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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.)

My question is does anyone know of any upper bounds, either published or unpublished?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Lagarias, and he said someone was claiming that they had proved something like $1 - 10^{-26}$, but that it was still unpublished.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

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Can the compactness argument be carried out in WKL0? If it can, then www.math.psu.edu/simpson/papers/hilbert.ps would seem to imply that an upper bound for regular packings could in principle be calculated from the proof. –  Ricky Demer Aug 13 '10 at 19:24
    
Compactness argument may be as follows: consider a small ball centered in each vertex of each tetrahedron. It can not be covered completely due to some irrationality or smth like it, also, not more then 100 tetrahedra touch it (since they are disjoint), hence fixed percentage of its volume remains uncovered by compactness. –  Fedor Petrov Aug 13 '10 at 19:35

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A paper just appeared on the arXiv that supports Jeff Lagarias's claim: "Upper bound on the packing density of regular tetrahedra and octahedra," 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.

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Thanks Joseph -- this is exactly what I was hoping to find out. –  Matthew Kahle Aug 25 '10 at 16:37
    
Does anyone know if there has been an improvement of this upper bound for a $\delta$ much greater than presented in the paper by Gravel, Elser, and Kallus? –  Samuel Reid Apr 30 '12 at 17:57

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