Two Classes of Regular Symmetric Fractals

The paper introduces new fractal families with annular and checkerboard structures that include the Sierpinski carpet and the Menger sponge as special cases. The complementary mapping is defined and a notation to represent the families is proposed.


Introduction
The importance of fractals to the study of natural, social and engineered systems is well established [1] [2]. They belong to the larger field of scale-invariant and selfsimilar systems [3][4] [5][6] [7] [8]. In a recent paper [9], novel fractals with dimension that is close to the optimal value from the perspective of information efficiency [10][11] [12] were presented. Designs were found for up to ten-way branching at each iteration and they included symmetric and asymmetric ones, as well as those where in addition to holes, a few layers of other regions are peeled. Only representative fractals were described with the hope that the method can be used to find many other designs that will provide insight into systems in biology, chemistry, polymers, and the earth sciences.
In more advanced investigations, the mathematical basis of noninteger spaces [13] [14] will have to be brought into the analysis. Some progress in that direction has been made in applications to the information dimension of physical space [15] which takes us to the view that space is not to be seen as a three-dimensional container. The evolutionary stages of a noninteger dimensional physical space were considered [16], and this has applications to astronomical systems as well as metamaterials [17] [18]. This paper lists new fractals that are motivated by examples of physical systems where influences flow out from the center in a manner that is captured by annular mapping as well as those given by checkerboard patterns. This work has many parallels with constructions in [9]. Since natural and engineered systems are likely to have a range of characteristics arising from the underlying physics associated with different values of information efficiency, the examples are of more than just a theoretical interest.

Iterated function systems
Iterated Function System (IFS) fractals are created on the basis of simple plane transformations that include scaling, dislocation and the rotation of the plane axes [19] [20]. Creating an IFS fractal consists of following steps: 1. Draw an initial pattern on the plane 2. Transform the initial pattern using appropriate contracting transformation 3. Combine initial and transformed patterns 4. Repeat as many times as desired In many cases, each iteration consists of one or more affine transformations of the type (with suitable constants (a, b, c, d, e, f): For example, the binary tree fractal of Figure 1 that has been drawn to 6 iterations requires the specification of the scaling factor r and the angle . We can also see the algorithm as one where each "cube" is replaced by the number ( ) at each iterative step, with 1 = , and dimension d equal to: This provides an easy procedure to compute the dimension for regular mappings as we will see in the examples given below.

Annular planar maps
We will first consider planar fractals (dimensions 1<d<2) which have annular square forms.
Definition. An annular planar fractal where at each iteration a square is replaced by × sub-squares with alternating bands of sides n, n-l, n-m… will be called 2 ( , − , − , … ), where the sub-squares at the layers of , − , − , … are retained and the others are removed.
Note that the layer counts of sub-squares from outside with count of n in decrement by 2.
Examples. The first iteration of two mappings based on 49 and 64 cell subdivision is presented in Figure 2. For each of these cases, we need to count the number of sub-squares that are retained in each iteration and use formula (2) to compute the dimension. In (a), the basic 7 × 7 plane is replaced by 25 smaller squares = 7 2 − 5 2 + 3 2 − 1 = 32 and this substitution is repeated. In the 8 × 8 example of (b), each iteration gives us Example. The complementary mapping for 7 2 (7, 3) will be 7 2 (7, 3) = 7 2 (5, 1).
In Figure 3, the iteration maps each square into 81 sub-squares a of which 49 are retained (shown in dark). The dimensionality of this mapping will be: � 9 2 (9,5,1)� = ln 49 9 ≅ 1.771 For a more general case, consider the alternating annular square rings of 11 × 11 given in Figure 4. The sub-squares that are retained will have the count 11 2 − 9 2 + 7 2 − 5 2 +3 2 − 1 = 2(20 + 12 + 4) = 72. Therefore, In general, for the × case, we can compute the retained sub-squares as where n -l is the least positive term in the sequence. Simplifying, we find that for even n, Proof. This follows from the fact d ( )= ln ln . It also follows that d ( ) will be less than 1, if < , the logarithm function is monotonically increasing.

Basic planar maps
We now consider basic annular maps for n = 3, 4, and 5 as in Figure 5. The dimensionality of these maps is As mentioned before, 3 2 (3) is the Sierpinski map [21], which is shown below in Figure 6 for its second, third, and fourth iterations:  This fractal is basically similar to the Sierpinski carpet excepting that the whites are bigger.

Related symmetric checkerboard maps
Another interesting regular fractal is given by the mapping of Figure 8 that represents part of the checkerboard. A mapping where C is rotated will naturally not change the dimensionality.
The next order symmetric checkerboard map will be 5 × 5 as shown in Figure 9: Its complement mapping (with three white squares at the top that is not shown here) will have dimensionality of ln 12 ln 5 ≅ 1.544. Since the difference in the counts of black and white squares will only be one, the two dimensionality figures will be quite close to each other.
One can also modify the checkerboard map in a variety of ways to get new fractals.

Annular fractals in three dimensions
For clarity, the generalizations of the annular mappings into three dimensions will be shown with the super-script 3, as in 3 (k).
Let us consider the generalization of 4 3 (4). In its first iteration it will look like Figure 10. The number of sub-cubes generated from a cube at each iteration is 32 and therefore The first iteration of 5 3 (5) in shown in Figure 11. Its dimensionality is: This dimensionality value is less than that for 4 3 (4) for it is less dense.

Conclusions
The paper introduced new fractal families with regular, square annular and checkerboard structures. The complementary mappings were defined and a notation to represent the families was proposed. This work may be extended by defining other classes that are either larger in the number of sub-squares at each iteration or are according to some other simple definition for the basic planar forms.