ABSTRACT
Traditional allometry measures scaling coefficients of laws using invariant (Euclidean) coordinate spaces as 'objective' frames of reference. The method proposed here measures coefficients of the coordinate space using invariant laws as 'objective' frames of reference. This strategy follows directly from the invariance postulate of natural law: Physical laws (or universal constants) constitute invariant standards for measuring objects and events. By postulating a physical law as an 'objective' frame, an allometric plot of the form X = ai(Physical law)ci can be used to derive metrics of the coordinate space that embeds the law. The metrical structure of the coordinate space is defined by the deviations of ai and ci from their canonical magnitudes. These indices are physically interpreted in terms of potentials.
Through the use of similarity analysis, dimensional analysis, and allometry a physical (scaling) law is identified relating periodic time to the (uncorrelated) mass and length of a wrist-pendulum system, namely, Ƭo = ai(M1/16L1/2)ci. This law of wrist-pendulum systems results from a nesting of transformations—two in scale and one in coordinate space—of a law that relates biological times to biological masses and lengths. Subjects are viewed as nonuniform coordinate spaces for the twice-scaled law; they differ in the strengths of certain scalar and vector 206 potentials indexed by ai and ci, respectively. Abiding by the invariance postulate of natural law, departures of Ƭo from the canonical form, (M1/16L1/2)1.0, are attributed to the properties of the coordinate space. These properties are measurable by casting the paired values (Ƭo, M1/16L1/2) into double logarithmic coordinates. (For example, the degree of curvature of a coordinate system, ci, is measured by the slope in a log by log plot.) Once measured, the contribution of these properties can be eliminated—through a renormalization of the coordinate space—and the invariance of the timing law thereby revealed.
The data of the experiment confirm Ƭ α M1/16L1/2, that is, the theoretically derived form of the basic scaling law subject to the mechanical constraints wrought by scale transformations of the type of biological system (of which the subject is a token) and the type of biological function (that the subject simulates). The law is also confirmed by two subsidiary experiments, one that directly manipulates one or more potentials of a coordinate space and one that compares in-phase with out-of-phase coordinations. These subsidiary experiments were motivated in part by the issue of whether or not there is a lawful basis for relating locomotory periods to animal size that holds invariant over changes in speed and gait.
