## usable tools for proofs

• Estimate the polyhedron's value
and combine it with already known polyhedra. In most cases it is usual to estimate isolateral triangles and special triangle-arrangements as:
• vertex of degree k
• vertex of degree k and the neighboured triangles
• two neighboured vertices of degree p and q, e.g. 4 and 4
• two neighboured vertices of degree p and q with an edge in common.

• Delaunay-Triangulation
The Delaunay-Triangulation of a pointset S has the following properties:
• three points are vertices of a reggion, if the circle defined by these points does not contain another point of S in its interior.
• pi and pj are an edge of the triangulation, if there is a circle containing pi and pj on its boundary which does not contain another point of S in its interior.
• minimal angle-vector: Let T be a triangulation of S and let the angles of T's triangles be ω1 ≤, ..., ≤ ω2n-4. Then Ω(T) := ( ω1 ≤, ..., ≤ ω2n-4 ) is known as angle-vector of T.
These vectors are ordered lexicographical, i.e.:
Ω(T)< Ω(T') gdw. ∃ i: ωi(T)< ωi(T').

• Edgeflip
The edgeflip removes the common edge of two neighboured triangles and replaces it by the connection of the opposite vertices. The edgeflip must not be done if the common edge connects at least one vertex of degree 3.

Theorem 1 (V.Eberhard)
Each simple polyhedral type with N vertices can be obtained by edgeflips from any simple polyhedral types with N vertices.

In other words: Let P be the set of all simple polyhedral types with N vertices. If we connect two types by an edge if the second type can be obtained by an edgeflip from the first one, and if K denotes the set of all these connections, the graph is 1-connected.