scaling
Think of how a coastline looks roughly the same whether you view it from a plane or crouch down to a few meters of shore: the same kind of jaggedness appears at every zoom level. Near a critical point matter behaves much like that — there is no single special size, and the same patterns repeat across scales. Scaling is the mathematical language for this no-favorite-size behavior.
Concretely, scaling means that near a critical point the physical quantities follow power laws — relationships where doubling one quantity multiplies another by a fixed factor, with no characteristic scale built in. Stretch or shrink your ruler and the laws keep their shape, only the units changing. This is why a single number, a critical exponent, can describe a whole quantity's behavior, and why far-flung quantities turn out to be linked by tidy scaling relations among those exponents.
Scaling matters because it is the precise expression of the self-similarity that reigns at criticality, and the foundation on which universality and the renormalization group are built. A subtle point: true scaling is an idealization that holds exactly only in an infinitely large system precisely at the critical point. In any real, finite sample it is an excellent approximation over a wide window, gently cut off when you get extremely close to the transition.
Right at the critical point, a magnet looks the same statistically at every magnification: zoom in on a patch of aligned spins and you find smaller patches inside it, with smaller patches inside those, like nesting dolls of order and disorder. No single patch size stands out — the very meaning of scaling.
At criticality a magnet is self-similar — patches of order within patches, with no special size.
Away from a critical point, systems do have a favorite size — the correlation length — beyond which different regions act independently. Scaling reigns precisely because, as you approach criticality, that favorite size swells to infinity and stops singling out any scale.