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Design
Published in Wanda Grimsgaard, Design and Strategy, 2023
Here is how the Fibonacci numbers work: Each number is the sum of the previous two numbers. If you add the first two numbers 0 and 1, you get 1, you add 1 and 1, you get 2, and so the number sequence goes on infinitely. The conditions of the golden ratio can also be found in the infinite Fibonacci numbers. By dividing two numbers in the Fibonacci sequence, one gets close to the golden ratio. Example: 144:89≈1,6179≈1.618 (last decimal rounded off to 8). Proportion can also be expressed in the aspect ratio 21:34 (Rannem, 2008). The relationship between the golden ratio and the Fibonacci numbers can be explained by a calculation, which is both long and complex. The fascinating thing about the connection between the Fibonacci numbers and the golden ratio is that the two mathematical theories were probably defined independently of each other.104
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 golden ratio is closely related to the Fibonacci numbers, Fn, that forms the Fibonacci sequence, in which the n-th number is equal to the sum of the previous two numbers (i.e., the (n – 1)-th and (n – 2)-th numbers) for n > 1. That is: F0=0,F1=1,andFn=Fn−1+Fn−2forn>1
Fibonacci and Lucas Riordan arrays and construction of pseudo-involutions
Published in Applicable Analysis, 2021
Candice Marshall, Asamoah Nkwanta
The Fibonacci numbers are given by the following recurrence relation, and for [6]. The first few Fibonacci numbers A000045 [1] are 1, 1, 2, 3, 5, 8, 13, …. The Fibonacci number is given by the following expression The generating function of the Fibonacci numbers is In this paper, we also present the Lucas numbers, sequence A000032 [1]. However, we introduce a modified version, denoted by , because the Lucas numbers begin with a leading 2. We prefer that the leading (constant) term be 1. The corresponding generating function of the modified Lucas numbers A000204 [1] is We now introduce the definition of Riordan arrays. The arrays are also known as Riordan matrices. Let denote the set of natural numbers (including 0) and the set of complex numbers.
The golden ratio and regular hexagons*
Published in International Journal of Mathematical Education in Science and Technology, 2020
This innocent-looking line division, which Euclid defined for some purely geometrical purposes, would have consequences in topics ranging from leaf arrangements in botany to the structure of galaxies containing billions of stars, and from mathematics to the arts [1]. Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties [1]. The golden ratio has a surprising connection to the Fibonacci numbers (1, 1, 2, 3, 5, …). The ratio of consecutive Fibonacci numbers gets closer and closer to the number 0.618034. For instance, 34/55 = 0.6182 which is already quite close. The limiting value is exactly , the golden ratio [2].
Modifications of the Robertson Method for Calculating Correlated Color Temperature to Improve Accuracy and Speed
Published in LEUKOS, 2023
Doug Baxter, Michael Royer, Kevin Smet
There are several ways to search a LUT that will all return the correct answer, but the number of calculations needed to reach the solution and type of operations being performed vary, which can affect the time needed to complete the calculation. Important search methods that may be used include: Linear: A linear search means the distance between the chromaticity of the test light source and the chromaticity coordinates in the LUT is calculated for each row of the LUT in sequential order until the point where the distance begins to increase (one row after the minimum). A linear search is simple to program and is used, for example, in Excel tools supplied with ANSI/IES TM-30 (IES 2020) and CIE 224 (CIE 2017)Binary (or bisection search): A binary search progresses by repeatedly dividing the array in half until the minimum distance is found. The first, last, and middle distance are first calculated, which allows identification of the portion of the array in which the minimum falls, producing the next interval to be searched. Executing a binary search provides an overall time savings compared to a linear search. Some implementations of CCT calculations, such as those in LuxPy (Smet 2020), implement a binary search.Golden section: A golden section search algorithm also divides the array into parts, but unlike the binary method, they are unequal. The method maintains four points, with the three intervals having the golden ratio. As with a binary search, the minimum is found to be iteratively reducing the region being searched. Zhang (2019) documented the use of a golden section search, in combination with a technique of iteratively creating new LUTs within a specified region, to estimate CCT at an accuracy determined by a stopping criterion.Fibonacci: The Fibonacci search algorithm is closely related to the golden section search. The relative size of the two parts of the searched array is the same as consecutive Fibonacci numbers – this ratio approaches the golden ratio. The size of the subintervals is not perfectly constant, however.