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Polytopes in Geometry.

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 generalize the idea to include such objects as unbounded polytopes (apeirotopes and tessellations), and abstract polytopes. When referring to an n-dimensional generalization, the term n-polytope is used. For example, a polygon is a 2-polytope, a polyhedron is a 3-polytope, and a polychoron is a 4-polytope…See more : History and Different approaches to definition at Polytope on Wikipedia.
In  “Universal constructors in polytopal graph theory”, a article about Polytopal graph theory, the author wrote:
Polytopal graph theory is concerned with the graphs formed by the edges and vertices of polytopes. The graph of a simple polytope contains all of the necessary information to recover its full combinatorial structure in polynomial time, and thus is equivalent in a strong sense to the object. These objects are both mathematically and aesthetically beautiful as well as practically relevant.  Properties of polytopal graphs are linked with a number of important algorithmic questions about polytopes such as the complexity of  linear programming and the convergence of randomized algorithms - Source.

Image: Polytope Movie Page (Hypercubes) by Komei Fukuda - A Catalog of Uniform Polytopes by Jenn &  Cubic Soap by  Jeff Buchbinder.

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