Assessments of bilateral asymmetry with application in human skull analysis.

As a common feature, bilateral symmetry of biological forms is ubiquitous, but in fact rarely exact. In a setting of analytic geometry, bilateral symmetry is defined with respect to a point, line or plane, and the well-known notions of fluctuating asymmetry, directional asymmetry and antisymmetry are recast. A meticulous scheme for asymmetry assessments is proposed and explicit solutions to them are derived. An investigation into observational errors of points representing the geometric structure of an object offers a baseline reference for asymmetry assessment of the object. The proposed assessments are applicable to individual, part or all point pairs at both individual and collective levels. The exact relationship between the developed treatments and the widely used Procrustes method in asymmetry assessment is examined. An application of the proposed assessments to a large collection of human skull data in the form of 3D landmark coordinates finds: (a) asymmetry of most skulls is not fluctuating, but directional if measured about a plane fitted to shared landmarks or side landmarks for balancing; (b) asymmetry becomes completely fluctuating if one side of a skull could be slightly rotated and translated with respect to the other side; (c) female skulls are more asymmetric than male skulls. The methodology developed in this study is rigorous and transparent, and lays an analytical base for investigation of structural symmetries and asymmetries in a wide range of biological and medical applications.