![]() ![]() The Fibonacci sequence even plays a role in the subtle spirals you can see in the seed head of a sunflower. Bananas have three sections whilst apples have five. If you cut into a piece of fruit, you’re likely to find a Fibonacci number there as well, in how the sections of seeds are arranged. No wonder rare four leaf clovers are seen as lucky! That is of course, until a petal falls off. Irises have three petals whereas wild roses and buttercups have five petals. Most flowers, for example, will have a number of petals which correspond with the Fibonacci sequence. Even though, it is seen that the curves in the complex plane displayed by the generalized Fibonacci functions during their evolution enable a better understanding of the behavior exhibited by the starting discrete model, such as the regimes of stability and instability, or the appearance of single and multiple fixed points.The mathematical sequence that governs natureįor starters, Fibonacci numbers can be found in the natural world all around us. The dynamics exhibited by the corresponding generalized Fibonacci functions are investigated and analyzed here, finding how apparently simple relations may describe relatively complex behaviors on the complex plane even in the case of regular or periodic solutions. This leads to a rich variety of dynamical behaviors in the complex plane depending on the value of the parameters involved in the associated characteristic equation. ![]() On the other hand, the discrete index of the usual Fibonacci sequence is replaced by a continuous (evolution) variable or parameter (although the Fibonacci initial conditions are kept). This work intends to go a step beyond, introducing a generalized Fibonacci function based, on the one hand, on the close resemblance between such an equation and the one resulting from recasting a general two-variable linear system of difference equations as an also single second-order homogeneous linear difference equation. In this regard, the renowned Fibonacci sequence constitutes an interesting example of iterative sequence that can be modeled in terms of such equations, more specifically a second-order homogeneous linear difference equation. That is, we not only must start at the initial values and iterate for the result, but also iterate the result to find the initial values using the inverse of f.ĭifference equations model the evolution of many processes of interest in physics, biology, economics, etc., in a discrete or iterative manner instead of a continuous parameter (e.g., time). In order to do so, it is necessary to view both directions of the Julia set. Finally, it will apply this process to a Julia set with two spi- rals from a single point and compare the results. The following sections will prove that the Julia Set exhibits a loga- rithmic spiral for small values, determine the growth rate, and analyze the error at larger values. This paper seeks to analyze one of the geometric features of the Julia sets: the spiral. Meanwhile, the presence of rotational symmetry owes itself to the rotational nature of complex numbers. This infinite complexity arises from the iterations of a function f : C → C. The fractal nature of the Julia sets causes spirals and circles appear mul- tiple times in sublimely symmetric patterns. ![]() Iterating a simple polynomial function infinitely leads to what is called a fractal, or a self-similar shape. Their popularity, on the other hand, comes not from the complexity of the results, but from their aesthetic appeal. In the study of chaotic systems, simple processes can lead to infinitely complex results. While considering the possibilities in an application of such creations (models), some optimal intersecting surfaces are discussed. 3D model presentation of dynamic spiral patterns is performed in engineering software Auto-CAD. Three types of regular polygons are here included: triangle, square, and pentagon. This investigation includes surfaces: cone, sphere, ellipsoid and elliptic hyperboloid. We observed the series of inscribed polygons as dynamical spiral patterns of scaled frames, according to the geometry of the basic quadric surface. Since the term "spiral" is directly connected to circles we aimed our investigation to quadric surfaces with circular sections, where inscribed polygons obtain the spiral form by "twisting". Acquainted with different approaches, present in the practical and theoretical sense, from empiric creations to parametric modeling, we chose to explore the dynamic patterns which appear in spiral shapes generating process. Spiral forms, nowadays actual, especially in the area of architecture and design, were the inspiration point for a creative geometrical research. ![]()
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