At about 15:49 in her 2011 talk Prof. Marjorie Senechal - "Quasicrystals Gifts to Mathematics":
But Hilbert understood that groups aren't everything and maybe not even the main thing. And so part of his problem which he just added on at the end as an afterthought was that one it's important to number three (referring to the slide) also maybe to physics and chemistry, how can one arrange most densely in space infinite number of given solids, spheres with a given radius or tetrahedra with given edges. The spheres, the most dense packing is the one that Walter (last name?) just showed you before, but that was conjectured by Kepler and only proved 300 years later which (was) just a few years ago.
The question of packing tetrahedra is still unsolved and this is one that's a very very hot problem today and literally every week or every other week there's a new publication about this. And one of the leaders in this field is Veit Elser, who played a big role in quasi crystal theory early on.
But anyway even though Hilbert had raised this question people did not pay attention to it until quasi crystals were discovered, because you really had to smash the paradigm to be able to think constructively about packings in space, and so on. So you really did demote group theory from people's thinking and realize to think more broadly more generally and to look at things that did not fit there. So thank you for that.
I believe the "thank you" is directed towards Daniel Shechtman, who won the Nobel Prize in Chemistry for work on quasicrystals, despite fellow chemist Linus Pauling's best efforts to squash the work.
Question: Senechal's talk was in 2011. Is the question of packing tetrahedra still unsolved?
A quick reading of Wikipedia's Tetrahedron packing suggests to me that it isn't based on the number of upper limits cited as work has progressed, ending with the following, but this work was published in 2015, eight years ago. And as far as I can understand, it's a lower limit, not a solution.
In late 2009, a new, much simpler family of packings with a packing fraction of 85.47% was discovered by Kallus, Elser, and Gravel.13 These packings were also the basis of a slightly improved packing obtained by Torquato and Jiao at the end of 2009 with a packing fraction of 85.55%, and by Chen, Engel, and Glotzer in early 2010 with a packing fraction of 85.63%. The Chen, Engel and Glotzer result currently stands as the densest known packing of hard, regular tetrahedra. Surprisingly, the square-triangle tiling[12] packs denser than this double lattice of triangular bipyramids when tetrahedra are slightly rounded (the Minkowski sum of a tetrahedron and a sphere), making the 82-tetrahedron crystal the largest unit cell for a densest packing of identical particles to date.15
13Kallus, Yoav; Elser, Veit; Gravel, Simon (2010). "Dense Periodic Packings of Tetrahedra with Small Repeating Units". Discrete & Computational Geometry. 44 (2): 245–252. arXiv:0910.5226. doi:10.1007/s00454-010-9254-3. S2CID 13385357.
15Jin, Weiwei; Lu, Peng; Li, Shuixiang (December 2015). "Evolution of the dense packings of spherotetrahedral particles: from ideal tetrahedra to spheres". Scientific Reports. 5 (1): 15640. https://doi.org/10.1038%2Fsrep15640
