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. We use this theorem to give, for all sets E that are analytic, i.e., Σ¹₁, a tight bound on the packing dimension of the hyperspace of E in terms of the packing dimension of E itself.Minglei Xu and Dr. (For a concrete computational example, the stages E₀, E₁, E₂, … used to construct a self-similar fractal E in the plane are elements of the hyperspace of the plane, and they converge to E in the hyperspace.) Our third main result, proven via our extended point-to-set principle, states that, under a wide variety of gauge families, the classical packing dimension agrees with the classical upper Minkowski dimension on all hyperspaces of compact sets. We demonstrate the power of our extended point-to-set principle by using it to prove new theorems about classical fractal dimensions in hyperspaces. Our first two main results then extend the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families. We first extend two fractal dimensions - computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions dim(x) and Dim(x) to individual points x ∈ X - to arbitrary separable metric spaces and to arbitrary gauge families. In this paper we extend the reach of the point-to-set principle from Euclidean spaces to arbitrary separable metric spaces X. These are classical questions, meaning that their statements do not involve computation or related aspects of logic. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces ℝⁿ. Go to the corresponding LIPIcs Volume PortalĮxtending the Reach of the Point-To-Set Principle When quoting this document, please refer to the following License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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