The Method of Mechanical Theorems convenient to use a little bit of coordinate geometry. If a sphere of radius 1 is placed with its center at "x" = 1, the vertical cross sectional radius formula_2 at any "x" between 0 and 2 is given by the following formula: The mass of this cross section, for purposes of balancing on a lever, is proportional to the area: Archimedes then considered rotating the triangular region between "y" = 0 and "y" = "x" and "x" = "2" on the "x"-"y" plane around the "x"-axis, to form a cone. The cross section of this cone is a circle
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