Discrete geometry

Apolinario G. Sanger III Submitted to : Professor Rody G. Balete
MT-31

Chapter I
The Problem and It’s Background

Introduction: Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.
Discrete geometry has large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory,toric geometry, and combinatorial topology.
Although polyhedra and tessellations have been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studied were: the density of circle packings by Thue,projective configurations by Reye and Steinitz, the geometry of numbers by Minkowski, and map colourings by Tait, Heawood, and Hadwiger.
László Fejes Tóth, H.S.M. Coxeter and Paul Erdős, laid the foundations of discrete geometry.
"This is an introduction to the field of discrete geometry understood as the investigation of combinatorial properties of configurations of (usually finitely many) geometric objects … . The book is written in a lively and stimulating but very precise style and contains many figures. It gives a good impression of the richness and the relevance of the field.

Statement of the Problem Discrete geometry is not in a scope of our study as mathematics students. Some questions need an answer. These are the questions to be asked in this topic. 1. What are the properties of Discrete Geometry? 2. What are the related topics on Discrete Geometry? 3. What are the use of this topic?

Theoretical Framework simplicialTopological combinatorics
Incident structures
Structural rigidity and Flexibility
Geometric graph theory
Topics on Discrete Geometry
Lattices and Discrete groups
Digital geometry
Discrete differential geometry Simplicial complexes
Polyhedral and Polytopes
Oriented matroids
Packings, Coverings and Tillings