Beryl over , and its dimensions were long and in diameter. The largest cut aquamarine gem is the Dom Pedro aquamarine, now housed in the Smithsonian Institution's National Museum of Natural History. The ancient Romans believed that aquamarine would protect against any dangers while travelling at sea, and that it provided energy and cured laziness. Emerald is green beryl, colored by trace amounts of chromium and sometimes vanadium. Most emeralds are highly included, so their brittleness (resistance to breakage) is classified as generally poor. The modern English word "emerald" comes via Middle English "Emeraude", imported from modern French via Old French
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