Geometry arose as the field of knowledge dealing with spatial relationships. Geometry is one of the two fields of pre-modern mathematics, the other being the study of numbers.
In modern times, geometric concepts have been extended. They sometimes show a high level of abstraction and complexity. Geometry now uses methods of calculus and abstract algebra, so that many modern branches of the field are not easily recognizable as the descendants of early geometry. (See areas of mathematics.) A geometer is one who works or is specialized in geometry.
A regular polytope is a geometric figure with a high degree of symmetry. Examples in two dimensions include the square, the regular pentagon and hexagon, and so on. In three dimensions the regular polytopes include the cube, the dodecahedron, and all other Platonic solids. other platonic solids include: the terahedron, the octahedron, the icosahedron, and the 5th dimensional hendecatope. There exist examples in higher dimensions also. Circles and spheres, although highly symmetric, are not considered polytopes because they do not have flat faces. The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians.
Many regular polytopes, at least in two and three dimensions, exist in nature and have been known since prehistory. The earliest surviving mathematical treatment of these objects comes to us from ancient Greek mathematicians such as Euclid. Indeed, Euclid wrote a systematic study of mathematics, publishing it under the title Elements, which built up a logical theory of geometry and number theory. His work concluded with mathematical descriptions of the five Platonic solids.