分形几何与斐波那契数列的对比

摘 要

分形是美籍法国应用数学家蒙德布罗特所提出的，它和英文中的

fracture(断裂）和fraction （分数）有一定联系，体现出蒙德布罗特创立这

个新的几何思想。分形几何作为一门新兴的交义学科，正在被越来越多的人

所认识和学习。据美国科学家情报所调查，八十年代，全世界有1257种重要

学术刊物所发表的论文中，有37.5%与分形有关。美国著名的物理学家Wheeler

说：“可以相信，明天谁不熟悉分形，谁就不能被认为是科学上的文化人”】16【。

传统的欧式几何主要研究对象是规则图形和光滑曲线，对自然景物的描述却

显得无能为力。而分形几何的创立，就是用来描述那些欧式几何无法描述的

几何现象和事物的，被誉为“大自然本身的几何学”，使自然景物的描绘得以

实现，这也是分形几何得到高度重视的原因之一。

斐波那契数列产生于一个关于兔子繁殖后代的问题：某人有一对兔子饲

养在围墙中，如果它们每个月生一对兔子，且新生的兔子在第二个月后也是

每个月生一对兔子，问一年后围墙中共有多少对兔子？斐波那契数列从问世

到现在，不断显示出它在数学理论和应用上的重要作用。如今，斐波那契数

列渗透到了数学的各个分支中。同时，在自然界和现实生活中斐波那契数列

也得到了广泛的应用。如一些花草长出的枝条会出现斐波那契数列现象，大

多数植物的花的花瓣数都恰是斐波那契数列等等。

斐波那契数列又被称为是黄金分割数列，而黄金分割本身就是一种分形

的例子。二者都可以解决一些传统数学所不能解决的问题，所不同的是分形

几何是通过几何的角度来解决问题，而斐波那契数列则是通过代数的角度来

解决实际问题。

作为一门新兴的对现实生活有重要影响的两个定义，研究两者的对比关

系，探讨如何更好地运用这两个定义来解决现实中的一些实际问题，具有重要

意义。

关键字：斐波那契数列；分形几何；应用；对比

ABSTRACT

Fractal is first put forward by French-American applied mathematician

Mandelbrot. It relates to the words “fracture” and “fraction”, reflecting Mandelbrot’s opinion on creating the new definition. As a rising interdiscipline subject, Fractal is being understood and learned by more and more people. According to the survey of

American Scientist Information Institution, in the 1980s, among all the papers published on worldwide 1257 important academic journal, 37.5% is related to Fractal. American famous physicist Wheeler said: “ I am confident that who is unfamiliar with Fractal, who will not be considered as the science intellectual in the future.” Traditional European-style geometry takes norm graph and smooth curve as the main researching object, and seems helpless to natural features. The foundation of Fractal is to describe the phenomenon and features that European-style geometry cannot, and so Fractal is honored as “geometry of the nature”. Being able to describe the nature features is one of the reasons that Fractal is highly valued.

Fibonacci Series comes from the problem of rabbits raising: a man has a couple rabbits raised within walls, if they give birth to a couple rabbits each month, and the new born will give birth to a couple rabbits in the next month, after one year, how many rabbits will be there within the walls? From established to today, Fibonacci Series continues to show its importance in mathematical theory and application. Nowadays, Fibonacci Series have permeated to each branches of mathematic. Meanwhile Fibonacci Series extensively applies to nature and real life. For example, flowers and plants’ branches appear Fibonacci Series phenomenon, and most plant’s peal is exactly Fibonacci Series.

Fibonacci Series is also named as Golden Section Sequence，and golden section itself is an example of fractal. Both of them can solve some problems that traditional mathematic cannot. The difference between them is that Fractal solve problems according to geometrical perspective, and Fibonacci Series according to algebraic perspective.

Two definitions as a new reality have an important influence on the real life, the study of contrast relationship between Fractal and Fibonacci Series and discussion of how to use the two definition to solve problems in real life has great significance.

Key words: the Fibonacci series; Fractal geometry; Application; contrast