The Relationship between the Stasheff Polytope and Painted Trees Using Tubings

Abstract

A polytope plays a central role in different areas of mathematics. It is used quite heavily in applied fields of mathematics, such as medical imaging and robotics, geometric modeling. A polytope has many users in modern science such as computer graphics, optimization, and search engine. It is intended for a broad audience of mathematically inclined. Therefore in this paper, we shall take a polytope with one kind of application, which is known as a Stasheff polytope. it has been applied in: moduli spaces, Erhart polynomial, physics- chemistry, and Hopf algebras. In this paper,a finite graph G with tubingis taken, the nodes of the graph are reducetopoints in R^n and the convex hull for them are simplex, permutahedron and associahedron (Stasheff polytope ) are studied. Also, a fan graph to the painted tree is also taken and reduces its nodes to points inR^n. The converse of this result is also given with different examples to consult our results.

Keywords

• stasheff polytope
• permutohedron
• tube
• tubing
• painted tree
• fan graph