2polytope

A 2-polytope, or two-dimensional polytope, is a polytope of dimension two. In standard definitions, a polytope is the convex hull of a finite set of points in the plane, or equivalently a bounded intersection of half-spaces. A 2-polytope therefore lies in the plane and is convex.

In two dimensions, a 2-polytope is precisely a convex polygon. Its boundary consists of a finite sequence of straight-line edges forming a closed polygonal chain. A 2-polytope has a finite set of vertices and edges, with the same number of vertices as edges, and at least three sides. The interior is connected and the boundary is homeomorphic to a circle.

2-polytopes are commonly classified by the number of sides, or equivalently by the number of vertices. Any polygon with n ≥ 3 sides is a 2-polytope. Regular polygons, such as the equilateral triangle, square, and regular pentagon, are notable examples. The set of symmetries of a 2-polytope reflects its geometric regularity and is described by its dihedral group.

Duality in two dimensions maps a 2-polytope to another 2-polytope. For polygons, the dual is another polygon with the same combinatorial structure (the same number of sides), illustrating that polygons form a self-dual class under this notion.

Relation to other concepts includes the connection to planar graphs: the boundary of a 2-polytope corresponds to a simple cycle, and the 2-polytope itself can be studied via its combinatorial and geometric properties. In summary, 2-polytopes are the convex polygons in the plane, fundamental objects in convex and combinatorial geometry.