This Curve Looks Simple At First, But It Gets Complicated Quickly

Calculus Level 3

Consider the above curve as the number of steps tends to infinity. Which of the following is true:

The curve does not go through the point

(\frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{3})

The curve does not go through the point

( \frac{1}{3},\frac{2}{3})

The curve does not go through the point

(\frac{1}{2},\frac{1}{2})

The curve goes through every point in the unit square