| Popis: |
There are infinitely many ways in which an object might rigidly move between any two positions in space. Experiments on real, apparent, and imagined motion indicate, however, that humans favor particular connecting motions. The favored motions may reflect internalizations of principles governing the transformations of objects in the external world. This first of two articles (Part I) offers formalizations and preliminary evaluations of two primary candidates for principles that may have been internalized: (a) laws of physics—especially the principle of least action, according to which, in the absence of external forces, the center of a rigid body traverses a straight line between any two positions while any rotation is about its rectilinearly translating center—and (b) principles of kinematic geometry—especially Chasles's theorem, according to which, for any two positions of an asymmetric object, there exists a unique axis in space such that the object can be carried from the one position to the other by a helical combination of a translation along and rotation about that axis. A one-parameter family of cases intermediate between the pure cases of physics and geometry is also considered. Each case is abstractly formulated in terms of the straightest transformational path (geodesic) that it prescribes in the six-dimensional manifold of possible positions of the object in space. The corresponding motions are concretely illustrated for asymmetric objects in two- and in three-dimensional space. Characterization of the additional geodesics permitted by objects possessing various symmetries is deferred to a second article (Part II). The formalization of internalized principles and associated geodesics may eventually lead to a mental mechanics governing how humans and other animals interpret events and plan and carry out actions in the world. |
