分形数集自相似新异类型

New Type of Self-similarity in Fractal Numbers Sets

按照集合特征把分形自相似划分为4类,以前分形数集中出现的自相似可归为第Ⅰ、Ⅱ、Ⅲ类,未见第Ⅳ类。发现一种新的三元数集,具有Ⅳ类自相似性。数集整体外观像3叶片叶轮,叶边有序排列着无数芽枝。芽枝上结有两种不同形状的自相似子集,结在枝梢段的子集具有全集形,结在枝干段的子集具有Mandelbrot集形,呈现一棵树结两种果的现象。两种子集各带有自己的自相似后代,自相似传递方式迥异。Mandelbrot集形子集的后代都是Mandelbrot集形;全集形子集的后代也分两种形,像全集那样遵循枝干段与枝梢段不同的规则。芽枝干梢两段有可辨的交接处,却是无穷精致的极限点。比较多种分形数集的运算规则和形象,得出数算规则小差别可以导致数集形态和自相似特性大差别的结论,也推测分形数集或有自相似的趋向性。文中给出了该三元数集的定义、样集参数和空间位置,附图21幅展示了整体三维形象、两种子集的剖面形态以及局部复杂的自相似结构。

The fractal self-similarities are divided into four types according to set characters.All examples of self-similarities ever seen in fractal numbers sets are ranged over type Ⅰ,Ⅱ,Ⅲ,except type Ⅳ.It is discovered that a new set of ternary numbers is of the nature of type Ⅳ.The numbers set has the shape of three blades vane-wheel in 3D.Unnumbered sprouts are orderly arranged on the vane edges.An interesting phenomenon shows that there are two different kinds of self-similar subsets growing on the same branches of sprouts,which is just like two different kind of fruits growing on a tree.The subsets on the trunks of the branches are of Mandelbrot set shape,and the subsets on the tips of the branches are of the whole set shape.Both of them have their own ways to transmit self-similarity.The subsets of Mandelbrot set shape have subsets of Mandelbrot set shape.The subsets of the whole set shape have two kinds of self-similar subsets just like their mother sets;they obey different rules at different position.The trunk and the tip of each branch have a visible joint but a limit point.By comparing some fractal numbers sets with their shapes and number operating rules,it can be seen that little difference of the number operating rules make much difference in shape and mark difference in self-similarity property.And it infers that there probably exist self-similarity tendency in fractal numbers sets.With the definitions,the parameters and coordinates are given;there are 21 pictures to show 3D shapes of the whole set,the section images for two kinds of subsets,and complex structures of self-similarity at location.