Cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.

More precisely, it is the solid figure bounded by a base in a plane and by a surface (called the lateral surface) formed by the locus of all straight line segments joining the apex to the perimeter of the base. The term "cone" sometimes refers just to the surface of this solid figure, or just to the lateral surface.

The axis of a cone is the straight line (if any), passing through the apex, about which the base (and the whole cone) has a circular symmetry.

In common usage in elementary geometry, cones are assumed to be right circular, where circular means that the base is a circle and right means that the axis passes through the centre of the base at right angles to its plane. Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the centre of the base. In general, however, the base may be any shape and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finite area, and that the apex lies outside the plane of the base).

Cone (category theory)

In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well.

Definition

Let F : J → C be a diagram in C. Formally, a diagram is nothing more than a functor from J to C. The change in terminology reflects the fact that we think of F as indexing a family of objects and morphisms in C. The category J is thought of as an "index category". (It may be the empty category.) One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well.

Let N be an object of C. A cone from N to F is a family of morphisms

for each object X of J such that for every morphism f : X → Y in J the following diagram commutes:

The (usually infinite) collection of all these triangles can be (partially) depicted in the shape of a cone with the apex N. The cone ψ is sometimes said to have vertex N and base F.

Conidae

Conidae (also previously referred to as Coninae), with the current common name of "cone snails", is a taxonomic family (previously subfamily) of predatory sea snails, marine gastropod molluscs in the superfamily Conoidea.

The 2014 classification of the superfamily Conoidea, groups only cone snails in the family Conidae. Some previous classifications grouped the cone snails in a subfamily, the Coninae.

Conidae currently (March 2015) contains over 800 recognized species. Working in 18th century Europe, Linnaeus knew of only 30 species which are still considered valid.

The snails within this family are sophisticated predatory animals. They hunt and immobilize prey using a modified radular tooth along with a venom gland containing neurotoxins; the tooth is launched out of the snail's mouth in a harpoon-like action.

Because all cone snails are venomous and capable of "stinging" humans, live ones should be handled with great care or preferably not at all.

Current 2014 taxonomy

In the Journal of Molluscan Studies, in 2014, Puillandre, Duda, Meyer, Olivera & Bouchet presented a new classification for the old genus Conus. Using 329 species, the authors carried out molecular phylogenetic analyses. The results suggested that the authors should place all living cone snails in a single family, Conidae, containing four genera: Conus, Conasprella, Profundiconus and Californiconus. The authors grouped 85% of all known cone snail species under Conus, They recognized 57 subgenera within Conus, and 11 subgenera within the genus Conasprella.