Lets look at circles with centers at points with integer coordinates and equal radii. When the radii are small, the circles will be disjoint. Something interesting first happens when the radius becomes 1/2, because then the circles touch.
When the radii grow, the circles will intersect, and interesting patterns emerge. These patterns change continuously,
but when a special intersection occurs, the complexity of the intersection pattern increases. The next special intersection after r=0.5 occurs at r=0.7071, when circles that are diagonally across touch, and then again at r=1.
Often, and due to the symmetry of things, whenever two of our circles touch, a second pair of circles must touch at the same point.
Then, at r=1.17851, we have true intersections of three circles at a single point (no touching!).
Mathematicians find this interesting because the special intersections (touch or triple cross) mark singular points in the space of all such circle configurations. Understanding them means understanding the whole space.
It is of course very satisfying that these singularities are also esthetically pleasing, as if they knew they are special and have dressed up for the occasion.