Symmetry of perception-action: extending Turvey's Ecological Foundations with Continuous Symmetry Groups

Perception and action are unified by invariants in energy arrays that directly specify (are information about) lawful agent-environment relations or affordances. Invariants are defined by symmetry: relational structures that persist across transformations. Building on this ecological foundation that Turvey championed, we propose a framework for perception-action that uses continuous symmetry groups to formalize how energy arrays-optic, acoustic, and haptic-are structured via action. Using the special Euclidean group SE(3) as a working exemplar, we demonstrate how the six generators of this group directly map invariants (symmetries) in ambient energy arrays to the motor-generator subspaces that actualize affordances. We do this across canonical cases of locomotion, catching, climbing, and grasping. By situating Ecological Psychology within the scientific foundations of symmetry and continuous symmetry groups, this framework potentially clarifies and extends Turvey's legacy, providing a formal and generalizable language for the direct theory of perception-action.