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In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions (such as a polychoron in four dimensions). Some theories further generalise the idea to include such things as unbounded polytopes (apeirotopes and tessellations), and abstract polytopes.
When referring to an ndimensional generalization, the term npolytope is used. For example, a polygon is a 2polytope, a polyhedron is a 3polytope, and a polychoron is a 4polytope.
The term was coined by the mathematician Hoppe, writing in German, and was later introduced to English by Alicia Boole Stott, the daughter of logician George Boole.^{[1]}
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Different approaches to definition
The term polytope is a broad term that covers a wide class of objects, and different definitions are attested in mathematical literature. Many of these definitions are not equivalent, resulting in different sets of objects being called polytopes. They represent different approaches of generalizing the convex polytopes to include other objects with similar properties and aesthetic beauty.
The original approach broadly followed by Schläfli, Gossett and others begins with the 0dimensional point as a 0polytope (vertex). A 1dimensional 1polytope (edge) is constructed by bounding a line segment with two 0polytopes. Then 2polytopes (polygons) are defined as plane objects whose bounding facets (edges) are 1polytopes, 3polytopes (polyhedra) are defined as solids whose facets (faces) are 2polytopes, and so forth.
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