Lattice-Packing by Spheres and Eutactic Forms

2019 ◽  
pp. 1-7
Author(s):  
Avner Ash ◽  
Robert Gross
Keyword(s):  
Lattice Packing

Some Sphere Packings in Higher Space

1964 ◽  
Vol 16 ◽  
pp. 657-682 ◽  
Author(s):  
John Leech
Keyword(s):  
Two Dimensions ◽  
Lattice Packing ◽  
Sphere Packings ◽  
Euclidean Spaces ◽  
Lattice Packings

This paper is concerned with the packing of equal spheres in Euclidean spaces [n] of n > 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.

Magnetic Nanocomposites Based on Opal Matrices

2018 ◽  
Vol 781 ◽  
pp. 149-154 ◽  
Author(s):  
Alexey Belyanin ◽  
Alexander Bagdasarian ◽  
Sergey Bagdasarian ◽  
Petr Luchnikov ◽  
Natalya Katakhova
Keyword(s):  
Raman Spectroscopy ◽  
Dielectric Constant ◽  
Hysteresis Loops ◽  
Lattice Packing ◽  
Magnetic Nanocomposites ◽  
Multiferroic Materials ◽  
X Ray ◽  
Temperature Range ◽  
Frequency Dependences

Features of obtaining magnetic nanocomposites based on the lattice packing of SiO2 nanoscale (opal matrices) with clusters of multiferroic materials (Li-Zn, Bi, Fe, Dy, Gd and Yb titanates) in their interstitial cavities have been considered. For magnetic nanocomposites creation opal matrices with SiO2 nanoscale of ~ 260 nm in diameter have been used. The composition of nanocomposites has been also studied using X-ray diffractometry and Raman spectroscopy. The results of the frequency dependences measurement for the dielectric constant of the nanostructures obtained have been presented. Hysteresis loops have been examined for the samples obtained in the temperature range from 2 to 400 K.

Organization of layer-type supramolecular nanostructures of a combined liquid-crystalline homopolymer toward lattice packing

Polymer ◽  
2011 ◽  
Vol 52 (18) ◽  
pp. 4114-4122 ◽  
Author(s):  
Yi-Fang Huang ◽  
Yan-Wei Chian ◽  
Jrjeng Ruan ◽  
Shi Jin ◽  
Kwang-Un Jeong ◽  
...  
Keyword(s):  
Liquid Crystalline ◽  
Lattice Packing

Six and Seven Dimensional Non-Lattice Sphere Packings

1969 ◽  
Vol 12 (2) ◽  
pp. 151-155 ◽  
Author(s):  
John Leech
Keyword(s):  
Sphere Packing ◽  
Lattice Packing ◽  
Sphere Packings ◽  
Euclidean Spaces ◽  
Lattice 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.

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