Universal Natural Intelligence

This is the document. Rendered from the repository at the commit above, with nothing rewritten for the web. A gate re-renders it on every deploy and fails the build if a single byte differs.

RENDERED FROM STRUCTURED DATA — this page is not a markdown file. Source: gpt/knowledge/K20-constants-ratios-and-nature-ledger.json in uni-cookbook at 575fc93d9d31. The sha256 recorded for this page is the digest of that JSON, not of this text. The renderer is generators/derive_docs.cjs; it copies fields and adds no facts. Every value below is a field of the source, verbatim, including the ones that read NOT RUN and PENDING.

The ratios that carry meaning without units.

23 rows.

name	symbol	formula	meaning	critical values	source
Reynolds number	Re	Re = ρuL/μ = uL/ν	ratio of inertial to viscous forces	Pipe transition: Re_c = 2040 ± 10 for the onset of SUSTAINED turbulence (Avila et al. 2011), where mean puff-decay time equals mean puff-splitting time; the textbook ~2300 (range ~2000–4000) answers a different question. Scope: smooth circular pipe only — it does not transfer to a wing or an artery.	Purcell (1977) Am J Phys 45:3–11; Avila et al. (2011) Science 333:192–196
Péclet number	Pe	Pe = uL/D	ratio of advection to diffusion	Pe ≪ 1 → stirring is useless. Crossover: advection beats diffusion beyond L* = D/u (~30 µm for typical D and bacterial v).	Purcell (1977) Am J Phys 45:3–11
Damköhler number	Da	Da_I = k·τ (τ = residence time); Da_II = kL²/D	ratio of reaction rate to transport rate	Da ≪ 1 → reaction-limited (stirring does nothing). Da ≫ 1 → transport-limited (the problem is delivery, not catalysis). Da ≈ 1 → both matter; model, don't estimate.	NA-07 (standard chemical-engineering definition)
Froude number	Fr	Fr = v²/(gL) — some authors use Fr = v/√(gL); one is the square of the other. A Froude number without its convention stated is defective.	ratio of inertial to gravitational forces; L = hip height (gait) or waterline length (hulls)	Walk→run transition at Fr ≈ 0.5 — scoped to ADULT HUMANS AT 1 g. It FAILED off-Earth: in actual lunar gravity the measured transition was Fr = 1.39 ± 0.45, with 6 of 8 subjects choosing Fr > 1.0 (De Witt et al. 2014). The inverted-pendulum mechanical ceiling is Fr = 1.	Alexander (1976) Nature 261:129–130; Alexander (1983) J. Zool.; De Witt et al. (2014) JEB 217:3200–3203; Prescott et al. (2025) Biol Lett 21:20250191
Strouhal number	St	St = fA/U	ratio of oscillatory to forward speed; f = beat frequency, A = peak-to-peak amplitude, U = cruise speed	0.2 < St < 0.4 AT CRUISE — the band of high propulsive efficiency (Taylor, Nudds & Thomas 2003). Scope is the science: not takeoff, not manoeuvre, not hovering. Individual scatter is large — only 44% of 248 odontocete values fall in 0.225–0.275 (Rohr & Fish 2004). Eloy (2012) models the OPTIMAL St as size-dependent, rising 0.15 → 0.8 from the largest cetaceans to the smallest tadpoles.	Taylor, Nudds & Thomas (2003) Nature 425:707–711; Bush & Hu (2006) ARFM 38:339–369; Rohr & Fish (2004) JEB 207:1633–1642; Eloy (2012) J Fluids Struct 30:205–218
Womersley number	α	α = R√(ωρ/μ), ω = 2πf	ratio of transient (oscillatory) inertia to viscous shear	α ≫ 1 (aorta ≈13.2, some studies ≈20.3): inertia-dominated, blunt plug-like profile lagging the pressure gradient. α ≪ 1 (capillary ≈0.005; all microcirculation < 1): quasi-steady, Poiseuille-like, parabolic and in phase — the capillary does not know the heart is beating.	Cardiovasc. Eng. Technol. (2024) doi 10.1007/s13239-024-00723-4
Weber number	We	We = ρU²w/σ	ratio of inertia to surface tension; σ = surface tension (air–water ≈ 0.07 N/m), w = leg width	We < 1 → the driving leg's meniscus survives — satisfied by all water-walking insects APART FROM the galloping fisher spider (carry the exception, it is real). Bush & Hu report the fit Bo ∼ We across water walkers.	Bush & Hu (2006) ARFM 38:339–369; Hu, Chan & Bush (2003) Nature 424:663–666
Bond (Eötvös) number	Bo	Bo = ρghw/σ	ratio of gravity to surface tension	Bo < 1 → supported by surface tension. The capillary length is the crossover: ℓ_c = (σ/ρg)^(1/2) ≈ 2.6 mm (CN-02 recomputes 2.71 mm at 25 °C). Below it, surface tension dominates gravity; above it, gravity wins.	Bush & Hu (2006) ARFM 38:339–369
Knudsen number	Kn	Kn = λ/L; λ ≈ 68 nm for air at 1 atm, 25 °C	ratio of molecular mean free path to system size — where 'fluid' stops being a fluid	Gas flows: continuum Kn < 0.01 (Navier–Stokes, no-slip); slip 0.01–0.1 (N–S usable IF slip boundary conditions are added); transition 0.1–10 (kinetic theory required); free-molecular > 10 (molecules collide only with walls).	standard rarefied-gas references; standard kinetic theory
Deborah number	De	De = t_relax / t_observe	ratio of the material's relaxation time to your observation time	De ≫ 1 → BEHAVES AS a solid. De ≪ 1 → BEHAVES AS a liquid. Not 'appears to' — behaves as.	Reiner (1964) Physics Today 17(1):62, doi 10.1063/1.3051374; mantle viscosity quoted in Purcell (1977)
Prandtl number	Pr	Pr = ν/α	ratio of momentum to thermal diffusivity	Pr ≪ 1 → heat outruns momentum; thermal boundary layer THICKER than the velocity layer.	standard property tables
Schmidt number	Sc	Sc = ν/D	ratio of momentum to mass diffusivity — Prandtl's mass-transfer twin	Sc ≫ 1 in liquids: momentum spreads far faster than solute.	NA-07 'The rest of the kit'
Nusselt number	Nu	Nu = hL/k_fluid	ratio of total to conductive heat transfer	Nu = 1 → pure conduction, no convective benefit.	NA-07 'The rest of the kit'
Sherwood number	Sh	Sh = k_c L/D	ratio of total to diffusive mass transfer — Nusselt's twin	Sphere in stagnant fluid: Sh = 2 — the diffusion-limited floor that Purcell's 4πaND expresses.	NA-07 'The rest of the kit'
Grashof number	Gr	Gr = gβΔT L³/ν²	ratio of buoyancy to viscous forces — free convection's Reynolds number	Gr/Re² ≫ 1 → buoyancy dominates forced flow.	NA-07 'The rest of the kit'
Rayleigh number	Ra	Ra = Gr·Pr = gβΔT L³/(να)	ratio of buoyancy to diffusive damping	Ra_c = 1707.762 (rigid–rigid boundaries), critical wavenumber ≈ 3.117, INDEPENDENT OF Pr AT ONSET. Below it, no convection; above it, Bénard cells.	Rayleigh–Bénard linear stability
Biot number	Bi	Bi = hL/k_solid	ratio of internal conductive to surface resistance	Bi < 0.1 → lumped-capacitance admissible. MODELED: an engineering convention, not a law.	standard heat-transfer texts
Mach number	Ma	Ma = u/c	ratio of flow speed to sound speed	Ma < 0.3 → density change <~5%, incompressible admissible (a convention). Ma = 1 → shocks.	standard gas dynamics
Rossby number	Ro	Ro = U/(fL), f = 2Ω sin φ; Ω = 7.2921 × 10⁻⁵ rad/s, f(45°) = 1.031 × 10⁻⁴ s⁻¹	ratio of inertial to Coriolis forces — whether rotation matters at all	Ro ≈ 0.1 → rotation DOMINATES → geostrophic balance, which is the whole reason weather maps work: pressure-gradient and Coriolis forces nearly balance, so wind blows ALONG isobars rather than across them. Ro ≫ 1 → rotation irrelevant. Inside the Hadley cells the vorticity-based LOCAL Rossby number is order 1 — a different definition and a different regime.	CN-03; Shapiro (1962) Nature 196(4859):1080–1081
Golden angle (phyllotactic divergence)	ψ / θ_φ / α_golden	ψ = 360/φ² = 137.507764050038…°, φ = (1+√5)/2	dimensionless divergence angle; the attractor of repulsion-driven sequential primordia	Clusters near 137.5° in spiral-phyllotactic vascular plants — but NOT universal: in the largest sunflower study with explicit inclusion criteria, 18% of scored counts (136/768) were NON-Fibonacci (Swinton & Ochu 2016). Scope: the attractor at LOW control parameter G — the same experiment yields OTHER angles at other G.	Douady & Couder (1992) Phys. Rev. Lett. 68(13):2098–2101, DOI 10.1103/PhysRevLett.68.2098; Swinton & Ochu (2016) R. Soc. Open Sci. 3:160091
Drag anisotropy ratio (resistive-force theory)	ζ⊥/ζ∥	ζ⊥/ζ∥ = 2(ln(2λ/a) − 0.5)/(ln(2λ/a) + 0.5)	ratio of normal to tangential drag on a slender filament — the asymmetry that makes flagellar thrust possible at all	≈2 ASYMPTOTIC ONLY; ~1.5–1.8 at realistic aspect ratios (ln(2λ/a) ≈ 3–5). Strictly < 2 for any real filament; reaches 2 only as aspect ratio → ∞. Set the ratio to 1 and thrust integrates to ZERO — travelling wave or not.	Gray & Hancock 1955, J Exp Biol 32:802–814
Buckingham Pi theorem	Π	Number of groups = n − j, where j = the RANK of the dimensional matrix (NOT the number of base dimensions — that is the common failure mode, and it is wrong whenever the dimensions are not independently represented). Then Π₁ = f(Π₂, …, Π_{n−j}).	the recipe for generating your OWN dimensionless groups	Worked example — drag on a sphere: variables F, ρ, U, D, μ; n = 5, rank j = 3 → 2 groups. Π₁ = F/(ρU²D²) → drag coefficient C_d; Π₂ = μ/(ρUD) → 1/Re. Result: C_d = f(Re). Five variables collapse to one curve.	Buckingham (1914) Physical Review 4(4):345–376, doi 10.1103/PhysRev.4.345 (not first: Vaschy 1892; Federman and Riabouchinsky 1911 have priority)
Dynamic similarity and the conflict you cannot escape	Fr vs Re	Fr matching requires v_m = v_s·√λ (model goes SLOWER); Re matching requires v_m = v_s/λ (model goes FASTER)	geometrically similar systems sharing every relevant group have identical dimensionless results — but you can rarely match all groups at once	At scale λ = 1/25: v_m = 0.2·v_s versus v_m = 25·v_s. INCOMPATIBLE BY A FACTOR OF 125. No cleverness removes this.	NA-07 'Dynamic similarity — and the conflict you cannot escape'

sha256 e443dca15319ea0b — of the original file, so what was ingested stays checkable.

Dimensionless numbers

How to read this page

Precise — the source document