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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
Design
Published in Wanda Grimsgaard, Design and Strategy, 2023
Constructed from the principles of the golden ratio, the golden rectangle has the classic proportions often associated with natural beauty and harmony. Consciously used, the golden rectangle can be a good starting point for defining proportions for formats, objects and spaces that are proportionately scalable, perceived as aesthetic, and that provide a sense of quality and harmony. The aspect ratio of the golden rectangle can be repeated and expanded within the same number ratios, both vertically and horizontally. This provides an optimal starting point for creating an aesthetic and functional modular grid system, with similarity between the format’s external proportions and the internal size ratios. The golden ratio can be found in many logo designs, including the logos of Toyota, Chevron, Nissan, National Geographic, even though this has probably not been the designer’s intention. And if we take a look around us, we will probably discover a golden rectangle in a table, a book, a window, a vase. But it is not always intentional, because it might have been the designer’s built-in talent for proportions that has led to the result being close to the golden ratio. The closer the ratio between the length and width of the rectangle is to 1.618, the more certain you can be that the ratio is placed there intentionally (Geelmuyden, 2013).
ce 500
Published in Paul Marsden, Digital Quality Management in Construction, 2019
The Greeks were masters of architectural building proportions. The proportion of length to width of buildings was centred on the preferred ratio of 1 : 1.6 (or more exactly 1.61803), the so-called ‘golden section’, referred to in Euclid (see Chapter 7 on data), based upon Phi Φ, and appearing in the numerical series called the Fibonacci sequence after the twelfth-century mathematician. Instinctively, many people seem to prefer this ratio and it may have been derived from many years of building design that seek to harmonise nature, the human appreciation for aesthetics and architecture. The number of petals in a flower usually follows the golden ratio and each petal is placed at 0.61803 per rotation, to maximise sunlight exposure. The human face, seed heads, pinecones and shells typically follow the golden ratio. Even the Parthenon, built 150 years before the golden ratio was formally recorded, seems to follow it in some parts of its design.12 While never an overriding design concept, it does seem to appear to be hard-wired into our aesthetic preferences.
Math and Art: An Introduction to Visual Mathematics, 2nd ed.,
Published in Technometrics, 2022
Chapter 1 of Euclidean Geometry starts with the proof of the Pythagorean theorem, describes the 5 axioms and several of 28 theorems presented in The Elements written by Euclid (325–265 BC). It shows how to construct geometrical objects by an unmarked ruler and compass, for example, bisecting a line segment, duplicating an angle, building a perpendicular and a parallel line, various drawings with circles, and more. It explains how to multiply numbers geometrically, discusses some ancient problems, including Heron’s problems. The golden ratio is defined and shown in building of the Parthenon in Athens, in the interiors of the Great Pyramid of Giza, and presented in golden triangles, spirals, and other figures, including our DNA two spiral coils of double helix. Fibonacci numbers are illustrated by various flowers structures.
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].
The Conceptual Design of Bridges: Form Finding and Aesthetics
Published in Structural Engineering International, 2021
The Golden Ratio is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. The same can be expressed in equation form as a/b = (a + b)/a = 1.618,033,988,749,89 … (etc.). This is also referred as “ϕ”, an irrational number. It is also known as the Golden Mean, the Golden Number, the Divine Proportion, the Divine Section and the Golden Proportion. This irrational ratio “ϕ” (Fig. 4) is inherent in one or another way in a rectangle, triangle, pentagram, spiral, etc. making them divine.