shown below for comparison.
  • $\mathbb{R}^{2}$ and $S^{2}$ are not homeomorphic
  • the letter $A$ is not homeomorphic to the letter $T$

A sphere

Sphere

A Torus

Torus

A Tautochrone is a curve described parametrically by the equations $x=a(\theta+\mathrm{sin}\theta)$ and $y=a(1-\mathrm{cos}\theta)$. Here is a graph of the curve for $\theta \in (-2\pi, 2\pi)$

A Tautochrone drawn using SAGE Math

The graph of a tautochrone drawn in SAGE math

This curve is not commonly encountered in mathematics, indeed, I only came across this curve for the first time about two months ago – but it has some very interesting properties. The curve itself is quite simple; however, if a particle were to be released from rest on one of the slopes of the tautochrone, then the time taken for the particle to reach the bottom of the slope is independent of its starting position assuming that the only force acting on it is the gravitational force. In other words, if you were to simultaneously set a ball rolling down the slope from the top of the slope and another ball from halfway down the slope then they would both arrive at the bottom of the slope at exactly the same time.

This property can be proved using energy considerations and some basic trigonometric identities to form the differential equation

$$a\dot{\theta}^{2}\mathrm{cos}^{2}\frac{\theta}{2}=g(\mathrm{sin}^{2}\frac{\theta_{0}}{2}-\mathrm{sin}^{2}\frac{\theta}{2})$$

where $g$ is the gravitational force – and then using integration by substitution we find that the time taken for the particle to reach the bottom is $T=\pi\sqrt{\frac{a}{g}}$ which is independent of starting position – you can download my full, detailed proof of this property here – Basic Properties of the Tautochrone. The derivation of the differential equation and then the integration that follows can all be done using techniques from the A-Level maths and further maths courses.

Since we are assuming that the only force acting on the particle is the gravitational force we can assume that all gravitational potential energy lost (remember that the particle will move down the slope and therefore lose gravitational potential energy) will be converted to kinetic energy. This is the starting point of the whole derivation of the above differential equation and although the resulting differential equation is non-linear, we are fortunate that it is nice enough to be able to solve – non-linear differential equations are notoriously difficult to solve and often impossible to solve analytically.

 

A function in mathematics is a very precise thing – one of the conditions is that a function can take a number in the domain to only one other number in the range at a time. For example $y=x^{2}$ will take 2 to 4 but no other number and $y=x^{3}$ will take $-3$ to $-27$ and no other number. This doesn’t mean that a number in the range can only have come from one number, for example $y=x^{2}$ will also take $-2$ to $4$.

We often see graphs of functions such as $y=x^{2}$ and $y=\mathrm{e}^{x}$ and it is a common mistake to think that every graph is the graph of a function – this is not true. For example take the graph of $y=\sqrt{x}$ as shown below

Graph drawn using SAGE Math

Graph of a curve that is not a function (?)

this graph is not the graph of a function because $x=4$ is taken to both $-2$ and $2$; this is not allowed by the definition of a function.

To resolve this we have to introduce a new independent variable (often $t$) called a parameter and make $x$ and $y$ functions of the parameter $t$. This is like adding a third axis to the usual $x$-$y$ plane – this new axis is the $t$-axis.

Here is a graph of the parametric equations $x=\mathrm{cos}t$ and $y=\mathrm{sin}^{3}t$

Graph of a parametric relation

2-dimensional graph of a parametric equation created in SAGE Math

This, however, appears not to satisfy the definition of a function since there are several values of $x$ that are taken to two different $y$-values. Lets look a little closer at what is going on. If we add a further axis – the $t$-axis – to the diagram then this is what we see,

SAGE Math 3D Parametric Plot

A 3-dimensional parametric plot created in SAGE_Math

As you can see from the plot each value of $t$ corresponds to exactly one point in three-dimensional space.

If we look at this plot from a certain angle – in particular, straight down the $t$-axis we see the following

SAGE Math 3D Parametric Plot

3-dimensional parametric plot created in SAGE Math

Now if we ignore the effects of perspective we see exactly the original graph that we started with – so even though we originally thought that this did not satisfy the conditions to be a function, it actually does. By using the trigonometric identity $\mathrm{sin}^{2}t+\mathrm{cos}^{2}t \equiv 1$ we can show that the Cartesian form of this curve is given by $y^{2}=(1-x^{2})^{3}$ which is much more difficult to deal with than the parametric form of $x=\mathrm{cos}t$ and $y=\mathrm{sin}^{3}t$

Similarly $y=\sqrt{x}$ can be represented by the parametric equations $x=t^{2}$ and $y=t$ and again looking at the three dimensional plot straight down the $t$-axis gives a graph like the one earlier in the post. So sometimes things aren’t always what they first seem. With the help of parametric equations we can produce graphs of some very interesting curves that we can now treat as functions – not only that but parameterisations of curves and paths are critical to being able to do higher level calculus.