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\begin{document}
\title{Platonic Packing}
\author{Dakota Blair}
\maketitle
Packing tetrahedrons has never been easier. Thanks to modern research it is now known
that up to just over 85\% of space can be filled with regular tetrahedrons, with some unpublished
results reporting that packing densities of more than 85.6\% are possible. This shows
that when it comes to luggage, tetrahedrons beat spheres which can only fill slightly more
than 74\% of space at best.
An article in the December 2009 issue of the journal Nature boasts a packing density of
85.03\%, the largest published value to date. This result is the combined effort of seven professors
following ``a conceptually different approach, using thermodynamic computer simulations
that allow the system to evolve naturally towards high-density states.''
A regular tetrahedron, or simply a tetrahedron, is a regular triangular pyramid, four
equilateral triangles glued together to enclose a volume. You might recognize it as the shape
of methane or a four-sided die used in games like Dungeons and Dragons. The packing density is
the ratio of volumes of a given arrangement of tetrahedrons to an enclosing volume. Then to
fill a 100 cubic centimeter cube with only tetrahedrons, water and what we know today one
would still need about 15 cubic centimeters of water after packing the cube with identical
tetrahedrons of any size.
Each packing has applications to high resolution imaging in microscopic medicine and
materials science. In particular new metamaterials, materials made by imposing a new
structure on a familiar one, are possible by taking advantage of these dense arrangements.
In a twist on the usual situation, these results are direct applications of concepts from
physics to mathematics. The papers are peppered with talk of quasicrystals, entropy, phase
transitions and Monte Carlo simulations which are more often associated with materials
science than geometry.
It may at first glance seem that stacking tetrahedrons would leave no space, resulting in
a perfect 100\% packing, but this belief, held even by Aristotle, is untrue. Regiomontanus
revealed the gaps in tetrahedral tilings in the 15th century through a straightforward calculation
correcting a mistake that lingered for close to two millennia. While this shows that
no perfect tetrahedral packing exists Salvatore Torquato, a chemist at Princeton, said in a
recent interview, ``I'd be shocked if what we have right now is the densest. It just happens to
be the densest known right now.''
On the other hand in the 17th century Johannes Kepler suggested the result above, namely
that the best packing of spheres resulted in a packing density just over 74\%. This remained
conjecture until 1998 when Thomas Hales, a mathematician at the University of Pittsburgh,
announced an incredibly intricate proof containing an unwieldy amount of computer data.
His proof was finally published in the Annals of Mathematics subject to the proviso that
the referees are ``99\% certain'' of the proofâ€™s correctness. The distrust of computer aided
proofs, specifically those which generate thousands of pages of material to be verified, has
existed since the publication of the Four-color theorem which shared a similar controversy.
This complexity is thus the impetus for a project devising a formal proof of Hales' Theorem.
A formal proof accounts for each logical step from its assumptions to its conclusion. Each
sentences is constructed with predetermined axioms and the rules of consequence determine
the conclusion. Therefore having a formal proof is similar to having all the pieces of the
puzzle, and the check is that the picture is complete. A missing piece will mar the picture
and the correct conclusion cannot be reached. A formal proof for the Four-color theorem
now exists which eases the controversy in the minds of many.
With the case of spheres almost certainly settled, mathematicians look to the packing
of other solids. In 2006 Dr. Torquato and John Conway, a mathematician at Princeton,
published results of a tetrahedral packing with a density less than 72\%. Remarkably it
posed the question of whether spheres pack better than tetrahedrons. However the new
record and previous results due to Elizabeth Chen, a mathematics graduate student at the
University of Michigan, Yang Jiao, a graduate student at Princeton, and Dr. Torquato
demonstrate that tetrahedrons do pack more densely than spheres. Announcements of these
and new results published this year reveal the increased interest in this problem by the
mathematical community. At this point it is still unknown whether any density strictly less
than 100\% is possible. Determining the limit, if there is one, will take more than finding
new arrangements of tetrahedrons, but will instead possibly be an undertaking comparable
to settling the Kepler conjecture.
\end{document}