Circle Patterns On Surfaces With Obtuse Intersection Angles And Sphere Patterns With Prescribed Combinatorial Types

Our main research object is the circle pattern.A circle pattern is a collec-tion of oriented circles.The circle pattern theory mainly studies the combinatorial relations between circles and the exterior intersection angles.The circle pattern theory has played significant roles in various subjects such as combinatorics,dis-crete and computational geometry,deformation theory,minimal surfaces.The Koebe-Andreev-Thurston circle pattern theorem is the bridge between the circle pattern theory and these subjects.The circle pattern theorem states as follows.For an arbitrary triangulation? of an oriented closed surface and a function ? defined on the edge set such that ?(e)is a non-obtuse angle for each edge e of ?,there exists a circle pattern whose combinatorial type is equal to ? and its intersection angles are given by ?.Moreover,the circle pattern is unique up to conformal transformations.In dimension two,using topological degree theory,we obtain the circle pattern theorem containing the case of obtuse intersection angles.Let S be a surface with finite topological type.For an arbitrary triangulation ? of an oriented closed surface and a function ? defined on the edge set of ?(?can take obtuse angles under some restriction),there exists a circle pattern whose combinatorial type is equal to ? and its exterior intersection angles are given by ?.In higher dimensions,using Cayley-Menger Determinant and combination,we obtain a generalized result of the classic circle pattern theorem.Let ? be an(n+1)-polytope constructed by repeatedly gluing new simplices or cross polytopes onto facets and let ? be an angle in (?) Then there exists an(n-1)-sphere pattern on the unit sphere Sn,unique up to Mobius transformations,such that its nerve is isomorphic to the 1-skeleton of? and each exterior intersection angle is ?.We also prove that there exists an(n-1)-sphere packing on Sn such that its nerve is isomorphic to the 1-skeleton of ?,where ? is the dual polytope of T.Furthermore,we proved that ? can be realized with all(n-1)-faces tangent to Sn and ? can be realized with all edges tangent to Sn.We prove that every non degenerate tetrahedron has a corresponded four sphere configuration.Hence we obtain the new volume formula and dihedral angle formula of Euclidean tetrahedra based on the radii of spheres and the intersection angles.We prove that the infinite iteration of several kinds of affine subdivision on a tetrahedron produce dense sets of shapes of smaller tetrahedra,respectively.