# Geometry

Geometry is a branch of mathematics concerning shapes and their relationships to one another. Because of its immediate practical applications it was one of the first to be developed, and was likewise the first field to be put on an axiomatic basis, by Euclid. The next most significant development had to wait until a millennium later, and that was analytic geometry, in which points are represented by as ordered pairs or triples of numbers. This sort of representation has since then allowed us to construct new geometries other than the standard Euclidean version.

The central notion in geometry is that of congruence. In Euclidean geometry, two figures are said to be congruent if they are related by a series of reflections, rotations, and translations. For instance:

```  *                *                   *
*                *         *        * *
*****        *****         ***      ***
*                *         *
*                *
```

The first two figures are congruent to each other. The third is a different size, and so is similar but not congruent to the others; the fourth is different altogether. Note that congruences alter some properties, such as location and orientation, but leave others unchanged, like distances and angles. The latter sort of properties are called invariants and studying them is the essence of geometry.

Other geometries can be constructed by choosing a new underlying space to work with (Euclidean geometry uses Euclidean space, Rn) or by choosing a new group of transformations to work with (Euclidean geometry uses the special orthogonal transformations, SO(n)). In general, the more congruences we have, the fewer invariants there are. As an example, in affine geometry any linear transformation is allowed, and so the first three figures are all congruent; distances and angles are no longer invariants, but linearity is.

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