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.

Following on from another post that I made recently on maths and music here is a video that I made of some wave-patterns that I created in SAGE Math of the common musical intervals up to an octave – I think it can be interesting to see what the waves look like so that you can see how the different frequencies interact with each other. There’s also a couple of wave patterns for some simple chords.

After playing around with SAGE-Math over the last week I discovered that it is possible to plot complex-valued functions. Unfortunately complex functions are functions from $\mathbb{C}$ to $\mathbb{C}$ which means that unless you are pretty good at visualising things in four dimensions it can be tricky to visualise them. A function from $\mathbb{R}$ to $\mathbb{R}$ such as $y=x^{3}-5x^{2}+4x+14$, which is a function from a one-dimensional space to another one-dimensional space, can be displayed on the familiar two-dimensional $x$-$y$-axes as in the diagram below.

A cubic equation

The graph of a cubic equation

However, with a function from $\mathbb{C}$ to $\mathbb{C}$ we are taking points from a two-dimensional space to another two-dimensional space so we would need four dimensions to plot the graph of the function. I don’t know about you but I’ve never really got the hang of seeing things in four dimensions so we have to have a different way of displaying the data. This is where SAGE comes in useful.

Here is a plot of the function $f(z)=\dfrac{z^{2}}{1+z^{4}}$ that I put into SAGE to plot.

Complex-Plot created in SAGE Math

Plot of a complex-valued function.

The code that I used to do this is surprisingly simple:

p = complex_plot(lambda z: (z^2)/(1+z^4), (-2, 2), (-2, 2));p

The plot shows what happens to each complex number after it has been transformed. The colours in the plot correspond to different behaviours of the function.

In the plot above, zero is the only point that is mapped by the function to zero – notice how the colours are quite dark near the origin; this means that these points are mapped to other points in the complex plane of small magnitudes. Darker colours correspond to points that are mapped to complex numbers of relatively small magnitudes and lighter shades represent points that are mapped to complex numbers of larger magnitudes. As you move out from the centre the colours become lighter and there are white spots at the points $\dfrac{1}{\sqrt{2}}(1+i), \dfrac{1}{\sqrt{2}}(1-i), \dfrac{1}{\sqrt{2}}(i-1), -\dfrac{1}{\sqrt{2}}(1+i)$ these are the singularities, or poles, of the function – these are the points that can be considered as being mapped to infinity.

The plot has a certain degree of symmetry – any points that are the same colour are mapped by the function to the same point. For example $z=1+\dfrac{3}{2}i$ and $z=-1-\dfrac{3}{2}i$ are mapped to the same point and are therefore the same colour on the plot. I’m sure that I will be able to use these plots in the future – here’s a few more that I managed to create of the functions $f(z)=\dfrac{\mathrm{sin}(3z)}{1+z^{4}},\; f(z)=z^{2}$ and $f(z)=\dfrac{z^{3}-3z^{2}+4}{z^{4}-2z^{2}+12}$

Complex-Plot created in SAGE Math Complex-Plot created in SAGE Math Complex-Plot created in SAGE Math

 

I have been taking quite an interest in Fourier Transforms lately. Fourier Analysis was one of the courses that I did during my final year at Warwick – I didn’t know what to expect at the time since I had heard about Fourier Series and Fourier Transforms but had never really worked with them or studied any of the theory. The course, like many of the other maths courses at Warwick, was very fast-paced and extremely demanding. I spent most of my time trying to keep up with all of the definitions, theorems and proofs not just from this course but from all of the other courses that I was doing at the time. I more or less just had to accept that there existed these things called Fourier Transforms but I never really got round to settling down to try and visualise them to really understand what they were (after all, there are only 24 hours in a day).

Well, a few years later and I have managed, to some degree, to do just that – better late than never; but I realise now how exciting and interesting Fourier Transforms are (no, really).

Fourier Transforms are a way of breaking a function down. The function can (and often does) represent a sound wave and the Fourier Transform enables you to analyse the frequencies that are involved – the transform tells you which frequencies are present and how much a particular frequency contributes to the overall sound wave being analysed.

From the transform it is then possible to see which frequencies are dominant in the sound wave and what gives it its characteristic feel. For example, a sound wave produced by a guitar will be very different from the sound wave produced by a saxophone – even though they may be playing the same note, neither one is playing a completely pure note; each will have various other frequencies present and in different amounts which all interact with each other to produce the sound that you recognise as either a guitar or a saxophone. This explains why it is possible to distinguish the sound of one instrument from the sound of a different instrument just by listening to them – as your ear receives the various frequencies all interacting with each other your brain is then able to recognise the characteristic pattern of the wave and determine whether you are listening to a violin or a piano.

Fourier transforms take the original function (which is in the time-domain) and converts it to a function in the frequency-domain.

Here is an example

Rectangle Function

Rectangle Function

Sinc Function

Sinc Function

The picture on the left is the rectangle wave (very important in digital signal processing) and the picture on the right is its Fourier Transform, the sinc function defined by $$f(x)=\dfrac{\mathrm{sin}(\pi x)}{\pi x}$$

The sinc function tells us that the dominant frequencies in the rectangle wave are between 0 and 1 and as the frequency gets higher then its contribution generally decreases. This is just the beginning – Fourier theory finds its way into many different and unexpected areas of science from pure mathematics, to signal processing, astronomy and music technology.

I have been playing around with Fourier Transforms on SAGE Math over the last couple of days so I will be posting a bit more about them over the coming weeks and going into a bit more detail with them.