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Nature's Rights in Permaculture
Published in Cameron La Follette, Chris Maser, Sustainability and the Rights of Nature in Practice, 2019
It is easy to observe spirals at night in the Milky Way and other constellations, and in the hurricane and tornado. The arrangement of leaves on a plant stem as well as the phylotaxy of the branches themselves shows a spiral pattern, often in the Fibonacci sequence (a numerical sequence in which each number is the sum of the previous two, on to infinity). Rivers create spirals by meandering, and are thereby oxygenated. Meanders throw the water from one bank to the other. The water flowing along the bank is thus slowed by friction, and creates a velocity differential in the river, causing it to turn over (spiral) as it flows downstream. This allows a greater percentage of the water to interface with the oxygen-filled surface atmosphere; meanwhile, higher in the terrain, the faster-flowing streams form plunge pools that create a more oxygen-rich environment and habitat for oxygen-loving fish such as trout. The downstream rivers have less oxygen available, so the fish that evolved in this environment are slower and larger, like catfish and carp.
Transform methods
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
We continue to analyze Fibonacci numbers, using methods from other parts of this book. By definition, the Fibonacci sequence solves the difference equation Fn + 2 = Fn + 1 + Fn with F0 = F1 = 1. As we did with ODE, we can guess a solution λn and obtain the characteristic equation λn+2=λn+1+λn.
Model updating by minimizing errors in modal parameters
Published in Heung-Fai Lam, Jia-Hua Yang, Vibration Testing and Applications in System Identification of Civil Engineering Structures, 2023
The Fibonacci sequence (and the golden ratio) is famous, as it describes many regular patterns in nature (e.g., the number of petals of flowers, the pattern of seeds within a sunflower, the nautilus shell). The Fibonacci spiral is shown in Figure 6.20, where the first 10 Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. Does it look like a nautilus shell?
Low voltage tunable liquid crystal Fibonacci grating
Published in Liquid Crystals, 2020
Swadesh Kumar Gupta, Zhibo Sun, Hoi-Sing Kwok, Abhishek Kumar Srivastava
Recently, Fibonacci gratings (FbG) have been explored extensively for its potential application in far-field super-resolution imaging system [1–3]. Circular FbG shows twin axial foci diffractive lenses with self-similarity features [4–7]. FbG have been studied for replicating of quasicrystals due to their unique aperiodic, self-repeating structure following the golden rule. Periodic gratings can only produce a fixed phase shift in the frequency domain, which leads to non-continuous frequency shift causing loss of some feature information particularly for the far-field imaging. FbGs provide a quasi-continuous shift of frequency spectrum, and thus, prevent the loss of certain part of subwavelength information [1,2]. The Fibonacci sequence is obtained by following the recurrence relation Fn = Fn–1 + Fn–2, n ≥ 2; with seed values F0 = 0 and F1 = 1 [6]. The next few generations can be written as F2 = 10, F3 = 101, F4 = 10110, and so forth. As n→∞, the ratio between successive numbers approaches to well-known golden ratio ϕ = (1+√5)/2 ≈ 1.618, which play an important role for diffractive properties of FbGs. A Fourier transform of FbGs does not show the broad peaks that are normally associated with a disordered structure. The transform from a Fibonacci array forms a dense set of sharp diffraction orders having different intensities. The positions of these orders are related to each other by the powers of the golden ratio [7,8].
Fibonacci Ideal Convergence on Intuitionistic Fuzzy Normed Linear Spaces
Published in Fuzzy Information and Engineering, 2022
The Fibonacci sequence of numbers and the associated ‘Golden Ratio’ are observed in nature. We examine that various natural things follow the Fibonacci sequence. It appears in biological settings such as branching in trees, the flowering of an artichoke and the arrangement of a pine cone's bracts etc. Nowadays Fibonacci numbers play a very significant role in coding theory. Fibonacci numbers in different forms are extensively applied in constructing security coding. The Fibonacci Numbers are also applied in Pascal's Triangle. Amazing applications can be examined in [16].
A parametric analysis of the “digitally-derived geometric design” of the façade of the Macau Holy house of Mercy
Published in Journal of Asian Architecture and Building Engineering, 2022
A Golden Rectangle could be divided into a subordinate golden rectangle and a square in succession, that is to say “A main golden rectangle = a subordinate golden rectangle + a square”. Such divisions bring about a series of mutually-perpendicular/mutually-paralleled diagonals and golden spirals (Figure 4). The mathematical expression of Golden Ratio is . The Fibonacci Sequence, also known as the Golden Section Sequence, is: 1, 1, 2, 3, 5, 8, 13, 21, 34, … …