Dimensional consistency and comparable scales

Hill’s proposal crosses enormous ranges of size, time and mechanism. That makes units a first-line diagnostic. Force cannot be added directly to energy, and a geometric length cannot determine an energy without additional parameters carrying the required dimensions. A spiral pitch angle is dimensionless; curve or principal curvature has units of inverse length, while Gaussian and spacetime curvature have units of inverse length squared; a frequency is an inverse time. Their meanings do not become interchangeable because all can be plotted. Buckingham’s dimensional-similarity work supplies a formal foundation for constructing dimensionless groups. The project should use that discipline before asking whether a shared visual motif signals a common law.

Check every term

Write each variable’s unit and reduce it to base dimensions. Every additive term in an equation must share dimensions. Arguments of exponentials, logarithms and trigonometric functions should be dimensionless, typically expressed as a ratio to a stated reference scale. If an equation relates curvature to energy, identify the coupling constant and its units rather than hiding it in a proportionality sign. Passing this check does not make a model true. It only removes a class of contradictions that would otherwise make numerical comparisons and parameter fitting meaningless.

Compare dimensionless groups

For an illustrative wave problem, frequency times length divided by propagation speed is dimensionless. Similar boundary shapes with similar conditions may then be compared without confusing metres with nanometres. The harder question is whether the governing equations share the same form and relevant dimensionless parameters. Atomic structure and a spiral galaxy involve different dominant physics; matching an image scale does not make them dynamically similar. The proposed cross-scale comparison must therefore list what is held fixed, what is rescaled and which physical terms become negligible or dominant.

Expose missing scales

A proposed law that works across every scale must explain characteristic lengths, times and energies already present in its domain. If a fitted exponent changes near a transition, that change can reveal a relevant scale rather than an analytical nuisance. Preserve these transitions in the dataset. Do not remove them simply to lengthen a straight segment on a log plot. For Alpha 1, an unspecified parent length or rotation time cannot generate a numerical prediction until its role in an observable equation is defined, constrained or explicitly marginalized over.

The mathematical lens

Turn it into a question.

Follow the original work.

These sources support the mechanisms and limits discussed here. They do not endorse Hill’s framework. Notes distinguish research proposals, reviews, and observations.

1. On Physically Similar Systems; Illustrations of the Use of Dimensional EquationsE. Buckingham · 1914 · Physical Review 4, 345–376Read original source ↗
2. The International System of Units (SI), 9th edition, updated 2026Bureau International des Poids et Mesures · 2026 · BIPM SI BrochureRead original source ↗