If you are looking for a fun science experiment to do at home, there are many kits that can be purchased that provide chemicals to grow amethyst crystals. If you don't want to buy a kit, however it, is also possible to grow them with items that can be found in your local grocery store.
Creating a Geode
Mix plaster of paris with water per the directions on the package.
Press the plaster into one section of the egg carton or other rounded object you have to use as a mold.
Let the plaster dry for about 30 minutes, then remove it from the egg carton to finish drying on its own.
Growing the Amethyst Crystals
Stir alum into a half cup of water in any container. You should stir the alum until it stops dissolving which happens when the alum starts gathering at the bottom of the cup. At this point, you can add purple food coloring to give your crystals the color of an amethyst.
Place the geode you made from the plaster of paris in a container that is big enough that your water mixture will just come over the top of your geode. Try not to allow any undissolved alum into the container.
Let the container sit undisturbed for two to three days while the crystals form. When you are satisfied with the size and appearance of your crystal, remove it from the container.