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.
