Final answer:
Amethyst is the correct gemstone with naturally occurring, euhedral crystals with six sides, as it is a type of quartz known for forming hexagonal crystals.
Step-by-step explanation:
Among the options given, amethyst is the gemstone that has naturally occurring, euhedral crystals with six sides. Amethyst is a form of quartz, which often forms hexagonal (six-sided) prismatic crystals. It is well-known for its beautiful purple color, which is a result of iron ions in its crystalline structure. This sets it apart from garnet, peridot, and aquamarine, which do not naturally form six-sided euhedral crystals. For example, garnets usually form in a more rounded shape with twelve or twenty-four faces, rather than six-sided crystals.