The Golden Ratio in the Creations of Nature Arises in the Architecture of Atoms and Ions

The Golden ratio, ϕ = a/b = (a + b)/a, where a and b are the Golden sections of their sum, has long been known to operate in a variety of the creations of Nature, ranging from the mollusks in the oceans to the spiral galaxies in the Universe. Recently, while researching for the exact values of ionic radii and for the significance of the ionization potential (I_H = e/2κa_B ) of hydrogen, it was found that a_B, the Bohr radius has two Golden sections pertaining to the electron and proton. Further, ϕ was also found to be the ratio of the anionic to cationic radii of an atom (A), their sum being the covalent bond length, d(AA). With these radii many bond lengths were shown to be sums of the covalent and or ionic radii, whether partially or fully ionic or covalent. For example, the crystal ionic distances of all alkali halides were shown to be sums of the ϕ-based ionic radii. When the ion-water distances were plotted against the above ϕ-based ionic radii, linear relations were found. This enabled the assignment of exact ionic radii in water, and to show that the hydration bond lengths of O (oxygen coordination bond) with cations and of H (hydrogen bond) with anions are constants. Simple integral multiples of the cationic radius, d(H+) = d(HH)/ϕ2, and the covalent radius of H, were found to quantitatively account for the lengths of the hydrogen bonds in many inorganic and biochemical groups.