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A cubic map is three-colorable iff each interior region is bounded by an even number of regions. A non-cubic map bounded by an even number of regions is not necessarily ...

In Euclidean space R^3, the curve that minimizes the distance between two points is clearly a straight line segment. This can be shown mathematically as follows using ...

A regular skew polyhedron is a polyhedron whose faces and vertex figures are regular skew polygons. There are only three regular skew polyhedra in Euclidean three-space ...

Oriented spheres in complex Euclidean three-space can be represented as lines in complex projective three-space ("Lie correspondence"), and the spheres may be thought of as ...

A matchstick graph is a simple graph which has a graph embedding that is planar, for which all distances between vertices have unit distance, and which is non-degenerate (so ...

The geodesics in a complete Riemannian metric go on indefinitely, i.e., each geodesic is isometric to the real line. For example, Euclidean space is complete, but the open ...

A subset of an algebraic variety which is itself a variety. Every variety is a subvariety of itself; other subvarieties are called proper subvarieties. A sphere of the ...

The hyperbolic octahedron is a hyperbolic version of the Euclidean octahedron, which is a special case of the astroidal ellipsoid with a=b=c=1. It is given by the parametric ...

In general, a triakis octahedron is a non-regular icositetrahedron that can be constructed as a positive augmentation of regular octahedron. Such a solid is also known as a ...

A Borel set is an element of a Borel sigma-algebra. Roughly speaking, Borel sets are the sets that can be constructed from open or closed sets by repeatedly taking countable ...