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The closed ball with center x and radius r is defined by B_r(x)={y:|y-x|<=r}.

A closed interval is an interval that includes all of its limit points. If the endpoints of the interval are finite numbers a and b, then the interval {x:a<=x<=b} is denoted ...

An axiomatic theory (such as a geometry) is said to be complete if each valid statement in the theory is capable of being proven true or false.

The five of Hilbert's axioms which concern geometric equivalence.

A set S in a vector space over R is called a convex set if the line segment joining any pair of points of S lies entirely in S.

A countable set is a set that is either finite or denumerable. However, some authors (e.g., Ciesielski 1997, p. 64) use the definition "equipollent to the finite ordinals," ...

A fractal curve of infinite length which bounds an area twice that of the original square.

The rotation vector of the trihedron of a curve with curvature kappa!=0 when a point moves along a curve with unit speed. It is given by D=tauT+kappaB, (1) where tau is the ...

A set of points which do not lie on any of a certain class of hyperplanes.

In two dimensions, the curve known as an "egg' is an oval with one end more pointed than the other.