*Geometry of position**Even in ancient times, geometers considered questions of relative position or spatial relationship of geometric figures and shapes. Some examples are given by inscribed and circumscribed circles of*

*polygons*

*, lines intersecting and tangent to*

*conic sections*

*, the*

*Pappus*

*and*

*Menelaus*

*configurations of points and lines. In the Middle Ages new and more complicated questions of this type were considered: What is the maximum number of spheres simultaneously touching a given sphere of the same radius (*

*kissing number problem*

*)? What is the densest*

*packing of spheres*

*of equal size in space (*

*Kepler conjecture*

*)? Most of these questions involved 'rigid' geometrical shapes, such as lines or spheres.*

*Projective*

*,*

*convex*

*and*

*discrete*

*geometry are three subdisciplines within present day geometry that deal with these and related questions.*

*A new chapter in Geometria situs was opened by*

*Leonhard Euler*

*, who boldly cast out metric properties of geometric figures and considered their most fundamental geometrical structure based solely on shape.*

*Topology*

*, which grew out of geometry, but turned into a large independent discipline, does not differentiate between objects that can be continuously deformed into each other. The objects may nevertheless retain some geometry, as in the case of*

*hyperbolic knots*

*.*

*Geometry beyond Euclid**For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of*

*space*

*remained essentially the same.*

*Immanuel Kant*

*argued that there is only one, absolute, geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was*

*synthetic a priori*

*.*

^{[3]}This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of*Gauss*

*(who never published his theory),*

*Bolyai*

*, and*

*Lobachevsky*

*, who demonstrated that ordinary*

*Euclidean space*

*is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by*

*Riemann*

*in his inaugurational lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based), published only after his death. Riemann's new idea of space proved crucial in*

*Einstein*

*'s*

*general relativity theory*

*and*

*Riemannian geometry*

*, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.*

*Symmetry**A uniform*

*tiling*

*of the*

*hyperbolic plane*

*The theme of*

*symmetry*

*in geometry is nearly as old as the science of geometry itself. The*

*circle*

*,*

*regular polygons*

*and*

*platonic solids*

*held deep significance for many ancient philosophers and were investigated in detail by the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the bewildering graphics of*

*M. C. Escher*

*. Nonetheless, it was not until the second half of nineteenth century that the unifying role of symmetry in foundations of geometry had been recognized.*

*Felix Klein*

*'s*

*Erlangen program*

*proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation*

*group*

*, determines what geometry is. Symmetry in classical*

*Euclidean geometry*

*is represented by*

*congruences*

*and rigid motions, whereas in*

*projective geometry*

*an analogous role is played by*

*collineations*

*, geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann,*

*Clifford*

*and Klein, and*

*Sophus Lie*

*that Klein's idea to 'define a geometry via its*

*symmetry group*

*' proved most influential. Both discrete and continuous symmetries play prominent role in geometry, the former in*

*topology*

*and*

*geometric group theory*

*, the latter in*

*Lie theory*

*and*

*Riemannian geometry*