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Design
Published in Wanda Grimsgaard, Design and Strategy, 2023
The golden spiral is constructed based on the golden ratio principle. We find it in many places in nature, such as in a conch shell, in a wave, in the human ear. The golden spiral has an outer shape with the same proportions as the golden rectangle. The same proportions are repeated in all the rectangles from which the spiral is built, all the way into the core. The golden spiral is proportional both vertically and horizontally, and can be built from the outside in or from the inside out infinitely. Or put in another way: It can be divided into smaller golden rectangles or more golden rectangles can be added. (4.8.5 The golden ratio, 4.9.9 Grid system)
Bioconstruction and Harmonic Complexity of Biomimicry Organisms
Published in Jacqueline A. Stagner, David S-K. Ting, Green Energy and Infrastructure, 2020
Gabriel Barbeta, Yomna Abdallah
One of the most widespread examples that illustrates it is the structure of the nautilus shell. This expresses the golden spiral, following the Fibonacci series. Each radius of the circle encompasses the sum of the previous two. Following the formula n = n − 1 + n − 2, the fundamental series is 1, 1, 2, 3, 5, 8, 13, 21, 34, as the numbers get larger, the quotient between two consecutive tends to approach the number ф. A similar fact occurs in the spiral of Archimedes.
The Conceptual Design of Bridges: Form Finding and Aesthetics
Published in Structural Engineering International, 2021
This ratio has certain algebraic and geometric properties and is a transcendental number similar to “π” (the ratio of the perimeter to the diameter of a circle). The proportions are infinitely divisible, where each subdivision retains its original proportion and is harmonically related not only to whole but to all subdivisions. For example, when a series of Golden Rectangles are assembled on each other and their outside corners are connected by a smooth curve as depicted in Fig. 4, the culmination is a Golden Spiral. Luca Pacoli, friend of Leonardo da Vinci, called this the “divine proportion”, and Kepler called the ratio 1.618 “one of the two jewels of geometry” expressing it as a positive root of x2 = x+1. Thus, the Golden Section satisfies the relationships ϕ2 = ϕ + 1 = 2.618, ϕ3 = ϕ2 + ϕ = 4.236 and so on. Thus it has a mathematical nature of equipartition (symmetry), succession (order) and continuous proportion (regularity), which can give rhythm to any art or physical form.