Please read Notes and Credits before clicking the green flag. Logistic Zoom https://www.youtube.com/watch?v=PtfPDfoF-iY
This project is one in a series of projects designed to build understanding of the concept of a strange attractor. In his book 'Chaos - The Making of a New Science', James Gleick devotes the Prologue and the 30 pages of the Universality chapter to Mitchell Feigenbaum, a physicist turned mathematician. He dug deep and found gold in a quadratic equation from Algebra II, y = Rx(1 – x). This equation was also known to biologists as a population model for the rise and fall of a fish population in a pond (see my project An Abstract Fish Population Model). The biologists did not know how to interpret the results when R > 3.5. https://scratch.mit.edu/projects/195253979/ The first bifurcation, R1, occurs at R = 3.00. The second bifurcation, R2, occurs at R =3.45. The third bifurcation, R3, occurs at 3.546. Note: Professor Feigenbaum used a computer with precision number routines to identify the bifurcation points with much greater accuracy than what's shown in this project. The point of the project is to show HOW Feigenbaum arrived at his constant. ********************************************************************* If you would like to compute these bifurcation points use my project: Chaos Lurks in y = Rx(1 – x) at https://scratch.mit.edu/projects/11173887/ ********************************************************************* Compute (R2 – R1)/(R3 – R2) to get 4.6875 as an approximation to Feigenbaum's constant, 4.669201609… This ratio predicts the NEXT bifurcation point. ********************************************************************* The iterated map of y = RX(1 – x) has an infinite amount of detail because there are an infinite number of Real numbers in the unit interval [0, 1]. Like the Mandelbrot set, one can zoom through it forever seeing new structure the deeper one goes.
