Polyhedra are the most familiar objects to us, so the main theme of this thesis is the category of polyhedra of Euclidean space, that is a union of polytope. _{1 2 k 1 2 k i i 0 0} ^{2 2}...[ Read more ]

Polyhedra are the most familiar objects to us, so the main theme of this thesis is the category of polyhedra of Euclidean space, that is a union of polytope.

Two polyhedra P and Q are said to be isomorphic, if they admit decompositions D = {p₁, p₂, · · · , p_k} and E = {q₁,q₂, · · · , q_k} respectively such that p_i and q_i are affinely isomorphic polytopes for all i. Roughly speaking, two polyhedra are isomorphic if they can be built up by two piles of isomorphic blocks.

The isomorphism classes of all bounded polyhedra is denoted by B₀(P) and the isomorphism classes of all polyhedra is denoted by B(P). They have nice structures. Schanuel have sketched the proof that B₀(P) is isomorphic to N[X]/X~2X+1 and B(P) is isomorphic to N[X,Y]/X~2X+1,Y~Y+X+1,Y²~2Y²+Y, we will prove this in detail.

We will prove some theorems on the characterization of polytopes, We will also prove some interesting formula.