GEOMETRY
Geometry is the study of shapes and configurations. It attempts to understand and classify spaces in various mathematical contexts. For a space with lots of symmetries, the study naturally focuses on properties which are invariant (remaining the same) under the symmetries.
HISTORY OF GEOMETRY
Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers.
Classic geometry was focused in compass and straightedge constructions. As they are the composition of five elemental constructions over a set of elements, as an algebra over an axiomatic system, the barrier between algebra and geometry began to fade out.
In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry.
The earliest recorded beginnings of geometry can be traced to cavemen, who discovered obtuse triangles in the ancient Indus Valley (see Harappan Mathematics), and ancient Babylonia (see Babylonian mathematics) from around 3000 BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras; the Egyptians had a correct formula for the volume of a frustum of a square pyramid; the Babylonians had a trigonometry table.
Geometry was thoroughly...