**LESSONS FROM CRYSTALLOGRAPHIC CLASSIFICATION** >• When, in the history of mathematics, have problems "like P vs. NP" arisen and then been solved?<br> > • In those cases, what were the resolutions? **The Crystallographic Classification Problem** In two and three dimensions, separate the set of ordered materials (that exhibit discrete Fourier diffraction patterns) from the set of disordered materials (that don't), and comprehensively classify the ordered materials on the basis of their Fourier diffraction patterns. **The Relevance to the Question Asked** We imagine that "an invisible fence" separates ordered materials from disordered materials. Viewed as a problem in tiling, we appreciate that abstract tile-sets and physical crystals alike are richly endowed with various (tricky!) reductions, as stipulated in the question-asked. **Historical Provenance** For centuries mathematicians believed that [the 230 crystallographic space groups](http://en.wikipedia.org/wiki/Space_group#History) rigorously and exhaustively classified the separation of crystallographically ordered from disordered materials... until the unexpected [experimental observation of quasicrystals](http://en.wikipedia.org/wiki/Quasicrystal#Mathematical_description) forced radical amendments to the mathematical postulates associated to the classification problem. **The Resolution** - Raphael Robinson's canonical "[Undecidability and nonperiodicity for tilings of the plane](http://math.stackexchange.com/q/40669)" (1971) established that the prediction of crystal diffraction patterns is undecidable in [both mathematical senses of "undecidable"](http://en.wikipedia.org/wiki/Undecidable_problem#Examples_of_undecidable_statements): first, the decision function associated to tile-set classification is not Turing-computable; second, given an axiom system at least as strong as PA, tile-sets exist for which the existence of periodic 2D tilings---and thus discrete Fourier diffraction patterns---is neither provable nor refutable in that system. - Dan Schectman's [quasicrystal discovery](http://www.nobelprize.org/mediaplayer/index.php?id=1731) mandated formal extensions to the traditional postulates of crystallography. - In retrospect, the infamously rancorous opposition to (now-Nobelist) Dan Schectman's extended crystallographic classification framework---*e.g.*, Linus Pauling's sardonic yet mathematically ill-grounded assertion "[There is no such thing as quasicrystals, only quasi-scientists](http://en.wikipedia.org/wiki/Dan_Shechtman#Work_on_quasicrystals)"---contributed insubstantially to progress in the classification problem **Conclusion** The Crystallographic Classification Problem teaches plainly that long-accepted mathematical postulates sometimes require radical reconsideration, and teaches too that sardonic rancor contributes little to mathematical discourse.