Patterns in Nature and in the Laboratory

这项工作将涉及两个基本主题。首先，我们将试图更精确地理解为什么植物茎尖分生组织附近的图案经常表现出斐波那契序列，也就是说，叶子、苞片和贴片位于斐波那契序列所列举的螺旋家族上。该研究将探索生物结构启动的机制和模式本身的性质。从迄今为止的工作来看，斐波那契模式似乎是环形几何形状中首选的模式，在这种情况下，模式是按顺序排列的，一个环接一个环，因此模式形成区域相当狭窄，并且先前铺设的结构的偏差很强。连接与最佳包装将是研究的一部分。其次，我们将以更好地理解点和线奇点的性质为特定目的重新审视模式的一般描述，当人们通过对模式波长进行平均，从系统的微观描述移动到宏观描述时，奇点和线奇点就会出现。有趣的是，这种模式的三维线缺陷具有自然定义的正、负1、2 / 3、正、负1的指标，而且前者的质量要大得多。此外，缺陷有一个额外的自然指数的一半。将探索与夸克和轻子的可能联系。到处都是这种模式。在沙漠的沙丘上，在长长的沙滩上，在云的形成中，在指尖和动物的皮毛上，在鸟的羽毛上，在鱼皮上，都可以看到它们。它们在流体对流、宽激光束和火焰前沿的实验室实验中也无处不在。在不同的微观环境中产生的许多模式具有许多共同的纹理，因此，通过研究一种环境中的模式，人们可以了解它们在其他系统中的行为，这些系统具有关键的对称特性。除了因其内在美而令人感兴趣外，图案也因其实际原因而值得研究。它们有许多潜在的用途，从开关到信息存储设备。在某些特殊情况下，它们也往往是结构，可以优化诸如流体中的热量传递和植物中的聚光能力。该项目将调查模式形成背后的基本问题，目的是对这一重要现象提供更深入的了解。

The work will address two basic themes. First, we will attempt to understand more precisely why it is that the patterns seen near the shoot apical meristems of plants so often exhibit Fibonacci sequences, that is, leaves, bracts, and stickers lie on families of spirals enumerated by Fibonacci sequences. The study will explore both the mechanisms through which the biological structures are initiated and the nature of the patterns themselves. From work to date, it would seem that Fibonacci patterns are the preferred pattern in annular geometries in circumstances where the pattern gets laid down sequentially, annulus by annulus, so that the pattern forming region is quite narrow and the bias from the previously laid down structures is strong. Connections with optimal packing will be part of the study. Second, we will revisit the general description of patterns with the particular aim of understanding better the nature of the point and line singularities which occur when one moves from a microscopic description to a macroscopic description of the system by averaging over the pattern wavelength. What is intriguing is that the three dimensional line defects of such patterns have naturally defined indices of plus and minus one and two thirds and plus and minus one and that the masses of the former are much larger. Also, the defects have an additional natural index of one half. Possible connections with quarks and leptons will be explored. Patterns turn up all over the place. One sees them on desert dunes, on long sandy beaches, in cloud formations, on fingertips and animal coats, on the feathers of birds, and on fish skins. They are also ubiquitous in laboratory experiments on convection in fluids, on broad laser beams, and on flame fronts. Many of the patterns arising in different microscopic contexts have textures with much in common, so that by studying patterns in one context one can learn about their behaviors in other systems that share key symmetry properties. Besides being of interest for their intrinsic beauty, patterns are also worth studying for practical reasons. They have many potential uses from switches to information storage devices. They also tend to be the structures which, in certain special circumstances, can optimize such things as heat transfer in fluids and light gathering capacity in plants. This project will investigate fundamental questions underlying pattern formation, with the goal of providing deeper insight into this important phenomenon.