The final optical element used on the table is a polarizer. As light can be seen as a wave in many cases, it also has an oscillation direction. This is similar to a rope, where the wave produced oscillates in the direction of the hand movement. A polariser is some sort of filter that only lets light that oscillates in a certain direction pass through. The analogy with the rope can help visualize this further. The polarizer is like two sticks through which the wave passes, which dampens oscillations in the direction perpendicular to them.
For light polarizers a substance that absorbs light oscillating in a certain direction is used. This usually works by using polymer chains. In such chains, valence electrons can only move along these chains, but not perpendicularly to them. For a substance to absorb light, the electrons must be able to to move in the direction it oscillates in. Since this is not possible for light polarized perpendicularly to these chains, that light is simply let through, while light polarized parallelly is absorbed. This has as an effect that when two polarizers are crossed, they don’t let any light pass through at all. All of this, while necessary to understand the puzzle, does not yet explain it on its own. Additionally, the concept of birefringence is required.
This is a rather complicated effect that is not easy to understand. The first thing that needs to be understood is that there are three different kinds of polarization. The first is linear polarization, as explained above, the second is circular, while the third is elliptical. The difference between them is the phase difference between the vertical and horizontal component of the polarization:
(source: Principles of Physical Optics)
In normal, linearly polarized light, the x and y components are precisely in phase. So, when they are added together, it results in an oscillation, exactly between both components. There are, however, certain materials in which both components are affected differently, introducing a phase shift. In order for this to happen, a material needs to have two different indexes of refraction, one for the fast axis and the second for the slow axis. So, in this case specifically, linearly polarized light, produced by the first polarization foil enters the scotch tape. The scotch tape is such a birefringent material with a fast and a slow axis. The polarized light can be split into two components, along the slow and fast axis respectively. Now, each of these components is affected by the different index of refraction and therefore introduces a phase shift. This phase shift depends on multiple aspects. Obviously, the indexes of refraction are vital, but also the thickness is important. That is why different colours are visible at different areas. That represents different amount of scotch tapes above one another. The final thing that needs to be explained is why different colours are visible at all. The reason for that is that any index of refraction depends on the wavelength and therefore the colour of light. So the precise phase shift depends on the colour. Under normal circumstances, nothing would be visible. The only reason something becomes visible is because of the second polarizer. As explained above, a polarizer only allows a certain direction of polarization pass through. Once again though, the light is made up of two components. One along the fast and the other along the slow axis. Thus, when it removes part of one component, this is weakened. However, the two components do not have the same colour composition, as explained above. therefore, the light that exits will actually have a colour. The shapes visible can be obtained by only using a certain amount of scotch tape everywhere, and therefore controlling the thickness well.