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→Mathematics
Sphere packings and coverings. Mathematics underlies the other pattern formation mechanisms listed.
Sphere packings and coverings. Mathematics underlies the other pattern formation mechanisms listed.
例如最密堆积（球填充）和覆盖。数学是其他斑图生成机制的基础。
In geometry, a '''sphere packing''' is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space.
In geometry, a '''sphere packing''' is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space.
在几何学中，球填充是在一个包含空间内不重叠的球的排列。球通常都具有相同的尺寸，填充空间一般指三维的欧几里得空间。然而，球填充问题可以推广到不相等的球、其他维度的空间(二维的问题变成圆填充，高维的问题变成超球填充)或非欧几里得空间，如双曲空间。
A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the density of the arrangement. As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume.
A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the density of the arrangement. As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume.
一个典型的球填充问题是找到一种安排，在其中球填充尽可能多的空间。球体所占空间的比例称为排列密度。由于在无限空间中填充的局部密度可能随测量的体积而变化，因此问题通常是使在足够大的体积上测量的平均密度或渐近密度最大化。
For equal spheres in three dimensions, the densest packing uses approximately 74% of the volume. A random packing of equal spheres generally has a density around 64%.
For equal spheres in three dimensions, the densest packing uses approximately 74% of the volume. A random packing of equal spheres generally has a density around 64%.
对于三维尺寸相等的球体，最密的填充使用大约74%的体积。随机排列的等量球体的密度通常在64%左右。
{{further|Gradient pattern analysis}}
{{further|Gradient pattern analysis}}