Circle to Square

I took many square roots of a semi-circle and it became a half-square. Pretty neat.

library(reshape2)
library(ggplot2)

by <- .01
x <- seq(-1, 1, by=by)

n <- 2^(0:10)

f <- function(x, r=1) {
    sqrt((r-x)*(r+x))
}

nRoot <- function(n, x=seq(-1, 1, by=.01)) {
    f(x) ^ (1/n)
}

data <- sapply(n, nRoot)
colnames(data) <- n
rownames(data) <- x
# data <- rbind(data, -data)

data <- melt(data, id="x")
names(data) <- c("x", "n", "y")
# data$n <- as.factor(n)

ggplot(data, aes(x, y, color=factor(n))) + geom_line() +
    ggtitle("Half-circle becoming a half-square as I took many square roots") +
    guides(color=guide_legend(title="# roots")) +
    theme_bw() +
    theme(legend.position="bottom")