A model of the growth curve of microorganisms was proposed, which reveals a relationship with the number of a ‘golden section’, 1.618[Formula: see text]…, for main parameters of the growth curves. The treatment mainly concerns the ratio of the maximum asymptotic value of biomass in the phase of slow growth to the real value of biomass accumulation at the end of exponential growth, which is equal to the square of the ‘golden section’, i.e., 2.618. There are a few relevant theorems to explain these facts. New, yet simpler, methods were considered for determining the model parameters based on hyperbolic functions. A comparison was made with one of the alternative models to demonstrate the advantage of the proposed model. The proposed model should be useful to apply at various stages of fermentation in scientific and industrial units. Further, the model could give a new impetus to the development of new mathematical knowledge regarding the algebra of the ‘golden section’ as a whole, as well as in connection with the introduction of a new equation at decomposing of any roots with any degrees for differences between constants and/or variables.

中文翻译：

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发酵中的 S 形生长曲线和黄金比例

提出了一种微生物生长曲线模型，该模型揭示了生长曲线主要参数与“黄金分割”的数量1.618[公式：见正文]...的关系。处理主要关注缓慢增长阶段生物量的最大渐近值与指数增长结束时生物量积累的实际值之比，等于“黄金段”的平方，即2.618。有一些相关的定理可以解释这些事实。考虑了新的但更简单的方法来确定基于双曲线函数的模型参数。与其中一种替代模型进行了比较，以证明所提出模型的优势。所提出的模型应该适用于科学和工业单位发酵的各个阶段。进一步，