Pink Floyd’s most famous album The Dark Side of the Moon is not only the UK’s seventh best selling album of all time – it also boasts one of the most instantly recognisable album covers in music history.
Whether it’s on the the vinyl cover or on a cool t-shirt, it’s hard to find people who haven’t at least seen the artwork somewhere in their lives.
As it turns out, there is a lot of mathematics behind the art of the cover, owing to its simple geometric design and rainbow pattern.
It shows a beam of white light hitting a triangular prism splitting into ‘constituent parts’ (which is the rainbow), according to the maths magazine Chalkdust.
It’s an example of an ‘optical phenomena’ known as refraction and dispersion, which may sound a little complicated, but it’s more understandable once you break it down.
The refraction and dispersion explains why the rainbow and the incoming beam are not parallel, and how the white light splits up as it passes directly through the prism.
Sean Jamshidi, a PhD student at University College London, broke down the significant mathematical points of the album cover and found three parts to focus on.
The refraction displayed in the album cover is all to do with the speed at which a beam of light travels (depending on the medium) which could be glass, air or water.
Given the medium, mathematics can compute refractive index:
N =c / v
In short, c is the speed of light in a vacuum, and v is the speed of light through the medium.
n is close to 1, while for glass it is about 1.5 – so light travels slower through glass than through air.
Light usually takes the fastest path from A to B, and the difference in speed between air and glass means that the shortest path in terms of distance may not be the fastest one.
Moving onto angles, θ1 and θ2 are measured from the normal to the material interface.
Since glass has a higher refractive index than air (n2>n1n2>n1), a light beam passing from air into glass will bend towards the normal (θ1>θ2θ1>θ2).
When the beam leaves the other side of the prism, the reverse happens and the angle increases again.
In terms of the change in angle on the prism, it can be computed by a quadrilateral (made up of the prism and the beam of light.)
It’s an example of ‘Snell’s law’ which in the mathematical world is described as ‘a formula used to describe the relationship between the angles of incidence and refraction’.
Sean continued: “Given an entry angle θ1, Snell’s law allows us to calculate θ2. We can then sum up the angles in the red quadrilateral, which gives θ3.
“In terms of θ1 and the refractive indices n1 and n2, we apply Snell’s law again at the point where the light exits to give us θ4.”
The multi-coloured beam is about the only glimpse of colour we get in this album cover, and although it’s minimal, it’s very strong in contrast with the black background.
In technical terms, Sean explained: “Where λ is the wavelength of the light, 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.”
It’s clear that album cover designer Hipgnosis knew a thing or two about maths, when combining the ideas of refraction and dispersion we can calculate the exact journey that the light beam takes.
So next time you listen to Pink Floyd, take a look at the album cover and prepare to have your mind blown from the maths in front of you.