Here's a more technical approach to my emerald study. Just for a recap, I was photographing these objects to demonstrate the behavior of light through translucent objects. The principles of refraction, reflection, dispersion and diffusion are easily exemplified when using emeralds. Later I'll get into some theory as to how they could be used as the power sources we see in the games. So If you are willing to read through, I can guarantee you that you will learn something new and interesting, whether it be about physics and optics or perhaps a deeper understanding of the game we've already come to know and love. ^^

So, just to clarify:

Refraction is the bending of light through translucent mediums such as glass and water. This occurs because the light passing through it is actually slowed down due to the properties of the material. This causes the path of light to bend slightly, subsequently causing the distortion and separation of images when seen through a medium. The simplest example of this is the loss of depth perception when looking at the bottom of the pool from above the water.

Dispersion is when a wave of light is split into it's spectral components. In other words, white light is split into different colors. This again happens because the speed of the light entering a medium slows down. Given the right conditions, the components making up the light will "outrun" the other components, causing the separation of color. Probably the easiest example of this is a rainbow caused by many droplets of water.

Diffusion is when the light is "spread" over surfaces due to that surface's texture properties. Even though an emerald itself has very little diffusive properties, a surface with diffusive properties must be used to observe the patterns of light the emerald produces. Fogged glass and paper are good examples of diffusive surfaces.

Finally, Reflection is when waves of light bounce off a surface. This is important because as you will see, the emerald acts partly as a mirror, taking in and bouncing back any light that may enter it from a certain angle.

When a strong light is shown through an emerald, all four principles occur at once and often times compound each other. For example the dispersed light is reflected back into the emerald which is then refracted to another facet to produce a distinct pattern of light which is diffused on a piece of paper. The end result is a spectacular light show, all due to a single source of light and a crystal.

Now then, we got the principles down, what about the shape itself? Why is a "chaos emerald" shaped like a "diamond"? Did Sega's designers choose them because they simply looked cool? Probably. XD As I see it, they chose, whether it be consciously or subconsciously, for the emeralds to resemble diamonds due to their refined quality and brilliance.

But there is a bit more to it. I did some digging and and it turns out that the shape of these chaos emeralds can actually be deeper than meets the eye, so to speak. You see, they are in the shape that jewelers call a "brilliant cut" It consists of 58 facets, each mathematically aligned to amplify, split and reflect light. Basically said, they were DESIGNED to take advantage of light that passes through it. Light seems to disperse near the bottom and concentrate near the top. If the light source were in the center, both instances would be happening at once. The dispersed light also creates distinct, quantized patterns of light that can be observed through refraction.

Now, here's the interesting twist story wise. It might be a bit sketchy but see if you can follow. If you were to replace light with plasma or another form of energy, say something that behaves like electrical charge, this design would be rather useful because the divergence and convergence flow will create what is known as a feedback loop, allowing the emerald to circulate and store large amounts of energy as long as it is kept resonating. The Emerald would act somewhat as an inductive battery and the energy it contains could easily be tapped using an apparatus, much like what Eggman would do or, if this form of energy permits, be fed directly into an organic body.
I'm sure there are more efficient designs if one was to use them in this manner but this one is certainly plausible if you look at it this way.

Anyway, sorry if anything was confusing. Feel free to ask questions as always! I'd love to hear your feedback! *bows*