Amethyst is a type of quartz that is mostly found in geodes and hollow agates. It is composed of silicon dioxide, and scores a 7 on the Mohs hardness scale. From different angles, amethyst occurs in primary hues from a light pinkish violet to a deep purple.
Agate is a cryptocrystalline variety of silica, chiefly chalcedony, characterized by its fineness of grain and brightness of color. Although agates may be found in various kinds of rock, they are classically associated with volcanic rocks and can be common in certain metamorphic rocks.