Orientation : from metaphysics to semantics

Less roughly: Two figures in a space1 S are are isometric iff there is a distance-preserving bijection between the points of one and the points of the other. It is common to talk as if we can ‘move’ a geometric figure around in a given space. That’s loose. Better to think of it like this: some regions of a given space are translations, rotations, or reflections of other regions of the space. There is a more general way of thinking about translations, rotations, and reflections. We can think of them as relations between possible ways of imposing a coordinate structure on a given space. This is made precise in linear algebra, but you can get a sense of the idea by thinking about a simple two-dimensional Euclidean plane space. Intuitively, • Translating a coordinization of the plane is like taking every coordinate and mapping it to a new coordinate at a fixed distance and direction; • Rotating it is like turning all the coordinates together around a fixed coordinate in the plane; • Reflecting it is like flipping the coordinization around a fixed line through the plane. More abstractly, we can think of translation, rotation, and reflection as possible symmetry properties of a space. Let’s take a figure to be a set of points in a metric space. We’ll restrict to metric spaces for the purposes of this handout, but little leans on this, and the discussion could be extended to manifolds. A metric space is a pair �S, μ� of a set S of points and a total function μ (the metric) from pairs of points x, y in S to nonnegative real numbers satisfying the following constraints: (1) μ(x, y) = 0 if x = y; (2) μ(x, y) = μ(y, x) (symmetry); (3) μ(x, y) + μ(y, z) ≥ μ(x, z) (triangle inequality).