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The Satisfaction of a Gauss-Bonnet-type Theore for Polyhedra with Some Faces Removed |
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The Descartes Law of Closure Deficit says that the sum of the angular deficits of the vertices of a simple polyhedron is equal to 4π radians. More generally for any polyhedron the sum of the angular deficits is equal to 2π times its Euler characteristic. This is just a special case of the Gauss-Bonnet Theorem extended to include polyhedral surfaces.
The Gauss-Bonnet Theorem says that the surface integral of the Gaussian curvature of a manifold plus the line integral of the geodesic curvature of the boundary of that manifold is equal to 2π times its Euler characteristic. In an extension of the theorem to include geometric figures with vertices where the Gaussian curvature is not defined and boundary curves with corner points the surface integral must be augmented by the sum of the angular deficits of the vertices and the line integral by the sum of the turning angles at the corner points.
In the case of polyhedra the Gaussian curvature is everywhere zero and there are no boundary curves. Thus the theorem reduces to the sum of the angular deficits at the vertices being equal to 2π times the Euler characteristic, a generalization of Descartes Law of Closure Deficits.
Let P be a polyhedron and χ(P) its Euler characteristic. Now consider the geometric object P* created by the removal of one face of P. The Euler characteristic of P* is (χ(P)−1). The sum of the angular deficits of P* is the same as that of P. But P* has a boundary which is the polygonal edge of the removed face. The geodesic curvature of that polygon is zero but the sum of its turning angles is simply 2π. Thus the sum of the angular deficits of the vertices of P* less the sum of the turning angles of its boundary polygon is given by
It is somewhat of a surprise that the line integral for the boundary must be subtracted to yield the proper result. However when the standard Gauss-Bonnet Theorem is applied to truncated spheres the contribution of the line integral of the geodesic curvature is positive where the truncated sphere is a small portion of a sphere, zero when the truncated shere is a hemisphere and negative where the truncated sphere is nearly a complete sphere. See Illustration of the Gauss-Bonnet Theorem for Truncated Spheres.
The previous result can be generalized. Let P* be created by the removal of k faces from a polyhedron P. Then χ(P*)=χ(P)−k. The boundary of P* is then the k polygons where the faces were removed and the sum of the turning angles of these polygons is just k(2π). Thus
If P* is created by the removal of two adjacent faces and the edge between them then χ(P*)=χ(P)−2+1=χ(P)−1. The boundary of P* is then the polygon created by adjoining the polygons of the the two removed faces and its sum of turning angles is just 2π. Thus the generalization of the Gauss-Bonnet Theorem is maintained.
If P has a pyramidal type vertex its removal would eliminate equal numbers of faces and edges so the Euler characteristic would be reduced by only one for the vertex. The sum of the angular deficits would be reduced by that of the vertex. However if a face F were added where the vertex and its impinging faces and edges is removed the result is another polyhedron P^. Then P* would be the same as the removal of face F from P^ and extended Gauss-Bonnet Theorem would hold for P*.
Thus a Gauss-Bonnet type theorem holds for the geometric objects created by the removal of faces from a polyhedron.
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