Triangulation in Geometry - Research Paper Example

Triangulation in geometry Outline Key definitions 2. Euler's theorem 3. Discovering and generalizations 4. Gauss-Bonnet theorem 5. Surfaces, theirs topology and triangulation 1. Key definitions To make our considerations of extensions of the Euler's theorem and triangulation concepts more pure, we need to preliminary define the key notions from the related topics. First of all, polygon is a closed plane figure with sides (Weisstein 2002). Then, a polyhedron is the union of a finite set of polygons such that: (i) any pair of polygons meets only at their sides or corners; (ii) each side of each polygon meets exactly one other polygon along an edge; (iii) it is possible to travel from the interior of any polygon to the interior of another; and (iv) it is possible to travel over the polygons which meet at the vertex from one to any other without passing through (Cromwell 1999). A polyhedron is the -D version of the more general polytope, which can be defined in arbitrary dimension (Weisstein 2002). In algebraic topology, polyhedron is defined as a space that can be built from the simplexes by "gluing them together" along their faces. More specifically, it can be defined as the underlying space of a simplicial complex. Here, a polyhedron can be viewed as an intersection of halfspaces (Webster 1994). Then, a convex polytope may be defined as the convex hull of a finite set of points (which are always bounded), or as a bounded intersection of a finite set of half-spaces (Webster 1994). More specifically, it can be defined as a finite region of -dimensional space enclosed by a finite number of hyperplanes. The part of the polytope that lies in one of the bounding hyperplanes is called a cell (Weisstein 2002). Simplex is the generalization of a tetrahedral region of space to -D. The boundary of a -simplex has -faces (vertices), -faces (edges), and -faces, where is a binomial coefficient. The simplex named because it represents the simplest possible polytope in any given space (Weisstein 2002). Simplicial complex is a space with a triangulation. Formally, a simplicial complex in is a collection of simplices in such that: (i) every face of a simplex of is in , and (ii) the intersection of any two simplices of is a face of each of them (Munkres 1993). Objects in the space made up of only the simplices in the triangulation of the space are called simplicial subcomplexes (Weisstein 2002). Usually, surface is a -D submanifold of -D Euclidean space. More generally, surface is an -D submanifold of an -D manifold, or in general, any codimension-1 subobject in an object like a Banach space or an infinitedimensional manifold. A surface with a finite number of triangles in its triangulation is called compact surface (Weisstein 2002). Genus is a topologically invariant property of a surface defined as the largest number of nonintersecting simple closed curves that can be drawn on the surface without separating it. In fact, it is the number of holes in a surface (Weisstein 2002). The geometric genus of a surface is related to the Euler characteristic by . Our final key definition is for Betti numbers. Betti numbers are topological objects which were proved to be invariants by Poincar, and used by him to extend the polyhedral formula to higher dimensional spaces. Informally, the Betti number is the maximum number of cuts that can be made without dividing a surface into two separate pieces. Formally, the -th Betti number is the rank of the -th homology group of a topological space (Griffiths 1976; Weisstein 2002). 2. Euler's theorem Euler's theorem states relation between the number of vertices , edges , and faces of a simply connected (i.e., genus ) polyhedron () or polygon (), i.e. it states the polyhedral formula . Initial Euler's proof of the polyhedral formula is not irreproachable (Cromwell 1999). This proof is based on the principle that polyhedrons can be truncated. Euler's proof begins with a polyhedron consisting of a large number of vertices, faces, and edges. By removing a vertex, at least 3 faces (while exposing a new face), and at least 3 edges will be removed. When proceeding in this fashion, more vertices are removed and the current values of , , and are computed. Euler's approach shows that truncated and augmented platonic polyhedra satisfy the polyhedral formula. However, eventually Euler's proof degenerates into an object that is not a polyhedron, and Euler's formula fails. For instance, let's begin with a regular octagon (). One vertex removal will decrease the number of vertices by 1, edges by 4, and faces by 3. The result of this truncation belongs to the family of pyramidal polyhedra. The polyhedral formula holds (). Another vertex removal will decrease the number of vertices by 1, edges by 3, and faces by 2. The polyhedral formula holds again (). One more vertex removal results in a degenerate object, non polyhedron. Indeed, by removing 1 vertex we would remove 3 edges, and have only 2 faces, which doesn't satisfy the definition of a polyhedron. Let's proof the polyhedral formula on the base of intuitive Von Staudt's approach (Cromwell 1999). This proof begins by picking any vertex on a convex polyhedron. From this vertex, we look for a vertex that has never been coloured, and connect the vertices by colouring along the edge that connects them. At the new vertex, we look for another vertex that has never been coloured. If we find one, we connect the vertices and colour along the edge that connects them again. We proceed in this fashion until there are no more vertices to be visited. If there are no more vertices to colour, then all vertices have been coloured. When traversing a polyhedron in this manner, we can see that this proof requires that there is always a path that touches all vertices only once, i.e. a Hamiltonian circuit. So, we initially connect two vertices, and for every other edge we shade (let's use red), we find another vertex that we previously had not visited. Therefore, by counting the number of edges that we shade red we can determine the number of vertices , i.e. . Furthermore, Von Staudt states that all the vertices have been coloured. To prove this, assume this is not the case. Then there would be a vertex that is not coloured, which means we were not done colouring our edges red. Next we will begin by examining faces. Pick a face and shade it green, as well as any edges that have not yet been coloured red. Proceed by finding another face who has only one green side, and shade green as before. Continue until there are either no more faces to colour that satisfy the condition. Since it is impossible that a face exists with all four sides coloured, all faces must be shaded green. So, the relationship between the number of green edges and the number of faces can be described as: . The total number of edges will equal the number of red edges and the number of green edges: . Therefore, and . The intuitive Von Staudt's approach can be formalized by so-called "interdigitating trees" approach (Eppstein 2005). For any connected embedded planar graph define the dual graph by drawing a vertex in the middle of each face of , and connecting the vertices from two adjacent faces by a curve through their shared edge . Note that . Any cycle in disconnects by the Jordan curve theorem. Any acyclic subgraph of is a forest and does not disconnect . So, it is possible to get from any face to any other face by detouring around trees in . Therefore, connectedness and acyclicity are dual to each other. This duality forms the basis of the following proof of the Euler's formula: Let's choose any spanning tree in ; this is by definition a connected acyclic subgraph. The dual edges of its complement, , form an acyclic connected subgraph of which is therefore also a spanning tree. The two trees together have edges. So, this is the formalized equivalent of the Von Staudt's proof for the polyhedral formula. 3. Discovering and generalizations The polyhedral formula of the Euler's theorem was discovered independently by Euler and Descartes, so it is also known as the Descartes-Euler polyhedral formula. This formula was discovered in around 1750 by Euler, and first proven by Legendre in 1794. Earlier, Descartes (around 1639) discovered a related polyhedral invariant (the total angular defect) but apparently did not notice the Euler formula itself (Eppstein 2005). Euler's work "Elementa doctrinae solidorum" (Elements of the doctrine of solids) was originally published in 1758. Here, Euler presents several results relating the number of plane angles of a solid to the number of faces, edges, and solid angles. The main theorem is that in all solid bodies confined by planes, the sum of the number of solid angles and the number of faces is two less than the number of edges, i.e. the polyhedral formula. Then, Euler wrote a sequel "Demonstratio nonnullarum insignium proprietatum, quibus solida hedris planis inclusis sunt praedita" (Proof of some of the properties of solid bodies enclosed by planes). These works and detailed historical analysis of discovering the polyhedral formula are at the on-line Euler Archive. There are almost 20 classical and modern proofs of the polyhedral formula, e.g. based on induction on faces, vertices, or edges, sum of angles or spherical angles in terms of spherical triangulations, or based upon usage of shelling or binary homology concepts, etc. Most of them are assembled by Eppstein (2005). The polyhedron formula can be generalized in many ways. First of all, let a closed surface have genus . Then the polyhedral formula generalizes to the Poincar formula , where is the Euler-Poincar characteristic. Here, the polyhedral formula corresponds to the special case . Then, the polyhedral formula was generalized to -D polytopes by Schlfli: , where , , and is the number of vertices , edges , and faces of a -polytope (Coxeter 1973). So, in terms of the combinatorial theory of polytopes Euler's relation states that, if a non-empty -polytope has -faces, then . The polyhedron formula is the more familiar form of this relation for -polytopes (Webster 1994). Although the Euler relation is the only linear equation satisfied by the numbers of faces of various dimensions of every polytope with a given dimension, there are other linear equations that are satisfied by the numbers of faces of various dimensions of every simplicial polytope with a given dimension, e.g. the Dehn-Sommerville equation (Webster 1994). Then, the Euler characteristic as a topologic invariant can be expressed in terms of the Betti numbers: , where is the -th Betti number of the space. For finite complex , the Euler number is defined by (Weisstein 2002). Finally, in terms of the integral curvature of the surface , the Euler characteristic , i.e. it is related with the integral of the Gaussian curvature over the entire surface. This is a result of the Gauss-Bonnet theorem. 4. Gauss-Bonnet theorem The Gauss-Bonnet theorem has several formulations (Weisstein 2002). The simplest one expresses the total Gaussian curvature of an embedded triangle in terms of the total geodesic curvature of the boundary and the jump angles at the corners. More specifically, if is any -D Riemannian manifold (like a surface in -space) and if is an embedded triangle, then the Gauss-Bonnet formula states that the integral over the whole triangle of the Gaussian curvature with respect to area is given by minus the sum of the jump angles minus the integral of the geodesic curvature over the whole of the boundary of the triangle (with respect to arc length), ; where is the Gaussian curvature, is the area measure, the are the jump angles of and is the geodesic curvature of ; with the arc length measure. The next most common formulation of the Gauss-Bonnet formula is that for any compact, boundaryless -D Riemannian manifold , the integral of the Gaussian curvature over the entire manifold with respect to area is times the Euler characteristic of the manifold, . Here, the total Gaussian curvature is differential-geometric in character, but the Euler characteristic is topological in character and does not depend on differential geometry at all. So, if the surface will be distorted at any location, the same total curvature is maintained regardless of surface distortion features. Another way of looking at the Gauss-Bonnet theorem for surfaces in -space is that the Gauss map of the surface has degree given by half the Euler characteristic of the surface ; which works only for orientable surfaces where is compact. This makes the Gauss-Bonnet theorem a simple consequence of the Poincar-Hopf index theorem. A general Gauss-Bonnet formula that takes into account both formulas can also be given. For any compact -D Riemannian manifold , the integral of the Gaussian curvature over the entire manifold with respect to area is times the Euler characteristic of the manifold, minus the sum of the jump angles and the total geodesic curvature of the boundary. Let's proof the Gauss-Bonnet theorem (Oprea 1997; Cromwell 1999; Weeks 2002). It was noted above that every surface may be triangulated. That is, the surface may be completely covered by shrinkable triangles in the surface which meet only along edges or at vertices. Furthermore, an orientation on a surface induces orientations on each of the triangles so that the edge orientations are opposite when considered in adjacent triangles: Suppose has a triangulation with total vertices, total edges and total triangles. If each triangle in a surface is shrinkable to a point, then (Oprea 1997): , where the angles are interior angles of a triangle. This applies to each triangle, so we can add up both sides of the formula over all triangles to get . Triangulated surface may have boundary curves. Let's denote surface boundary by . We may split up the sets of vertices and edges according to whether they lie on the boundary or in the interior of . Let and , where the subscripts and refer to the interior and boundary respectively. Then , since the sum of the interior angles at a particular vertex in , for all triangles intersecting that vertex, is and the sum of the interior angles at a particular vertex in , for all triangles intersecting that vertex, is the angle of the tangent line to the boundary at that vertex, . Furthermore, each triangle has edges surrounding it and two triangles share a common edge, except for edges in which meet only one triangle. This leads to the equality . Replacing by , we then have , since . Then , , . Therefore, if is a compact oriented surface with boundary made up of a finite number of smooth closed curves, then . So, if is a compact oriented surface without boundary, then . The quantity may not be characteristic of , but may depend on which triangulation is chosen. That this is not the case is a result which goes back to Euler. In fact, the result is more general. A graph for surface consists of finite sets of vertices and edges where denotes the unique edge joining and . Here, we considered connected graphs only. For these graphs there is an edge path joining any two vertices of triangulated surface . 5. Surfaces, theirs topology and triangulation Finally, let's construct some connected sums and consider orientability of some surfaces. By definition, the surface is orientable if the triangles can be given an orientation so that all neighbouring triangles are coherently oriented, a clockwise or anti-clockwise (Weeks 2002). From triangulations of the following surfaces we can conclude that (Ward 2001): 1) the Mbius band is not a closed and non-orientable surface: 2) the -torus is orientable: 3) the Klein bottle is non-orientable; here, the shaded region is a Mbius band, and the space left after removing this Mbius band is again a Mbius band; therefore, is "twice" as non-orientable; moreover, this surface cannot be represented as a submanifold of (see below): To show that the Klein bottle is homeomorphic to , we need to split the square in the following figure along a diagonal, flip one of the resulting triangular pieces over, and paste the two pieces together along the edge labelled (Munkres 2000): Alternatively, we can start with two projective planes, cut a disk out of each to get two Mbius bands, and then glue the Mbius bands' edges together to get a Klein bottle (Weeks 2002); 4) the projective plane is non-orientable; here, the shaded Mbius band, when removed, leaves surface that has no more Mbius bands in it: 5) the sphere is orientable; in general, an -dimensional submanifold of is orientable if it has a unit normal vector field, and the choice of unit determines the orientation of the submanifold (Weisstein 2002): Then, the unit -sphere is orientable; indeed, admits a non-zero normal vector field, namely, the restriction to of the unit vector field on pointing radially outward (Singer & Thorpe 1987). A variety of simple (even trivial, , , , etc.) and complex connected sums (e.g. Mbius, Mbius, which are topologically the equivalent objects) is illustrated by the Weeks (2002) and Francis (1987). Bibliography 1. Coxeter, H.S. (1973). Regular Polytopes. New York: Dover. 2. Cromwell, P.R. (1999). Polyhedra. Cambridge: Cambridge University Press. 3. Eppstein, D. (2005). Proofs of Euler's Formula. Retrieved July 20, 2006 from Geometry Junkyard, ICS, UC Irvine at http://www.ics.uci.edu/eppstein/junkyard/. 4. Francis, G.K. (1987). A Topological Picturebook. New York: Springer-Verlag. 5. Griffiths, H.B. (1976). Surfaces. Cambridge: Cambridge University Press. 6. Munkres, J.R. (1993). Elements of Algebraic Topology. New York: Perseus Press. 7. Munkres, J.R. (2000). Topology. Upper Saddle River, NJ: Prentice Hall. 8. Oprea, J. (1997). Differential Geometry and Its Applications. Upper Saddle River, NJ: Prentice Hall. 9. Singer, I.M. & Thorpe, J.A. (1987). Lecture notes on elementary topology and geometry. New York: Scott & Foresman. 10. Ward, T. (2001). Topology Lecture Notes. UEA. 11. Webster, R. (1994). Convexity. Oxford: Oxford University Press. 12. Weeks, J.R. (2002). The shape of space. New York: Marcel Dekker. 13. Weisstein, E.W. (2002). Concise Encyclopaedia of Mathematics. New York: CRC Press. Read More