Densest Lattice Packing Research Articles

Dense lattice packings of spheres in Euclidean spaces of dimension n⩽16 *

We present a number of lattice packings of equal spheres in ℝn for n⩽16 For n⩽15, these packings have the same density as the densest known lattice packings. For n=16, the packing described here is denser than the known ones.

The Packing Problem for Twisted Pairs

When wires are packed together in a bundle, as in a cable or on a shelf of a main distribution frame, the packing fraction f is the fraction of cross-sectional area of the bundle occupied by wire. With wires all the same radius, packing fractions as high as 0.90690 can be achieved. However, when the wires are pairs that have been twisted to avoid crosstalk, the packing fraction is much smaller. The largest obtainable packing fraction depends on other properties of the packing. For example, with pairs twisted by machine, all pairs twist at the same rate, and that influences the packing fraction. Several packing problems are considered, but most attention is given to a particularly regular kind of packing in which pairs twist about straight parallel axes located in a lattice arrangement. The densest lattice packing has packing fraction 0.56767. The densest lattice is a complicated one in which each wire touches 10 wires belonging to 6 other pairs. The numbers 10 and 6 cannot be increased even with nonlattice packings of pairs with straight parallel axes. These other packings are also conjectured to have packing fractions less than 0.56767, although only f < 0.62240 is proved.

Six and Seven Dimensional Non-Lattice Sphere Packings

The densest lattice packings of equal spheres in Euclidean spaces En of n dimensions are known for n ⩽ 8. However, it is not known for any n ⩾ 3 whether there can be any non-lattice sphere packing with density exceeding that of the densest lattice packing. W. Barlow described [1] a non-lattice packing in E3 with the same density as the densest lattice packing, and I described [6] three non-lattice packings in E5 which also have this property.

Five Dimensional Non-Lattice Sphere Packings

The densest lattice packings of spheres in Euclidean spaces En of n dimensions are known for n ≤ 8 (for full n — references see [6]). However, it i s not known for any n ≥ 3 whether there can be any non-lattice sphere packing with density exceeding that of the corresponding densest lattice packing.

Packing and Covering.

Introduction 1. Packaging and covering densities 2. The existence of reasonably dense packings 3. The existence of reasonably economical coverings 4. The existence of reasonably dense lattice packings 5. The existence of reasonably economical lattice coverings 6. Packings of simplices cannot be very dense 8. Coverings with spheres cannot be very economical Bibliography Index.

Corrosion Of Iron In An H2S-CO2-H2O System: influence Of (Single Iron) Crystal Orientation On Hydrogen Penetration Rate

Abstract The rates of hydrogen penetration through single iron crystals in contact with an H2S-CO2-H2O environment were related to the specific crystallographic orientation of the crystals using a novel, highly reliable penetration cell. The rate of hydrogen penetration was shown to be a function of the lattice packing density of the exposed crystal plane. The rate decreased with the orientation of the single crystal planes according to the order: (310) > (100) > (210) > (211) > (111). The (310) plane is the most loosely packed and the (111) plane is the most closely packed.

Some Sphere Packings in Higher Space

This paper is concerned with the packing of equal spheres in Euclidean spaces [n] of n &gt; 8 dimensions. To be precise, a packing is a distribution of spheres any two of which have at most a point of contact in common. If the centres of the spheres form a lattice, the packing is said to be a lattice packing. The densest lattice packings are known for spaces of up to eight dimensions (1, 2), but not for any space of more than eight dimensions. Further, although non-lattice packings are known in [3] and [5] which have the same density as the densest lattice packings, none is known which has greater density than the densest lattice packings in any space of up to eight dimensions, neither, for any space of more than two dimensions, has it been shown that they do not exist.