Disorienting patterns of motion

Abstract

The spatial orientations of a biological organism can be identified with the elements of its rotation group. By physically realizing a group-invariant integration scheme described in this article, the growth of the organism can be made orientation independent. The integration was effected by means of a path in the group, ergodic with respect to the invariant probability distribution. The simply connected covering group, furnished with its invariant metric, is isometric to the hypersurface of a hypersphere in Euclidean four-space. The path was approximated by a mesh of points uniformly distributed on the hypersurface. Regular four-dimensional polytopes were used to construct meshes of arbitrary fineness, and graph theory was used to find the desired Hamiltonian paths. A magnetic tape was programmed to direct servo-controlled gimbals realizing the path as continuous motion of the organism.