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Nothing in Nature is random … A thing appears random only through the incompleteness of our knowledge (Benedict Spinoza)
This is one of my favorite mathematical mysteries. In 1991 David Boll was trying to confirm that the neck of the Mandelbrot Set is 0 in thickness. Neck is located at -0.75+0i (where two biggest circles meet each other).
Before doing it, I thought I was going to find some pattern in the graphic. Apart from the mirror effect produced by the sign of ε, there is nothing recognizable. Convergence is chaotic. Here you have the code. This example is also nice to practice with ggplot2 package, one of the totems of R:
i<-0 # Counter of iterations x <- 0 # Initialization while (Mod(x) <= 2) { x <- x^2+(c+complex(real = 0, imaginary = e)) i <- i+1 } i } results <- as.data.frame(c(NULL,NULL)) for (j in 1:length(epsilons)) {results <- rbind(results, c(epsilons[j], testMSConvergence(epsilons[j])))} colnames(results) <- c('epsilon', 'iterations') dev.off() p <- ggplot(results, aes(epsilon,abs(epsilon)*iterations))+ xlab("epsilon")+ ylab("abs(epsilon)*iterations")+ opts(axis.title.x=theme_text(size=16)) + opts(axis.title.y=theme_text(size=16)) + ggtitle("How to Estimate Pi Using Mandelbrot Set's Neck")+ theme(plot.title = element_text(size=20, face="bold")) p <- p + geom_ribbon(data=results,aes(ymin=abs(epsilon)*iterations-abs(epsilon),ymax=abs(epsilon)*iterations+abs(epsilon)), alpha=0.3) p <- p + geom_abline(intercept = pi, , slope = 0, size = 0.4, linetype=2, colour = "black", alpha=0.8) p <- p + geom_line(colour = "dark blue", size = 1, linetype = 1) p <- p + geom_text(x = 0, y = pi, label="pi", vjust=2, colour="dark blue") p <- p + geom_point(x = 0, y = pi, size = 6, colour="dark blue") p + geom_point(x = 0, y = pi, size = 4, colour="white")
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