>[!abstract] >The Feigenbaum constant ($\delta \approx 4.6692$) is a mathematical constant that describes the limit of the ratio of the distances between successive bifurcation points in a period-doubling cascade that leads to chaos. It is a fundamental constant in chaos theory, indicative of a universal behavior found in various dynamical systems, such as the [[logistic map]]. Other maps also reproduce this ratio; in this sense, the Feigenbaum constant $\delta$ in bifurcation theory is analogous to $π$ in geometry and $e$ in calculus. >[!note] >What I find fascinating about the Feigenbaum constant is not only that it introduces regularity and predictability into chaos (specifically, into the doubling periods), but also universality, i.e., the underlying mathematical function itself is irrelevant. It is found in both quadratic formulas (such as $x_{t+1} = r(x_t - x_t^2)$) and trigonometric formulas (such as $x_{t+1} = r \sin \pi x_t$), for instance. >[!quote] >Imagine that a prehistoric zoologist decides that some things are heavier than other things —they have some abstract quality he calls weight— and he wants to investigate this idea scientifically. He has never actually measured weight, but he thinks he has some understanding of the idea. He looks at big snakes and little snakes, big bears and little bears, and he guesses that the weight of these animals might have some relationship to their size. He builds a scale and starts weighing snakes. To his astonishment, every snake weighs the same. To his consternation, every bear weighs the same, too. And to his further amazement, bears weigh the same as snakes. They all weigh 4.6692016090. Clearly weight is not what he supposed. The whole concept requires rethinking ([[Gleick, 1987]], p. 174). >[!related] >- **North** (upstream): [[Bifurcation theory]] >- **West** (similar): [[Logistic map]] >- **East** (different): — >- **South** (downstream): —