Generalized golden mean and the efficiency of thermal machines

We investigate generic heat engines and refrigerators operating between two heat reservoirs, for the condition when their efficiencies are equal to each other. It is shown that the corresponding value of efficiency is given as the inverse of the generalized golden mean, ${\varphi }_p$, where the parameter $p$ depends on the degree of irreversibility of both engine and refrigerator. The reversible case ($p=1$) yields the efficiency in terms of the standard golden mean. We also extend the analysis to a three-heat-resrervoir setup.