Children learn the language of "spatial orientation" only through repeated experience. At around age two, they begin to understand words such as "up" and "down," "on" and "under." They grasp "in front" and "behind" at around three to five years old, whereas the more difficult notions of "right" and "left" are not acquired until around five to seven.
All these notions of orientation arise from broken symmetries in our everyday space, which single out certain directions. In mathematics, orientation was first associated with familiar geometric objects—the line, the plane and space—before being defined more rigorously for curves and surfaces.
The decisive sign
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Choosing an arbitrary origin O on a line, a one-dimensional mathematical object, divides it into two rays. Orienting the line means selecting one direction as preferred. Let denote a unit vector pointing in the chosen direction. The coordinate x of a point M on the line then satisfies
Positive coordinates correspond to the reference ray, and negative coordinates to the other ray. Reflection through the origin takes one ray to the other. Rotation is excluded because it takes place in the plane, outside the line.
