WIT Press

Characteristic Equations With Solutions That Contain Nature’s Golden Ratio


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153 - 163




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Paper DOI



WIT Press


B. T. Rosson


It has been shown that certain characteristic equations of eigenproblems produce results in which the golden ratio is revealed. The characteristic equations that produce these results are developed for application to a broad range of physical systems. The resulting equations provide the background to understand why these results are obtained and the physical system requirements needed to produce them. Two physical systems that typically involve eigenproblems in structural mechanics are used to illustrate the use of the equations. The analysis of natural frequencies and principal stresses involve characteristic equations of the general form presented in the paper. Examples are presented that demonstrate how the equations can be used to determine the physical conditions that are necessary to produce golden ratio solutions, and also how they may be utilized to study the dimensions of the natural world. Keywords: golden ratio, characteristic equation, eigenproblem, Fibonacci series, frequency, principal stress, dimensions. 1 Introduction A few physical systems have been discovered that reveal the golden ratio in the solution of the equations that predict their natural behavior or response to external action. Since the vast majority of physical systems do not reveal such results, one seems left to discover these unique systems somewhat serendipitously. It is to be expected that systems with physical properties of golden proportion will reveal these results, but for physical systems that do not, finding the conditions to produce them are not so obvious. For instance, the structural system in Fig. 1 has no physical properties that contain the golden proportion, but Moorman and Goff [1] have shown that the system’s natural


golden ratio, characteristic equation, eigenproblem, Fibonacci series, frequency, principal stress, dimensions