Hermit crab on physical characteristics, genetic information, and combined data, support the longstanding hypothesis that the king crabs in the family Lithodidae are derived hermit crabs and should be classified as a family within Paguroidea. Other researchers have challenged this, asserting that the Lithodidae (king crabs) should be placed with the Hapalogastridae in a separate superfamily Lithodoidea. Six families are formally recognized in the superfamily Paguroidea, containing around 1100 species in total in 120 genera. The fossil record of "in situ" hermit crabs using gastropod shells stretches back to the Late Cretaceous. Before that time, at least some hermit crabs used ammonites'
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