Emerald Emerald Emerald is a gemstone and a variety of the mineral beryl (BeAl(SiO)) colored green by trace amounts of chromium and sometimes vanadium. Beryl has a hardness of 7.5–8 on the Mohs scale. Most emeralds are highly included, so their toughness (resistance to breakage) is classified as generally poor. Emerald is a cyclosilicate. The word "emerald" is derived (via and ), from Vulgar Latin: "esmaralda"/"esmaraldus", a variant of Latin "smaragdus", which originated in (smaragdos; "green gem"). Emeralds, like all colored gemstones, are graded using four basic parameters–the four "C"s of connoisseurship: "color", "clarity," "cut" and "carat weight". Normally, in 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