Volume 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes. The volume of a sphere is the integral of an infinite number of infinitesimally small circular disks of thickness "dx". The calculation for the volume of a sphere with center 0 and radius "r" is as follows. The surface area of the circular disk is formula_10. The radius of the circular disks, defined such that the x-axis cuts perpendicularly through them, is or where y or z can be taken to represent the radius of a disk at a particular x value. Using y
Filling radius imbedding is impossible if the ambient space is a Hilbert space, even when "X" is the Riemannian circle (the distance between opposite points must be , not 2!). We then set formula_23 in the formula above, and define Filling radius In Riemannian geometry, the filling radius of a Riemannian manifold "X" is a metric invariant of "X". It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating systolic geometry in its modern form. The filling