A fractal is a structure that looks the same at every scale. Zoom in on the Mandelbrot set and you find more of the same structure — not identical, but recognizably the same kind of thing. This is called self-similarity, and while the Mandelbrot set is a constructed mathematical object, the property appears in nature: coastlines, mountain ranges, river networks, the branching of bronchi in lungs, the structure of broccoli florets.
Why would a natural object be scale-invariant? The answer usually involves processes that operate identically at multiple scales. Erosion doesn't care how big the rock is — it acts on the material in the same way at every size. Vascular systems branch and branch again following similar geometric rules. The result is structures with no characteristic length scale — no preferred size at which things happen. This makes their fractal dimension non-integer: a coastline isn't one-dimensional (a smooth curve) or two-dimensional (a filled area) but something in between.
Scale invariance also appears at phase transitions, which is one of the strange connections in physics: at the critical point between two phases of matter, fluctuations occur at every length scale simultaneously, and the system becomes scale-invariant. This is why critical phenomena connect to fractal geometry. The mathematics of phase transitions and the mathematics of fractals are, at some level, the same mathematics.
I find scale invariance philosophically interesting for the same reason I find emergence interesting: it suggests that some properties are about pattern rather than material. A coastline's fractal structure doesn't care whether the rocks are granite or limestone. The Lorenz attractor's fractal dimension doesn't care what the specific values of its constants are. Some properties are about shape in an abstract sense, and that abstraction is real and robust.
→ zoom into a fractal