Parabolic patterns appearing in the graph of $ \{x^2\} $.

Question

I draw the graph of $ \{x^2\} $ in desmos; where $ \{\space.\} $ represents fractional part function. Here is what I saw,

I kept the thickness and opacity of lines 1 in desmos.

Now, I know that the lines are bent so they appear thicker at some points in the screen but why do they form a particular pattern which looks similar to its graph? The pattern seems to be continuous when you go away from origin. The lines form an acute angle with the positive x-axis but in the following picture, the pattern is forming an obtuse angle.

How can I explain the formation of these patterns?

Answer 1

This is an artifact of a pixel based display. Let’s say that there are $R$ pixels per unit along the $x$-axis. Then working with pixel offset $p$ we have $x=p R^{-1}$ and in pixels the function becomes $p \mapsto p^2 R^{-2} \pmod 1$. Now observe that for $p_0 = \tfrac12 R^2$ $$(p+p_0)^2 R^{-2} = p^2R^{-2} + p + \tfrac14 R^2 = p^2 R^{-2} + \tfrac14 R^2 \pmod 1.$$ So the function is “periodic” with period $p_0$ upto a vertical offset $\tfrac14 R^2 \pmod 1$. In original units ($x=pR^{-1}$) this period is $x_0 = \tfrac12 R$.

Your picture shows a period that is close to $x=19$, which would imply that $R \approx 38$, so about $38$ pixels per unit. It is a bit difficult to count, but it seems about right (zoom in far enough to count individual pixels in the $x$ direction).