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Given a map with genus g>0, Heawood showed in 1890 that the maximum number N_u of colors necessary to color a map (the chromatic number) on an unbounded surface is N_u = ...

The moment of inertia with respect to a given axis of a solid body with density rho(r) is defined by the volume integral I=intrho(r)r__|_^2dV, (1) where r__|_ is the ...

An algebraic surface of surface order 4. Unlike cubic surfaces, quartic surfaces have not been fully classified. Examples of quartic surfaces include the apple surface, ...

A closed planar quadrilateral with opposite sides of equal lengths a and b, and with four right angles. A square is a degenerate rectangle with a=b. The area of the rectangle ...

A solid of revolution is a solid enclosing the surface of revolution obtained by rotating a 1-dimensional curve, line, etc. about an axis. A portion of a solid of revolution ...

Thurston's conjecture proposed a complete characterization of geometric structures on three-dimensional manifolds. Before stating Thurston's geometrization conjecture in ...

The trefoil knot 3_1, also called the threefoil knot or overhand knot, is the unique prime knot with three crossings. It is a (3, 2)-torus knot and has braid word sigma_1^3. ...

The set of roots of a polynomial f(x,y,z)=0. An algebraic surface is said to be of degree n=max(i+j+k), where n is the maximum sum of powers of all terms ...

Topology

An algebraic manifold is another name for a smooth algebraic variety. It can be covered by coordinate charts so that the transition functions are given by rational functions. ...