◄ WORLD V · SONNY 5DART 403 · a helldive at the net

SCALE INVARIANCE the same at every size — f(λx) = λⁿ f(x), the symmetry of no yardstick

A thing is scale-invariant when zooming changes nothing but the units: f(λx) = λk f(x) (a homogeneous function of degree k). Area scales as λ², volume as λ³; a fractal looks the same at every magnification; a power law xk has no natural scale. It is the symmetry of having no yardstick — no special size — and it governs critical phenomena (at a phase transition, fluctuations of every size appear and the system becomes scale-free) and the renormalization group that tamed them. Euler's homogeneous-function theorem ties the degree to the derivatives: x·f′(x) = k·f(x).

THE TECHNIQUE f(λx) = λk f(x) — homogeneous of degree k; no natural scale

The demo checks that f(x)=x² is degree-2 homogeneous: scaling the input by k scales the output by k²: live demo







  
HISTORY & CREDIT Euler · the renormalization group

  
“Everything has a characteristic size.” — scale-invariant systems do not: fractals, power laws, and critical points look the same at every zoom. The symmetry of no yardstick. cited

  
the relation · f(λx)=λkf(x) — homogeneous of degree k.
Euler · the homogeneous-function theorem: x·∇f = k·f.
criticality · at a phase transition the system is scale-free; the renormalization group (Wilson, Nobel 1982) exploits it.

  
The same at every magnification — no special size, only a power of the zoom. The symmetry of the scale-free. Euler / RG






  
RECOMMEND FOR I-13 degree-2 homogeneity, on the compiler

  
On the canonical compiler, f(x)=x²: scaling x=3 by k=2 gives f(6)=36 = 4·f(3)=4·9 — degree-2 homogeneous:

  
$ i13 run sy_scaleinvariance.i13   # f(kx) == k^2 f(x) ?
RUN OK · 24 step(s) · peak stack 3 · call depth 0
fx = 9   fkx = 36
homogeneous_deg2 = 1     -- f(2x) = 4 f(x) : no natural scale

  
Recommend: scale invariance is the symmetry of no yardstick — the same at every zoom, up to a power — and i13 confirms x² is degree-2 homogeneous. Not a keeper (homogeneity is a property a function has, witnessed; and it rhymes with the self-similarity B43's attractor already touched). The dart that names why fractals, power laws, and critical points all look alike under magnification: none of them owns a scale.