Iterates of the sine function (blue), in the first half-period.     Half-iterate (orange), i.e., the sine's functional square root; the functional square root of that, the quarter-iterate (black) above it; and four integral iterates below it, starting with the second iterate (red). The green envelope triangle represents the limiting null iterate, the sawtooth function serving as the starting point leading to the sine function. From the general pedagogy web-site.