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:

blog.wolfram.com/2010/08/30/te

(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.

@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.

Follow

@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.

Sign in to participate in the conversation
Qoto Mastodon

QOTO: Question Others to Teach Ourselves
An inclusive, Academic Freedom, instance
All cultures welcome.
Hate speech and harassment strictly forbidden.