Are crystals, like amethyst, an example of igneous rock or metamorphic rock?

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Amethyst is actually classified as a mineral.  Most rocks are made of minerals. Minerals have a specific chemical composition with atoms arranged in a definite structure giving them a crystalline structure. Amethyst can be found in igneous and metamorphic rock and sometimes even in sedimentary rock.

Amethyst is made up of mainly quartz, a silicon ore,  which is a naturally occurring inorganic solid with a crystalline structure. Quartz is the most common mineral in earth's crust. Quartz can be found in all types of igneous, sedimentary, and metamorphic rocks, for example granite, rhyolite, sandstone, shale, schist, and gneiss. Quartz forms six-sided crystals generally with pyramids on the ends. On the Moh's Hardness scale, quartz is a 7 with a specific gravity of 2.6.

Amethyst is a deep purple to pale blue type of quartz due to the small amounts of iron in it. It is used as an ornamental stone.

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