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Modern geometry is expressed with group theory.

Let X be a set of points and S a set of subsets of X. For example, s in S may represent a line or a circle or some other characteristic feature of X. Consider A a set of axioms about X and S. Finally, let P be a proposition expressing a feature of elements of S.

Suppose b is a bijection of X with itself. The proposition Pb is obtained from P by exchanging all mentions of elements of S in P by their images under b. Now consider the set of all bijections that respect the property represent by proposition P:

If c is in G, then so that bc is in G. Also the inverse of b is in G, so G is a group.

Definition: The geometry is the transformation group determined by property P.

First consider the plane X = R2 with the property P given by the distance between two points:

Distance is invariant under rotation, translation, or inversion in a line. Therefore G is the group generated by these transformations: the Euclidean group of the plane.

In space X = R3, the distance between a pair of points represents P and generates the corresponding Euclidean space group. Consider a screw displacement given by a rotation about an axis and a translation parallel to that axis. According to a theorem in kinematics (attributed to Mozzi and Chasles), any motion in the Euclidean space group may be represented as a screw displacement.

Affine geometry[edit]

Returning to the plane X = R2, let P be the property of parallel lines. Thus b is in G when two parallel lines are taken by b to another pair of parallels. Then G is the affine group, which contains the Euclidean group but also includes squeeze mappings that transform a square to a rectangle of the same area as the square. This group has found application in flat-space cosmology where light rays trace lines through spacetime. In fact, a squeeze mapping in the affine group corresponds to a leap from a frame of reference determined by one velocity to one with another velocity.

The conversion of geometry, from properties of configurations of points, lines and other features of geometric space, to group theory, was accomplished by Felix Klein. He was put on the path to this conversion by Arthur Cayley's piece "On the theory of distance" (1859), which obtained a metric space known as the elliptic plane from the real projective plane by use of logarithm of crossratio. Klein ran with the idea, demonstrating the models for the hyperbolic plane, and he established non-Euclidean geometry as a well-founded branch of mathematics. He articulated a philosophy of geometry via group theory, where when property P implies Q, then GP is contained in GQ, as in the case of Euclidean and affine geometries.

The interior of the unit disk represents the hyperbolic plane in one model. Any circle intersecting D orthogonally on the boundary represents a line in this hyperbolic plane. Evidently, for a point in D but not on a given line, there are many lines through the point that do not intersect the given line in D. This model, with its expression of motion by Mobius transformations leaving D stable, gave evidence of the consistency of the earlier theoretical hyperbolic plane described by Bolyai and Lobachevskii. Thus geometry was expanded beyond Euclid to the non-Euclidean, and the classical field became a branch of group theory.

It was in 1872 at Erlangen, Germany, that Felix Klein first articulated his group-philosophy of geometry. Since that time the subsequent movement has been labeled the w:Erlangen program.