Maths has strong connections to art and music, but what about to both at the same time?

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For the last fifty years, one of the most reliable ways of making sure that your artwork is seen by thousands of people around the world has been to stick it on the front of a (successful) album. Images used by artists like The Beatles, The Velvet Underground and The Clash are so famous not only because they adorn the fronts of CDs and LPs, but also because of the huge amount of branded merchandise that they have spawned. T-shirts, tote bags, coasters and Sex Pistols Virgin Money credit cards have ensured that album covers provide some of the most enduring modern works of art.

Given this, it might be easy to dismiss album art as a purely commercial exercise. But many artists also look to link their cover art with their music by visually representing some of the central themes. Perhaps it is not surprising, then, that some of the best album covers can be looked at through a mathematical lens. After all, mathematicians have always had ways to visualise abstract ideas, and there are easy connections to be made between physics and the psychedelic and cosmic imagery that was popular during the ‘golden era’ of the concept album. Let’s take a look at one of the most famous examples…

The dark side of the moon — Pink Floyd (1973)

(Artwork by Hipgnosis, George Hardie, Storm Thorgerson)

Pink Floyd’s 8th studio album has sold 45 million copies, featured on every ‘best albums’ list since its release and been through numerous re-masters and cover versions, so we can safely say that it is one of the most popular albums, with one of the most famous covers, of all time. The simple, geometric design with the iconic rainbow pattern on a black background is striking, and fits the sonic landscape perfectly. But it is also a classic childhood physics experiment in the field of optics.

The image, which shows a beam of white light hitting a triangular prism and splitting into its constituent parts (the rainbow), is a neat example of two related optical phenomena known as refraction and dispersion. The first of these explains why the rainbow and the incoming beam aren’t parallel, and the second tells us how the white light splits up as it passes through the prism.

Speak to me (about refraction)

First, let’s investigate some of the properties of light. The speed at which a beam of light travels depends on the medium (for example, glass, air or water) that it’s travelling through, and for a given medium we can compute the refractive index

$$ n = c/v $$

where $c$ is the speed of light in a vacuum, and $v$ is the speed of light through the medium. For air $n$ is close to 1, while for glass it is about 1.5—so light travels slower through glass than through air. Light (nearly) always takes the fastest path from A to B (that is, of course, how it gets everywhere so quickly) and the difference in speed between air and glass means that the shortest path in terms of distance may not be the fastest one.

The usual analogy is a lifeguard trying to reach a drowning swimmer: the lifeguard has to run across a beach and then swim through the sea to reach a swimmer, and they can run faster than they can swim. As in the diagram above, a straight line might not be the quickest route, since the lifeguard can travel the same distance along the beach in less time than they can in the water. If the optimal path isn’t straight, then the lifeguard will change direction when they go from sand to water. The same applies to light, and the change in angle is governed by Snell’s law:

$$ \frac{\sin{(\theta_1)}}{\sin{(\theta_2)}} = \frac{n_2}{n_1} $$

The angles $\theta_1$ and $\theta_2$ are measured from the normal to the material interface. Since glass has a higher refractive index than air ($n_2 > n_1$), a light beam passing from air into glass will bend towards the normal ($\theta_1 > \theta_2$). When the beam leaves the other side of the prism, the reverse happens and the angle increases again.

Time (to calculate the angle)

The exact change in the angle from entry to exit can be computed by drawing a quadrilateral (shown red below) made up of the prism and the beam of light.

Given an entry angle $\theta_1$, Snell’s law allows us to calculate $\theta_2$. We can then sum up the angles in the red quadrilateral, which gives $\theta_3$ in terms of $\theta_1$ and the refractive indices $n_1$ and $n_2$. Then, we apply Snell’s law again at the point where the light exits to give us $\theta_4$. This turns out to be the expression

$$ \sin{(\theta_4)} = \frac{\sqrt{3}}{2} \sqrt{ \left( \frac{n_2}{n_1} \right)^2 – \sin{(\theta_1)}^2} – \frac{\sin{(\theta_1)}}{2} $$

Any colour you like

So far, we’ve not mentioned anything about the most dramatic effect of the cover; the splitting of the white beam into its constituent parts. In fact, the refractive index $n$ depends on the wavelength of the light beam, or the colour of the light. For both glass and air, the refractive index is given by

$$ n = A + \frac{b}{\lambda^2} $$

where $\lambda$ is the wavelength of the light, and $A$ and $b$ are constants that depend on the material. In air, the value of $b$ is very small (about $10^{-18}$) and so the refractive index can be set to a constant, which means that our light beam arrives at the prism with all the colours hitting at the same place and time. Think about how weird this would be if it weren’t true, and different colours travelled at different speeds through air!

Once inside the prism, the refractive constants for glass are $A =1.5$ and $b= 0.05$, so we expect to see the light separating into colours as the angle is different for different wavelengths. This is known as dispersion.

Notice also that the refractive index is a decreasing function of $\lambda$, so longer wavelengths (the red end of the visible spectrum) have a lower refractive index. Turning back to our equation for $\theta_4$, we see that the exit angle is an increasing function of $n_2$—that is, longer wavelengths have smaller $n_2$, and thus smaller $\theta_4$.

In other words, the red end of the spectrum is the one that bends the least, which is exactly what we see on the cover of the album! In fact combining the ideas of refraction and dispersion allows us to calculate the exact journey that the light beam takes, and we can recreate our own version of the album cover by just picking an angle of entry $\theta_1$ and computing the different exit angles.

Not bad, eh? The mathematically accurate image matches up pretty well with the album art, so clearly the designers at Hipgnosis knew something about optics. There is one obvious area where they’ve used their artistic licence, however. From the reproduction above, we can see that the light should actually split inside the pyramid. Snell’s law tells us that $\theta_2$ will vary with the refractive index, and so the colours should split up straight away, before dispersing again (by a larger amount) when they leave the prism.

Another nice thing about this problem is that it’s not too hard to generalise. We can calculate the path of the light beam for any given prism just using Snell’s law and the angle between the two sides of the shape. The diagram above suggests that the equilateral triangle was a good choice. Repeating the calculations with a rectangular prism gives $\theta_1 = \theta_4$ … the angle doesn’t change at all! This would be much less memorable, and so Pink Floyd can thank a judicious choice of prism shape, and the mathematics of dispersion, for at least some of their fame.

Now, all that remains is to enjoy the music and… Breathe.