This could also be illustrated with a vector editor like AI or a raster editor like PS, like Rafael mentioned. It is quite similar to a "low poly" illustration, which are quite trendy nowadays. This link has a tutorial on how to create a low poly portrait that can give you an idea of the method. It is quite time intensive and not for the faint of heart, but if you are like me and suffer of Mayaphobia (irrational fear of 3D editor software) it is a viable solution.
The basic idea, using AI, would be like this
- Draw your general outline of the shape. I have simplified it to a single gemstone in my example.
- Triangulate: start drawing triangles (or polygons) on top of the shape. Make sure to add triangles for the reflections and refractions (things you see throw the shape).
- After all triangles are done and tidy, start colouring.
- Select a triangle using the direct select (a.k.a. black arrow)
- Change its colour either by picking it from a reference image using the eyedropper or by selecting a different fill colour
I really don't have time to finish the whole gem right now, but if I have time later I will finish it. It would look something like this:

Tips:
- You can do the triangulation on top of the reference image and use a contrasting stroke colour so you can see the shapes you are creating. After you colour them, you can remove the stroke colour.
- Don't be too tidy while triangulating. Actually, be extra non-tidy and overlap the polygons if you want. After you are done all of them, select them all and using the pathfinder apply "divide". This will get rid of overlapping areas and simplify the triangulation.
- After colouring and removing the stroke you might notice that some of the polygons do not align perfectly. You can fix this two different ways:
- Select the points that are not aligned and using the align window select "align to selection" then "vertical align center" and finally "horizontal align center". All the points will end up in the same spot.
- Select one of the corners of the polygons and move it so it overlaps the adjacent polygon.
- For extra effect, fill some of the polygons with gradients instead of flat colours.