Fractals with Optimal Information Dimension

Abstract

We present novel fractals with dimension that is close to the optimal value of e. The examples are for up to 10-way branching at each iteration and they include symmetric and asymmetric ones, as well as those where in addition to holes a few layers of other regions are peeled. Only representative fractals are described and the method can be used to find many other designs. Since natural systems are likely to be optimal, the examples have potential applications in both natural or engineered systems.

The e-Dimensionality of Genetic Information

“…Consider the rank order associated with the number of codons to amino acids which is shown in Table 3. The same pattern repeats to a different scale as we trace from left to right in Figure 1, which is the hallmark of a fractal or noninteger dimensional process [20][21][22][23] [24].…”

New Classes of Regular Symmetric Fractals

“…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.…”

“…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.…”

New Classes of Regular Symmetric Fractals

“…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.…”