In this brief note, we introduce the new, emerging sub-discipline of iso-fractals by highlighting and discussing the preliminary results of recent works. First, we note the abundance of fractal, chaotic, non-linear, and self-similar structures in nature while emphasizing the importance of studying such systems because fractal geometry is the language of chaos. Second, we outline the iso-fractal generalization of the Mandelbrot set to exemplify the newly generated Mandelbrot iso-sets. Third, we present the cutting-edge notion of dynamic iso-spaces and explain how a mathematical space can be iso-topically lifted with iso-unit functions that (continuously or discretely) change; in the discrete case examples, we mention that iteratively generated sequences like Fibonacci's numbers and (the complex moduli of) Mandelbrot's numbers can supply a deterministic chain of iso-units to construct an ordered series of (magnified and/or de-magnified) iso-spaces that are locally iso-morphic. Fourth, we consider the initiation of iso-fractals with Inopin's holographic ring (IHR) topology and fractional statistics for 2D and 3D iso-spaces. In total, the reviewed iso-fractal results are a significant improvement over traditional fractals because the application of Santilli's iso-mathematics arms us an extra degree of freedom for attacking problems in chaos. Finally, we conclude by proposing some questions and ideas for future research work.
We derive the (d+2)-linear forms that compute the moment of degree d of the area enclosed by a subdivision curve in the plane. We circumvent the need to solve integrals involving the basis function by exploiting a recursive relation and calibration that establishes the coefficients of the form within the nullspace of a matrix.
For demonstration, we apply the technique to the dual three-point scheme, the interpolatory C1 four-point scheme, and the dual C2 four-point scheme.
Authors: Jan Hakenberg
Comments: 25 Pages.
We list examples of 2-dimensional domains bounded by subdivision curves together with their exact area, centroid, and inertia. We assume homogeneous mass-distribution within the space bounded by the curve. The subdivision curves that we consider are generated by 1) the low order B-spline schemes, 2) the generalized, interpolatory C^1 four-point scheme, as well as 3) the more recent dual C^2 four-point scheme.
The derivation of the (d + 2)-linear form that computes the area moment of degree p + q = d based on the initial control points for a given subdivision scheme is deferred to a publication in the near future.