Curious natural patterns on the surface of basalt blocks that make up the sidewalk
These are most likely manganese dendrites. It is not a fossil, and not organic. These usually form in cracks in rocks, and most likely this slab was broken along an existing crack.

You can read more about it now: https://www.mindat.org/min-26645.html

One comment: this is not basalt. Basalt is black. Most likely some form of limestone.