Gemstone usually found in. For example, diamonds, which have a cubic crystal system, are often found as octahedrons. Gemstones are classified into different "groups", "species", and "varieties". For example, ruby is the red variety of the species corundum, while any other color of corundum is considered sapphire. Other examples are the emerald (green), aquamarine (blue), red beryl (red), goshenite (colorless), heliodor (yellow) and morganite (pink), which are all varieties of the mineral species beryl. Gems are characterized in terms of refractive index, dispersion, specific gravity, hardness, cleavage, fracture and luster. They may exhibit pleochroism or double refraction. They may have luminescence
History of manifolds and varieties History of manifolds and varieties The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Certain special classes of manifolds also have additional algebraic structure; they may behave like groups, for instance. In that case, they are called Lie Groups. Alternatively, they may be described by polynomial equations, in which case they are called algebraic varieties, and if they additionally carry a group structure, they are called algebraic groups. The term "manifold" comes from German "Mannigfaltigkeit," by Riemann. In English, "manifold" refers to