Hard water
Water's hardness is determined by the concentration of multivalent cations in the water. Multivalent cations are positively charged metal complexes with a charge greater than 1+. Usually, the cations have the charge of 2+. Common cations found in hard water include Ca and Mg. These ions enter a water supply by leaching from minerals within an aquifer. Common calcium-containing minerals are calcite and gypsum. A common magnesium mineral is dolomite (which also contains calcium). Rainwater and distilled water are soft, because they contain few ions.
Knot thickness
The thickness is defined using the simpler concept of the local thickness τ("x"). The local thickness at a point "x" on the link is defined as
where "x", "y", and "z" are points on the link, all distinct, and "r"("x", "y", "z") is the radius of the circle that passes through all three points ("x", "y", "z"). From this definition we can deduce that the local thickness is at most equal to the local radius of curvature.