the self-similarity of Brownian motion
A fractal looks the same at every magnification — zoom into a coastline and the small bays mimic the big ones. Brownian motion is the most fundamental random fractal. If you zoom into any tiny piece of a Brownian path with the RIGHT aspect ratio, you get back a curve statistically identical to the whole thing. There is no 'natural scale': a Brownian path has the same jagged character whether you look at it over a microsecond or over a century. This scale-free roughness is its self-similarity.
The precise statement fixes the aspect ratio. For any positive number c, if you stretch the time axis by c and the space axis by sqrt(c), you recover a standard Brownian motion: the rescaled process (1/sqrt(c)) times B(ct), as a function of t, is again a standard Brownian motion. The mismatched exponents — time by c but space only by sqrt(c) — are the same square-root law that runs through everything Brownian: variance grows like time, so distance grows like the square root of time. To keep the picture statistically unchanged you must shrink space more gently than time, by the square root. This is why a Brownian path photographed at any zoom, with the axes scaled in this 1-to-square-root proportion, is indistinguishable from any other.
Self-similarity is more than a pretty picture: it is a computational lever. Because the law is invariant under this scaling, many quantities must be 'scale-covariant' — for instance the maximum of B over [0, t] must scale like sqrt(t), and the first hitting time of level a must scale like a^2. You can read off these exponents instantly from the scaling, without solving anything. Self-similarity also makes Brownian motion the canonical model whenever a phenomenon has no built-in length or time scale. The honest note: this exact self-similarity is special to driftless Brownian motion; add a drift term (a steady trend) and the symmetry is broken, because a constant drift does have a preferred scale.
Take a Brownian path over [0, 100] and zoom into the window [0, 1] — but also stretch its height by a factor of sqrt(100) = 10. The blown-up snippet has exactly the same statistical look as the original 100-second path. With c = 100, (1/10) B(100 t) is again a standard Brownian motion.
Stretch time by c and space by sqrt(c) and Brownian motion looks identical — a random fractal.
The scaling is 1-to-square-root (time by c, space by sqrt(c)), not 1-to-1. Adding a constant drift breaks this exact self-similarity, since a drift introduces a preferred scale.