A Dense Packing of Regular Tetrahedra

We construct a dense packing of regular tetrahedra, with packing density $D > >.7786157$.Comment: full color versio

Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra

The determination of the densest packings of regular tetrahedra (one of the five Platonic solids) is attracting great attention as evidenced by the rapid pace at which packing records are being broken and the fascinating packing structures that have emerged. Here we provide the most general analytical formulation to date to construct dense periodic packings of tetrahedra with four particles per fundamental cell. This analysis results in six-parameter family of dense tetrahedron packings that includes as special cases recently discovered "dimer" packings of tetrahedra, including the densest known packings with density $\phi= 4000/4671 = 0.856347...$. This study strongly suggests that the latter set of packings are the densest among all packings with a four-particle basis. Whether they are the densest packings of tetrahedra among all packings is an open question, but we offer remarks about this issue. Moreover, we describe a procedure that provides estimates of upper bounds on the maximal density of tetrahedron packings, which could aid in assessing the packing efficiency of candidate dense packings.Comment: It contains 25 pages, 5 figures

A Picturebook of Tetrahedral Packings.

We explore many different packings of regular tetrahedra, with various clusters & lattices & symmetry groups. We construct a dense packing of regular tetrahedra, with packing density D > .7786157.Ph.D.MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/75860/1/bethchen_2.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/75860/2/bethchen_1.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/75860/3/bethchen_3.pd

On the origin of second-peak splitting in the static structure factor of metallic glasses

It is proposed that the splitting of the second peak of the total static structure factor, S(k), of many metallic glasses is essentially the same feature as the indentation at kσ = (9/2)π in the function (sin k σ + α−1 sin kασ), caused by the coincidence of the fourth minimum of the second term with the third maximum of the first term when α ≈ 5/3. Together with the strong-weak relation of the split peak components of S(k), this feature indicates the splitting to be direct evidence for face-sharing of regular tetrahedra (α = 2√2/3) dominating the topological short range order; increasing the number of face-sharing tetrahedra in local structural units indeed increases the amount of peak splitting in S(k); a dense random packing of well defined identical structural units (DRPSU), with neighbouring units linked together by a shared icosahedron, is described in detail. The packing fraction in a homogeneous, isotropic 1078-atom model is 0.67, after static relaxation under a two-body Lennard-Jones potential.\ud \u