From the original vinyl to t-shirts and merchandise, it is hard to think of a more iconic image in the music industry. Yet surprisingly, there is a lot of mathematics behind the simple geometric design and rainbow patterned artwork.
As reported by Chalkdust Magazine, the album cover shows a beam of white light hitting a triangular prism splitting into “constituent parts” (which form the rainbow).
It’s an example of an “optical phenomena” known as refraction and dispersion. This explains why the rainbow and incoming beam are not parallel and how the light splits as it passes through the prism.
Sean Jamshidi, a PhD student at University College London, broke down three significant mathematical elements of the album cover.
The refraction within 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 ‑ therefore light travels slower through glass than through air.
Light normally takes the fastest path 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 maths is described as “a formula used to describe the relationship between the angles of incidence and refraction”.
Sean said: “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 the only glimpse of colour we get on this album cover, and although the rainbow is minimal, it is contrasting to the black background.
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.”
Designer Hipgnosis knew a thing or two about maths when designing the album cover. By combining the ideas of refraction and dispersion we can calculate the exact journey that the light beam takes.
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.
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