Today I was completing my self-study through spivak calculus book, and I see this beautiful curve:
I tried to answer one of the questions mentioned by the writer, which is to find the linear function corresponding to the curve in an interval. Then I noticed something (I claim that it is a good thing): this curve consists of triangles and if we drew a base for those triangles and then a semicircle above it, after we project this semicircle on the horizontal axis, we can see a nice figure (I drew it on paper, but I think images are not allowed on this site, so I cannot publish it here, anyway it's easy to imagine it) we can see the diameter of the first semicircle starts on point $1$ and finishes on point $\frac{1}{3}$, and the diameter of the second semicircle starts on point $\frac{1}{2}$ and finishes on point $\frac{1}{6}$, and so on. (note that those semicircles do not have a similar diameter and each of the semicircles start on the half of the diameter of the previous semicircle)
Then I wondered if we could find a curve and if we applied to it the same thing we did for the previous curve, would we get semicircles of equal diameter? It prompted me to the next question; Is it possible to find linear functions which by their curve we can get a curve similar to the sin function curve ?
(the functions of above curve it that:
$f(\frac{1}{n})=(-1)^{n+1}$,
$f(-\frac{1}{n})=(-1)^{n+1}$,
$f(x)=1$ for $|x|\geq 1$