There have been several recent advances on packing regular tetrahedra in $\mathbb{R}^3$. All the results I've seen have been lower bounds -- first John Conway and Sal Torquato showed that there exists an arrangement of tetraheda filling about 72% of space. This has been improved in a series of papers, and the latest result of which I am aware is Elizabeth Chen's record of 85.63%. (A NYTimes article summarizing the history of the problem can be found [here](http://www.nytimes.com/2010/01/05/science/05tetr.html?_r=1).)


My question is does anyone know of any upper bounds, either published or unpublished? I saw a colloquium by Jeff Lagarias, and he said someone was claiming that they had proved something like $1 - 10^{-26}$, but that it was still unpublished.


(A compactness argument gives that since regular tetrahedra don't tile space the maximum volume is strictly less than one, but this argument does not give a quantitative bound.)



Upper bound for tetrahedron packing?
A paper just appeared on the arXiv that supports Jeff Lagarias's claim: "[Upper bound on the packing density of regular tetrahedra and octahedra](http://arxiv.org/abs/1008.2830)," by Simon Gravel, Veit Elser, and Yoav Kallus (medicine and physics researchers at Stanford and Cornell):



> 
> In this article, we obtain an explicit bound to the packing density of regular tetrahedra, namely
>  $\phi < 1-\delta$ with $\delta = 2.6...\times 10^{-25}$. 
>  [...]
>  In order to obtain a bound to the packing density, we show the existence, in any tetrahedron packing, of a 
>  set of disjoint balls whose intersection with the packing is particularly simple, and whose density can be 
>  bounded below. The construction is such that the density of the packing within each of the balls can be 
>  bounded away from one. The combination of these two bounds gives the main result. 
> 
> 
> 


They modify their argument to obtain a $10^{-12}$ bound on regular octahedra.


