Clearly, common fractal features such as non-linearity and symmetry are present in these pictures. 3 shows some of the experimental results. Also, to eliminate numerical errors, the rewrite rules for the Koch curves have been modified as depicted in Fig. Due to space complexities, the implementation is limited to a 4-D Koch curve with only 100 pixels per dimension, which still takes as much as 100MB disk storage. Software has been written to validate and visualize (in 2-D sections) these n-D algorithmic fractals. Line segment divided by the length of the parent line segment. Where a is the number of child line segments spawned from the parent line segment at eachĪpplication of the rewrite rules, and s is the contraction factor, i.e., the length of one child Possibly, D will not be a whole number, and it can be computed as follows: Generation of n-D Koch Curves and Other Algorithmic Fractalsįor every n-D algorithmic fractal, the IFS theory predicts a fractal, or self-similarity, dimension D different than its Cartesian dimension n. Since 3-D and n-D translation, scaling, and rotation matrices are known, the formula, in a short hand, !’ = × ! is practical and convenient to extend to n-D spaces. Then !’ for the 3-D Koch curves can be computed by expanding ! to 4×4 matrices, and then left-multiplying by. Let ! be a set of 3 × 3 affine transformation matrices used by the IF S corresponding to the 2-D Koch curve L-system. In other words, F % (2-D rewrite rules of the same Lsystem), where % denotes concatenation. Using formal languages, this generalization can be represented as an L-system: F F F-F F, where =. Intuitively, instead of replacing the mid-one- third line segment with two line segments in the 2-D plane, replace it with eight segments into four directions, thereby expanding the three dimensions. Consider a possible generalization of the classical construction of 2-D Koch curves into 3-D spaces (Fig. In order to understand how 3-D algorithmic fractals can be generalized to 4-D or even higher dimensional fractals, a process which is certainly difficult for humans, it is essential to understand how 2-D fractals can be generalized into 3-D. In this research, these techniques have been augmented and extended to study n-D algorithmic fractals. Algorithmic fractals are equally interesting since they serve as bridges connecting chaos theory to well studied formal language theory.1 2-D and 3-D algorithmic fractals are conventionally generated and analyzed using L-systems and iterated function systems (IFS). However, all these discoveries have been limited to algebraic fractals such as the Mandelbrot set and the Julia sets. Notable works in this area include the uses of quaternions, commutative hyper-complex calculus, and, most recently, doubling processes. Multi-dimensional space filling curves, for instance, are a focus for contemporary topologists. N-D fractals, where n> 4, have long been under scrutiny for their mathematical properties and artistic values.
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