Euclidean geometry

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Euclidean geometry is the geometry described by Euclid in the Elements. Euclid based his work on five axioms, including as fifth the parallel postulate, stating that through a point not on a given straight line, one and only one line can be drawn that never meets the given line. The parallel postulate seems less obvious than the others and many geometers tried in vain to prove it from them.

In the 19th century, though, it was discovered that this could not be done by constructing non-euclidean geometries where the parallel postulate is false, and either an infinite number of parallel lines can be drawn through a point not on a straight line (Lobachevsky geometry, also called hyperbolic geometry), or none can (Riemannian geometry, also called parabolic geometry). That other geometries could be logically consistent was one of the most important discoveries in mathematics, with vast implications for science and philosophy. Indeed, Einstein's theory of general relativity shows that actual space can be non-Euclidean.

Another thing that was noticed at the time was that Euclid's five axioms are actually fairly incomplete. For instance one of his theorems is that any line segment is part of a triangle, which he constructs in the usual way, by drawing circles around both endpoints and taking their intersection and them as three corners. However, his axioms do not guarantee that the circles actually do intersect! David Hilbert gave a revised list containing no fewer than 23 separate axioms.

Nowadays Euclidean geometry is usually constructed rather than axiomatized, by means of analytic geometry. A rectangular coordinate system maps each point in Euclidean space with a unique list of n real numbers (x1,...,xn), so we can define it to be the set of all such lists (Rn). We also define our metric (distance function) d by

d(x,y)2 = (x1 - y1)2 + ... + (xn - yn)2

which you might recognise as an application of the Pythagorean Theorem. This turns Rn) into a metric space. Maps that preserve the distance between all pairs of points are called isometries, and include reflections, rotations, translations, and compositions thereof. In matrix notation any of these have the form

x' = Ax + b

where A is an orthogonal matrix and b is a column vector. Isometries are taken as the congruences of Euclidean geometry - that is, we only consider properties preserved by them. That way we do not have to worry about the precise origin or axes, but still consider distances, angles, and so forth.