Where does hardness collide?

In materials science, hardness is a solid's resistance to localized permanent deformation, usually measured by pressing a hard indenter of defined shape into a surface under a known load and gauging the size of the dent it leaves. Broitman's 2017 Indentation Hardness Measurements at Macro-, Micro-, and Nanoscale: A Critical Overview traced how the common scales, Vickers, Brinell, Rockwell, and the older scratch-based Mohs ranking, each define hardness through a particular indentation or scratch test, so a single number depends on the method used. The referent is a mechanical property of a solid, reported as load over the indented area, and it has nothing to do with how hard a computation is, how much calcium sits in water, or how energetic a star's X-rays are.[1]

Unit

A hardness number for a solid, as load divided by the indented area

Recovery means

The measured value sets aside elastic springback and reflects the residual plastic dent

Measured by

Measured by pressing a standard indenter under load and reading the dent size, as in Vickers, Brinell, Rockwell, or Mohs tests

In computational complexity theory, hardness measures how intractable a problem is: a problem is hard for a class of problems when an efficient algorithm for it would give an efficient algorithm for every problem in that class, a link established by a reduction that translates one problem into another. Karp's 1972 Reducibility Among Combinatorial Problems showed that twenty-one classic problems are all NP-complete, meaning each is among the hardest in the class NP, so a fast solution to any one would solve them all. The referent is a property of a problem and the algorithms that might solve it, ranked by how badly the effort grows with size, not a property of a metal surface, a glass of water, or a beam of starlight.[2]

Unit

A classification of a problem by the class it is hard for, such as NP-hard

Recovery means

Not applicable

Measured by

Established by a reduction proving that an efficient algorithm for the problem would efficiently solve a whole class of problems

In water chemistry and environmental engineering, hardness is the concentration of dissolved calcium and magnesium ions in water, conventionally expressed as an equivalent number of milligrams of calcium carbonate per litre. Hard water forms scale in pipes and boilers and reacts with soap to leave a curd, and Catling and colleagues' 2008 A Systematic Review of Analytical Observational Studies Investigating the Association Between Cardiovascular Disease and Drinking Water Hardness weighed the evidence on whether these dissolved minerals affect heart health. The referent is a chemical content of water, measured by titration, and it bears no relation to how hard a solid is to dent, how hard a problem is to compute, or how hard a source's radiation is.[3]

Unit

Dissolved calcium and magnesium, in milligrams of calcium carbonate per litre

Recovery means

Not applicable

Measured by

Measured by titration with a chelating agent, or by adding calcium and magnesium concentrations expressed as calcium carbonate equivalent

In x-ray astronomy, hardness describes the shape of a source's spectrum, the balance between higher-energy (hard) and lower-energy (soft) photons, most often summarized as a hardness ratio comparing the counts detected in a hard band to those in a soft band. Park and colleagues' 2006 Bayesian Estimation of Hardness Ratios: Modeling and Computations set out how to estimate this ratio properly when only a handful of photons are collected. The referent is a colour-like property of astrophysical radiation, computed from photon counts in energy bands, and it has nothing in common with the resistance of a solid, the difficulty of a computation, or the mineral content of water.[4]

Unit

A hardness ratio, comparing photon counts in a hard band and a soft band

Recovery means

Not applicable

Measured by

Computed from the counts detected in defined hard and soft energy bands, often as their ratio or a normalized difference