Golden ratio and bond-valence parameters of hydrogen bonds of hydrated borates

Structural Chemistry of Akdalaite, Al10O14(OH)2, the Isostructural Aluminum Analogue of Ferrihydrite

“…H bonds (Table A8). Previous studies found that hydrogen bonding is unlike typical cation-anion interactions and the hydrogen bonding interactions have to be treated differently [38][39][40]. The aluminum sites show signs of over-and underbonding as the BVS of the tetrahedral site (Al3) is underbonded at 2.71 Å, below the ideal value of 3 for aluminum ( Figure 6) but conforming to the result reported for the MM model (Figure 1).…”

Phidias numbers as a basis for Fibonacci analogues

“…Relations (9) express the main properties of the golden section. This task was developed in many works, the results of which are systematized, for example, in monographs [4,8,18] et al The golden numbers Ф and  are currently used in modern research, quantum [21] and spin [1] physics, in new field of quasicrystals [14], fractal geometry [6], algebra of numbers [5,16,20]. The American mathematician G. Bergman [2] built a system of calculus, in which the number of Phidias plays the role of the basis.…”

Morphic Angles

“…with k and l 2 N. While the golden ratio is already known to mankind since antiquity and still is a subject of study in art, architecture (Xu et al 2012;Huylebrouck and Labarque 2002) and various scientific domains such as mathematics, physics, biology and chemistry (Gonzlez 2010;Tung 2007;Yu et al 2006;Swartzendruber et al 1985;Boeyens and Thackeray 2014), the plastic number w was first defined and studied mathematically in 1924 by the engineer Gerard Codonnier. The first publication about this number, however, dates from 1960.…”