A metre stick is not a metre long to everyone, says relativity
Length seems like the most solid quantity there is, the thing a ruler measures. Yet Einstein's special relativity says a rod one metre long in its own frame measures differently for an observer moving past it, and mathematicians have stretched the idea to curved spaces, networks and strange sets of points.
In everyday use, length usually means the longest dimension of an object, though that can change with how the object is placed. Height is vertical extent measured up from a base, width and breadth normally name a shorter side, and depth covers the third direction. A plank lying down has length, height and depth; stand a thin rectangle on its short edge and its area becomes height times width rather than length times width. Length covers one dimension, area two and volume three, which is why they are counted in units, squares and cubes.
Most measurement systems pick one base unit of length and derive the rest from it, and in SI that unit is the metre. The need for standards grew once people stopped roaming, began building and dividing land, and traded with neighbours, and it sharpened as commerce spread. Modern technology demands far finer precision, from microelectronics to measuring distances between planets.
Geometry offers several versions. Euclid's length runs along straight segments, underpinning results such as Pythagoras's theorem on right triangles, while arclength follows a curve. A triangle's height is the length of an altitude, the perpendicular dropped from a corner to the opposite side. Perimeter adds up a polygon's sides, and circumference is the length of a disk's boundary. On a sphere, distance is measured along a great circle, taking the shorter arc cut by the plane through both points and the centre, and the Riemannian geometry of general relativity uses curved paths called geodesics.
Beyond geometry the word takes on new meanings. In a graph, a path's length is the number of edges it crosses, or the total of their weights if edges carry weights, and it defines shortest paths, longest paths and girth, the length of the shortest cycle. Measure theory generalises length to arbitrary sets of real numbers through the Lebesgue measure, starting from the lengths of open intervals.
Source: Length
