分形——数学与后数学 因此，我们需要做的就是找出究竟是哪些数学被用来创造了宇宙，那么我们将能够了解我们是怎么来的，我们又将去往何方。因为我们正试图辨别环境的模式，特别是当它们涉及到生物圈时，我们需要发现自然用来将物理结构放入空间的数学。 这样的任务需要使用几何学，因为根据定义，这一数学分支专门关注空间中的结构的特性，量度和关系。几何学对于宇宙的组织而言具有如此的根本性，以至于在伽利略的觉悟之前很久，柏拉图就认为，“几何学在创世之前就已存在。” 直到 1975 年，普通公众仍然只熟悉欧几里德的几何学原理，它们总结在那 13 卷大约成书于公元前 300 年左右的古老希腊文本，《 欧几里德原理 》中。这就是我们大多数人在学校里学到的几何学，我们用它在绘图纸上画出各种结构，如立方体、球体、锥体等等。欧几里德几何使我们得以预测天体的运动，建造宏伟的建筑和园林，甚至建造各种飞船和尖端武器。 然而，欧几里德几何的数学公式并不能马上用于自然界中。比如说，使用标准欧几里德几何的完美形状，你能创造出一棵怎样的树呢？回想一下你在幼儿园画过的树，一个圆圈坐在一根细长的矩形顶上。毫无疑问，你的幼儿园老师承认它是树的某种表达，但是它无论如何也描述不出树到底是什么，就像用火柴棒摆出的小人描述不了真正的人一样。 在欧几里德几何和圆规的帮助下，你可以画出一个完美的圆。但你没法用欧几里德几何去画一棵完美的，或者至少是一棵真实的树。欧几里德几何也同样无法画出像甲虫、山、云、或者其他任何自然界中那些常见形体的结构。当需要描绘生命的结构时，欧几里德几何就相形见绌了。那么，我们到哪里去找柏拉图和伽利略所说的那种数学，那种可以描述自然界固有的设计原理的数学呢？ 大约 90 年前，一位年轻的法国数学家加斯顿·朱利叶发表了一篇论文，报告了他关于迭代函数的研究工作。这篇论文为我们提供了一个线索。他所用的是一个相对简单的公式，只使用乘法和加法，无限地重复下去。要实际地把他的数学公式所编码的图象可视化，朱利叶将不得不解出该公式上百万次的迭代结果，这个过程会花掉他几十年的时间。因此，尽管朱利叶在数学意义上已经构想出了一个分形，但他实际上从来也没有看到过。 只有到了 1975 年，当朱利叶的公式在计算机的帮助下求出结果之后，其深远的意义才得以显现。法裔美籍数学家波努瓦·芒德勃罗在 IBM 计算实验室中分析了混沌系统的模式，他第一个观察到了这种朱利叶只能想象的东西。面对着由分形公式所产生的具有惊人的美丽、充满生机、并且无限复杂的图像，芒德勃罗充满了敬畏。他第一个观察到，分形图像具有重复的自相似模式，无论在何种尺度下研究时均是如此。他越是放大图像，这些结构看起来就越相同。 内在于分形图像的混沌复杂性之中的，是不断重复、相互嵌套的模式。那种国际流行玩具，手绘俄罗斯嵌套娃娃，为分形的重复图像本质提供了一个粗略的观念。每个更小版本的娃娃都与它外边嵌套的那个较大的娃娃相似，但并不一定完全相同。芒德勃罗引入了 自相似 这一名词来描述他在这种新的数学当中所观察到的对象，他把这种数学称为 分形几何 。 图 11-2. 俄罗斯嵌套娃娃代表了分形的重复图像。 芒德勃罗在他分形图像的复杂性中，看到了各种类似于自然界中常见形状的生动模式，如昆虫，贝壳和树木。在历史上，科学多次记录了在自然界结构中的不同尺度上出现的自相似组织模式。然而，在芒德勃罗引入分形几何学之前，这些自相似的模式都被视为仅仅是奇妙的巧合。 分形几何学强调的是整体结构中的模式与其各部分中的模式之间的关系。回想一下前文所说的关于海岸线的例子和关于枝叶、树枝和树干的例子。自相似的模式在自然界中随处可见，特别多见于人体的结构中。例如，在人的肺脏中，气道沿大支气管分支的模式在小的支气管，甚至更小的细支气管的气道分支模式中不断地重复。循环系统中的动脉和静脉血管，以及人体的周围神经网络也都显示重复的，自相似的分支模式。 由于分形几何是真正的自然界设计原理，生物圈本身在其组织的每个层面都显示出相互嵌套的自相似模式。因此，当我们观察并发现了一个组织在较高或者较低水平上的结构模式时，我们就可以像使用地图一样地使用分形原理。分形可以帮助我们洞悉该组织在任何其他水平上的模式。在生物圈中，人类进化的分形模式可以内在地显示出某种与自然界的组织在其他水平上的结构所经历的进化自相似的模式。 恩斯特·海克尔是与达尔文同时代的著名胚胎学家， 1868 年，他在不经意间首次报道了进化过程中自相似分形模式的端倪。海克尔出版了一套现在已经闻名天下的显微图像，它们比较了若干物种和人类的胚胎发育阶段。他指出，所有脊椎动物胚胎，包括人类胚胎在内，都通过了一系列类似的结构阶段。海克尔提出，各种有机体在通过他们的早期发育阶段时，实际上重新追踪了它们的祖先进化的每一个阶段。 海克尔的理论，隐晦地定义为 个体发育重演系统发育 ，其字面意思是“发育是某种对祖先的重演。”不幸的是，这个狂热的海克尔在推广他的想法时，篡改了他的图片，使胚胎的早期阶段看上去比它们实际上更为相似。 尽管他的报告有瑕疵，但人类胚胎在最终获得人形之前的确发生了一系列形变。在这些转变当中，人类的胚胎采取了一系列有序的自相似结构模式，在其中它很像脊椎动物进化早期阶段的那些胚胎。 发育中的人类胚胎形状从一个酷似鱼类的胚胎变形为类似两栖动物的胚胎。然后它继续变形，采纳了爬行动物的胚胎外观，然后是哺乳动物的胚胎外观，最后才获得了人形。通过沿袭其生物圈祖先的胚胎阶段演变，人类胚胎为分形性自相似提供了一个动态实例。 Fractals—Math and Aftermath Consequently, all we need to do is findout which mathematics was used to create the Universe and we will be able tounderstand how we got here and where we are bound. Because we are trying todiscern environmental patterns, specifically as they relate to the biosphere,we need to discover the math Nature used to put physical structure into space. Such a mission invokes the use of geometry because, by definition, this branch of mathematicsis specifically concerned with the properties, measurement and relationships ofstructure in space. Geometry is so fundamental to the organization of the Universethat long before Galileo’s realization, Plato concluded, “Geometry existedbefore creation.” 4 Until 1975, the general public was onlyfamiliar with the principles of Euclidean geometry, summarized in thethirteen-volume ancient Greek text, The Elementsof Euclid, written around 300 b.c.e. This is the geometrymost of us learned in school to plot structures such ascubes and spheres and cones onto graph paper. Euclidian geometry has enabled us to projectthe movement of heavenly bodies, construct great edifices and gardens, and evenbuild spaceships and sophisticated weapons. However, the mathematical formulae of Euclidiangeometry are not readily applicable to Nature. For example, what kind of tree can you createusing the standardized perfect forms of Euclidean geometry? Think back to thetree you drew in kindergarten, a circle sitting atop an elongated rectangle.Your kindergarten teacher, no doubt, recognized it as a representation of atree, but in no way does it describe what a tree really is, no more than astick figure describes a human. WithEuclidean geometry and a compass, you can draw a perfect circle. But you cannotuse Euclidean geometry to draw a perfect or, at least, a realistic tree. Norcan Euclidian geometry describe the structure of a beetle, a mountain, a cloud,or any other familiar patterns found in Nature. Euclidean geometry falls shortwhen it comes to describing the structure of life. So where do we find the typeof mathematics referred to by Plato and Galileo, the math that describes thedesign principles inherent in Nature? We wereoffered a clue about 90 years ago when a young French mathematician namedGaston Julia published a paper on his work with iterated functions. His was a relatively simple equation that used only multiplicationand addition, repeated ad infinitum . To actually visualize the imageencoded in his mathematical formula, Julia would have had to solve millions ofiterations of the formula, a process that would have taken him decades. Therefore,even though he conceived of a fractal in mathematical terms, Julia neveractually saw one. Theprofound implications of Julia’s formula were only revealed when his equationwas solved with the aid of computers in 1975. Benoit Mandelbrot, a French–Americanmathematician who analyzed patterns in chaotic systems at an IBM computing lab,was the first person to observe what Julia could only imagine. Mandelbrot wasawestruck by the strikingly beautiful organic and infinitely complex imagesgenerated by fractal formulae. He was the first toobserve that fractal images possessed repeated self-similar patterns,regardless of the scale on which they were examined. The more he magnified theimages, the more the structure appeared to be the same. Inherent within the chaotic complexity offractal images is the presence of ever-repeating patterns, nested within oneanother. The internationally popular toy, hand-painted Russian nesting dolls,provides a rough idea of the nature of a fractal’s repetitive images. Eachsmaller version of the doll is similar to, but not necessarily an exact versionof, the larger doll in which it is nested. Mandelbrot introduced the term self-similar to describe such objects that heobserved in the new math, which he called fractalgeometry. Withinthe complexity of his fractal images, Mandelbrot observed vivid patterns that resembleshapes common in Nature, such as insects, seashells and trees. Historically,science had frequently documented the presence of self-similar organizationalpatterns at different scales of Nature’s structure. However, until Mandelbrotintroduced fractal geometry, these self-similar patterns were deemed to bemerely curious coincidences. Fractal geometry emphasizes therelationship between the patterns in a whole structure and the patterns seen inits parts. Recall the examples of the coastline and of the twigs, branches andtree trunks cited earlier. Self-similar patterns are found throughout Natureand especially within the structure of the human body. For example in the humanlung, the pattern of branching along the large bronchus air passages isrepeated in the branching structure of the smaller bronchi and even smallerbronchiole passages. Arterial and venous vessels of the circulatory system aswell as the body’s network of peripheral nerves also display repetitive,self-similar branching patterns. Because fractal geometry is truly thedesign principle of Nature, the biosphere inherently reveals nestedself-similar patterns at every level of its organization. Consequently, as weobserve and become aware of patterns at higher or lower levels of anorganization’s structure, we can use fractals in the same way we would usemaps. Fractals can help us gain insight into the organization at any otherlevel. In the biosphere, the fractal pattern of human evolution can inherentlydisplay a self-similar pattern of evolution experienced by structures at otherlevels of Nature’s organization. Ernst Haeckel, a famous embryologist andcontemporary of Darwin, inadvertently reported the first inkling of aself-similar fractal-like pattern in evolution in 1868. Haeckel published a nowfamous sequence of microscopic images that compares the stages of embryonicdevelopment of a number of species with that of the human. He noted that allvertebrate embryos, including the human embryo, pass through a series ofsimilar structural stages. Haeckel argued that, in transitioning through theirearly development, organisms actually re-trace every stage of theirevolutionary ancestry. Haeckel’s theory, cryptically defined as ontogeny recapitulates phylogeny, literallymeans “development is a replay of ancestry.” Unfortunately, when promoting hisideas, an overzealous Haeckel fudged his drawings to make the early stages ofembryos appear more alike than they actually are. Regardlessof his flawed presentation, human embryos do morph through a variety of shapesbefore acquiring human form. In these transitions, the human embryo assumes asequential series of self-similar structural patterns wherein it resembles embryos from earlier stages of vertebrate evolution. Thedeveloping human embryo shape-shifts from one that resembles a fish embryo toone that resembles an amphibian embryo. It continues morphing until it takes onthe appearance of a reptilian embryo and, later, that of a mammal beforefinally assuming a human shape. Evolving through the embryonic stages of its biosphericancestors, human embryos offer a dynamic example of fractal-likeself-similarity.
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