Polytopes

A polytope is a multi-dimensional geometric shape formed by flat surfaces called faces. In two dimensions, polygons like triangles and squares are polytopes; in three dimensions, shapes like cubes and pyramids are polytopes. Extending beyond three, polytopes can exist in higher dimensions, still composed of flat "faces" called hyperfaces. They are defined mathematically by linear inequalities and are fundamental in areas such as optimization, computer graphics, and mathematics. Essentially, polytopes are generalizations of familiar shapes into higher dimensions, characterized by their flat sides and bounded volume.