Search

# Triangulation in Geometry - Research Paper Example

Summary
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)…

## Extract of sample"Triangulation in Geometry"

Download file to see previous pages 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 vertice ...Download file to see next pagesRead More
Cite this document
• APA
• MLA
• CHICAGO
(“Triangulation in Geometry Research Paper Example | Topics and Well Written Essays - 3000 words”, n.d.)
Triangulation in Geometry Research Paper Example | Topics and Well Written Essays - 3000 words. Retrieved from https://studentshare.org/science/1533727-triangulation-in-geometry
(Triangulation in Geometry Research Paper Example | Topics and Well Written Essays - 3000 Words)
Triangulation in Geometry Research Paper Example | Topics and Well Written Essays - 3000 Words. https://studentshare.org/science/1533727-triangulation-in-geometry.
“Triangulation in Geometry Research Paper Example | Topics and Well Written Essays - 3000 Words”, n.d. https://studentshare.org/science/1533727-triangulation-in-geometry.
Click to create a comment or rate a document
CHECK THESE SAMPLES - THEY ALSO FIT YOUR TOPIC
The objective of this assignment is to present information about a mathematician: Euclid
in Alexandria of ancient Egypt by the mouth of the Nile River while Tyre was known to be his original place of birth and Damascus witnessed how he was reared to adulthood. Having taught king Ptolemy in his time as a renowned mathematician in Alexandria, he ought not to be confused with the philosopher Euclid of Megara who was around 100 years prior to his existence.
4 Pages(1000 words)Research Paper
Curriculum Staff Development Plan
Consequently, there is a great need for devising a staff development plan to enhance performance in this subject. Establishing a skilled teaching staff can enhance the comprehension of the subject, thus improving the performance of the students. This paper presents a tactical staff development plan that will enable teachers to handle the subject successfully, thus helping the students pass the state exam.
5 Pages(1250 words)Research Paper
Infidelity and Divorce
alists in infidelity and divorce issues may have varying insights about how much should be revealed, but majority of them claim that it is imperative for the person who had the affair to inform the spouse and to disclose the facts (Olmstead, Blick, & Mills, 2009). This common
2 Pages(500 words)Research Paper
Pythagoras and the Musical Ratios
He founded a school in his period and promoted studies in “music, geometry, and astronomy” (Svarney and Svarney, p. n.p.). Pythagoras’ development of music ratios is traced to a real life experience in which he heard varying
1 Pages(250 words)Research Paper
ACL Injuries Prevention, Rehabilitation and Risk factors
Current research on ACL suggests that causation is likely multifunctional. Commonly used knee geometry is a notch width index. Gender differences in tibial geometry may contribute to the fact. The current research on ACL injuries suggests that their
2 Pages(500 words)Research Paper
BIOMETRICS: Hand Geometry and Vein Check
surements taken; the measurements range from the length of the fingers, to the distance between knuckles, to the height or thickness of the hand and fingers” (NSTC, 2006, p. 8). Unlike the hand geometry, vein geometry is a fairly recent development. Vein network differ from
2 Pages(500 words)Research Paper
MGT Week 9 SA
During every research, quantitative and qualitative methodologies are very important factors, to come up with the best results. It provides a clear way to understand the behavior and characteristic of others. When the purpose
1 Pages(250 words)Research Paper
The artist M.C. Escher
He became famous from creating mathematically challenging artwork. The tools he used in his drawings were simple, and naked eyes, however he managed to create stunning mathematical work. He mainly focused on division of plane and played around with spaces
2 Pages(500 words)Research Paper
The Future of the Universe: Cold and Dark, or a Big Crunch
The two top theories are the big Crunch theory and the cold dark model. The big Crunch theory claims that if there is enough matter, then the
5 Pages(1250 words)Research Paper
We use cookies to create the best experience for you. Keep on browsing if you are OK with that, or find out how to manage cookies.
Let us find you another Research Paper on topic Triangulation in Geometry for FREE!
+16312120006