**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 -

Polytope on Wikipedia&

Cubic Soap by Jeff Buchbinder.