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 spaces with a differentiable or topological structure, while "variety" refers to spaces with an algebraic structure, as in algebraic varieties. In Romance languages, manifold is translated as "variety" – such spaces with a differentiable structure are literally translated as "analytic varieties", while spaces with an algebraic structure are called "algebraic varieties". Thus for example, the french word "" means topological manifold. In the same vein, the Japanese word "" (tayōtai) also encompasses both manifold and variety. ("" (tayō) means various.) Ancestral to the modern concept of a manifold were several important results of 18th and 19th century mathematics. The oldest