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The Kummer surfaces are a family of quartic surfaces given by the algebraic equation (x^2+y^2+z^2-mu^2w^2)^2-lambdapqrs=0, (1) where lambda=(3mu^2-1)/(3-mu^2), (2) p, q, r, ...

A graph is planar if it can be drawn in a plane without graph edges crossing (i.e., it has graph crossing number 0). The number of planar graphs with n=1, 2, ... nodes are 1, ...

A projective plane, sometimes called a twisted sphere (Henle 1994, p. 110), is a surface without boundary derived from a usual plane by addition of a line at infinity. Just ...

An embedding of a 1-sphere in a 3-manifold which exists continuously over the 2-disk also extends over the disk as an embedding. An alternate phrasing is that if a knot group ...

Geometry is the study of figures in a space of a given number of dimensions and of a given type. The most common types of geometry are plane geometry (dealing with objects ...

Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates that are natural for describing positions ...

There are many formulas of pi of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations. pi is ...

An ambient isotopy from an embedding of a manifold M in N to another is a homotopy of self diffeomorphisms (or isomorphisms, or piecewise-linear transformations, etc.) of N, ...

Roughly speaking, a tangent vector is an infinitesimal displacement at a specific point on a manifold. The set of tangent vectors at a point P forms a vector space called the ...

A C^infty (infinitely differentiable) manifold is said to be a submanifold of a C^infty manifold M^' if M is a subset of M^' and the identity map of M into M^' is an ...