Aristotle claimed that identical regular tetrahedra could be packed in a way that fills space. It seems unlikely if you actually try it - but only in 1925 was it proved impossible. And only in 2010 did people prove a nontrivial upper bound on the density of the packing you can achieve, namely
0.9999999999999999999999974
On the other hand, the densest known packing has density about .8563 - there's a discussion and picture on Wolfram's website:
https://blog.wolfram.com/2010/08/30/tetrahedra-packing/
(1/n)
@johncarlosbaez Wow, extremely cool! Though I'll give it to Aristotle, with the things he might have had to pack, such a density does equate complete packing.
@rastinza - while the best upper bound people have been able to prove is
0.9999999999999999999999974
nobody has found a way to pack equal-sized regular tetrahedra with a density of more than
0.8563
The gap between these numbers is mainly because it's hard to prove stuff.
@johncarlosbaez Oh, damn. Could this packing problem be improved by experimental findings? I know we discuss packing a lot in chemistry, but that's mostly from the observation of crystal structures. And generally they use spherical models for the atoms. I know nothing about it, but I guess someone might be studying the packing of some compound forming tetrahedras like quartz, maybe a little bit less interconnected.
@johncarlosbaez It is a highly interconnected network, this makes is much less dense than what it could be.
I have not worked in this field, but I do imagine there are materials which are much more packed than this.
I imagine nobody ever observed packings denser than that, but there may be different types of packing available or some materials might be designed specifically to increase the packing.
@rastinza - the SiO₄ tetrahedra in quartz are less densely packed than the tetrahedra I showed in part (1/n). I imagine that any packings of tetrahedra in nature that have already been seen are no denser than that one.