Gemstone few as one in a million atoms. These so-called impurities are sufficient to absorb certain colors and leave the other colors unaffected. For example, beryl, which is colorless in its pure mineral form, becomes emerald with chromium impurities. If manganese is added instead of chromium, beryl becomes pink morganite. With iron, it becomes aquamarine. Some gemstone treatments make use of the fact that these impurities can be "manipulated", thus changing the color of the gem. Gemstones are often treated to enhance the color or clarity of the stone. Depending on the type and extent of treatment, they can affect the
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