Mineralogy
"Hardness" is determined by comparison with other minerals. In the Mohs scale, a standard set of minerals are numbered in order of increasing hardness from 1 (talc) to 10 (diamond). A harder mineral will scratch a softer, so an unknown mineral can be placed in this scale by which minerals it scratches and which scratch it. A few minerals such as calcite and kyanite have a hardness that depends significantly on direction. Hardness can also be measured on an absolute scale using a sclerometer; compared to the absolute scale, the Mohs scale is nonlinear.
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.