In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap.

In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a lattice packing is

2022年11月13日 · Show how alternative ways of stacking three close-packed layers can lead to the hexagonal or cubic close packed structures. Explain the origin and significance of octahedral and tetrahedral holes in stacked close-packed layers, and show how they can arise.

A circle packing is an arrangement of circles inside a given boundary such that no two overlap and some (or all) of them are mutually tangent. The generalization to spheres is called a sphere packing.

Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible.

2025年1月31日 · In hexagonal close packing, layers of spheres are packed so that spheres in alternating layers overlie one another. As in cubic close packing, each sphere is surrounded by 12 other spheres. Taking a collection of 13 such spheres gives the cluster illustrated above.

2022年11月24日 · The Hexagonal Close-Packed (HCP) unit cell can be imagined as a hexagonal prism with an atom on each vertex, and 3 atoms in the center. It can also be imagined as stacking 3 close-packed hexagonal layers such that the top layer and bottom layer line up.

Hexagonal close packing, or hcp in short, is one of the two lattice structures which are able to achieve the highest packing density of ~74%, the other being face centred cubic (fcc) structure. This packing structure is found in metals such as zinc, cadmium, cobalt and titanium.

1999年5月26日 · In 3-D, there are three periodic packings for identical spheres: cubic lattice, face-centered cubic lattice, and hexagonal lattice. It was hypothesized by Kepler in 1611 that close packing (cubic or hexagonal) is the densest possible (has the greatest ), and this assertion is known as the Kepler Conjecture.

In transition metal compounds, d electron effects such as crystal field stabilization energy (CFSE) can be important in determining structure. The larger CFSE of metal ions in octahedral sites is sometimes an important factor in determining spinel structures (normal vs inverse).