# Discrete geometry

Discrete geometry, a fundamental branch of mathematics, focuses on the study of geometric structures that are not continuous, but rather consist of distinct, separable elements. This intriguing field delves into patterns, shapes, and configurations that emerge in discrete spaces, playing a pivotal role in computer science, cryptography, and combinatorics. To encapsulate its essence, imagine discrete geometry as the exploration of the geometric beauty found in the jigsaw puzzle of discrete objects, offering a mathematical framework that bridges abstract concepts with tangible applications.

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## What is Discrete Geometry?

Discrete Geometry focuses on the study of geometric objects and their properties in discrete settings. It diverges from the smooth curves and surfaces typical of classical geometry, concentrating instead on distinct, often countable shapes like points, lines, and polygons. This branch of mathematics finds applications across various fields, from computer science to architectural design.

### Discrete Geometry Definition for Beginners

Discrete Geometry is a subfield of mathematics dealing with the study and analysis of discrete and combinatorial geometric structures. Unlike traditional geometry, which considers continuous entities, Discrete Geometry focuses on objects that can be separated clearly and distinctly such as vertices, edges, and other geometric figures.

An example of discrete geometry in action is the study of graph theory. Consider a network of computers connected by cables; each computer can be represented by a vertex, and each cable by an edge. This setup forms a graph, a fundamental concept in discrete geometry, facilitating the analysis of networked systems.

### The Importance of Discrete Geometry in Maths

Discrete Geometry holds a significant place in mathematics for several reasons. Firstly, it provides a foundation for understanding complex networks and systems, such as social networks or transportation systems, through a mathematical lens. Secondly, its principles are crucial in solving real-world problems in computer science, such as network design, coding theory, and cryptography. Lastly, discrete geometry helps in the development and understanding of algorithms, which are essential for computer operations and data processing.

Many modern technologies, including GPS and virtual reality, rely on principles of discrete geometry.

Exploring further into discrete geometry reveals its influence on tessellations and tilings, which are methods of covering a plane using shapes without gaps or overlaps. These concepts are not only mathematically interesting but also have practical applications in art, design, and architecture. Tessellations can efficiently use space and materials, showcasing the blend of mathematical theory and practical utility.

## Exploring Examples of Discrete Geometry

When diving into the fascinating world of discrete geometry, you'll find its applications woven into the fabric of daily life as well as the cutting-edge field of computer science. This segment aims to shed light on how discrete geometric concepts are not just academic exercises but practical tools that shape our world.

### Everyday Examples of Discrete Geometry

Discrete geometry manifests in several familiar scenarios. For instance, when you play games like chess or Tetris, you are engaging with discrete geometry. Moreover, architectural designs often leverage principles of discrete geometry to create aesthetically pleasing and structurally sound buildings.

Consider the simple yet profound example of tiling a bathroom floor. Here, the tiles represent discrete geometric shapes that must fit together without gaps or overlaps. The process involves not only aesthetics but also mathematical precision to ensure optimal use of space and materials, a principle directly rooted in discrete geometry.

Discrete geometry can also be observed in the layout of pedestrian walkways in parks, where the paths are designed to connect discrete points of interest efficiently.

Going deeper into everyday applications, consider the planning and design of urban areas. Urban planners utilise geometrical shapes and principles to design cities that are easy to navigate, ensuring that roads, buildings, and public spaces are organised in a logical and accessible manner. This meticulous planning involves discrete geometry at every step, from the layout of city blocks to the design of public transportation systems.

### Discrete Geometry in Computer Science

In the realm of computer science, discrete geometry plays a pivotal role. It underpins the development of algorithms, data structures, and optimisation techniques that are core to effective problem-solving and computational efficiency.

A prime example of discrete geometry in computer science is the rendering of 3D graphics. Consider a complex 3D model in a video game, consisting of polygons that must be efficiently processed and rendered. This process relies on algorithms that are deeply rooted in discrete geometry, ensuring the model is visually cohesive and performs well under real-time constraints.

Diving deeper into the computer science application, algorithms for pathfinding in networks or graphs are heavily grounded in discrete geometry. These algorithms, like Dijkstra's or A*, are critical for network routing, AI movement in games, and solving mazes. Here's a simple representation of Dijkstra's algorithm in Python:

def dijkstra(graph, start):
shortest_path = {}
unvisited = set(graph)
while unvisited:
node = min(unvisited, key=lambda node: shortest_path.get(node,float('inf')))
unvisited.remove(node)
for neighbour, distance in graph[node].items():
path = shortest_path.get(node,float('inf')) + distance
if path < shortest_path.get(neighbour, float('inf')):
shortest_path[neighbour] = path
return shortest_path

## Core Areas of Discrete Geometry

Exploring the vast landscape of discrete geometry reveals a variety of core areas, each with its unique characteristics and applications. From the study of simple shapes to the complexities of computational systems, discrete geometry offers a rich field ripe for exploration.

### Fundamentals of Discrete Geometry Explained

The fundamentals of discrete geometry lay the foundation for understanding this intricate branch of mathematics. This section promises to unlock the basics, covering essential concepts and their significance in various fields.

Polyhedra: A central concept in discrete geometry, a Polyhedra is a solid figure bounded by plane polygons or faces. These shapes are not only fascinating geometrically but also form the basis for many advanced theories in the field.

An example of polyhedra is a cube, which has 6 square faces, 12 edges, and 8 vertices. The simplicity of a cube serves to illustrate the essence of polyhedra in discrete geometry.

Tessellation, the process of covering a plane with polygons without gaps or overlaps, is an application of polyhedra concepts.

### Convex and Discrete Geometry: A Closer Look

Convex geometry, a subset of discrete geometry, focuses on the study of convex sets and their properties. This area significantly contributes to solving optimisation problems and is closely linked with linear programming.

Convex Set: A set in a vector space is considered convex if, for any two points within the set, the line segment connecting them lies entirely within the set. This concept is crucial for understanding the geometry of shapes and spaces.

An example of a convex set is a circle. Regardless of the two points chosen inside a circle, the line segment connecting these points will always remain inside the circle, showcasing the set's convex nature.

Exploring the concept of convex hulls offers an exciting excursion into applied discrete geometry. A convex hull of a set of points is the smallest convex set containing all the points. This concept has critical applications in computational geometry, such as designing algorithms for pattern recognition, data analysis, and image processing.For instance, in computer vision, determining the convex hull of a set of points can help in identifying the outer boundary of an object within an image, aiding in object recognition and classification.

### Discrete and Computational Geometry: How They Interact

Discrete and computational geometry intersect to solve complex problems in computer science, robotics, and more. This synergy enables the development of algorithms that can process and analyse geometric data efficiently.

Voronoi Diagram: A partitioning of a plane into regions based on distances to points in a specific subset of the plane. Each region consists of all points closer to a particular point than to any other. Voronoi diagrams have widespread applications in fields such as meteorology, aviation, and urban planning.

An example application of Voronoi diagrams is in cell phone network design. By treating cell towers as points, the area covered by each tower is a Voronoi cell, helping network designers optimise coverage and minimise interference.

Delving deeper, computational geometry's reliance on discrete geometry is evident in algorithms for 3D object recognition. The ability to precisely define and manipulate geometric shapes within a computer requires the discrete representation of these shapes, illustrating the symbiotic relationship between these areas.For instance, 3D object recognition algorithms often rely on discretely approximating the shapes of objects using polygons or polyhedra, allowing computers to efficiently process and analyse their geometric properties for recognition tasks.

## Advanced Topics in Discrete Geometry

Delving into the advanced topics of discrete geometry opens up a world where mathematics meets practicality in an exciting interplay. These topics extend the fundamental concepts of discrete geometry and bridge them with real-world applications, particularly in modern technology.

### Discrete Differential Geometry: An Overview

Discrete Differential Geometry (DDG) stands at the fascinating crossroads between classical differential geometry and the discrete world of computational mathematics. By applying the principles of differential geometry to discrete sets and structures, DDG offers powerful tools for the analysis and simulation of geometric shapes.

DDG is particularly relevant in fields such as computer graphics, where it aids in the realistic rendering of curved surfaces and dynamic simulations. It strategically adapts continuous mathematical concepts to discrete computational frameworks, enabling the efficient processing of complex geometrical data.

Discrete Differential Geometry is a branch of geometry that deals with discrete spaces (such as graphs) with the goal of mirroring the properties and behaviours found in smooth differential geometry. It focuses on the study of geometric properties and structures that arise from discretising smooth geometries.

An example of DDG in action is the simulation of cloth dynamics in computer graphics. Fabric is modelled as a mesh of interconnected particles, where discrete differential geometric methods are used to calculate how the mesh deforms under various forces, simulating realistic cloth behaviour.

Discrete Differential Geometry enables the smooth transition from theoretical mathematical models to practical computational algorithms.

### How Discrete Geometry Shapes Modern Technology

Discrete Geometry plays a pivotal role in the foundation of several technologies that drive the modern world. From the digital realms of computer science to the tangible innovations in engineering, discrete geometric concepts underpin significant advancements.

In computer graphics, discrete geometry provides the mathematical framework for modelling and rendering 3D environments. Similarly, in telecommunications, it aids in network design and optimisation, ensuring efficient data transmission across complex networks.

Delunay Triangulation and Voronoi Diagrams are two key concepts from discrete geometry that find extensive applications in technology, ranging from GIS (Geographical Information Systems) to autonomous vehicle navigation.

Consider the application of Voronoi diagrams in mobile network tower placement. By representing potential tower locations as points, Voronoi diagrams can help determine the optimal areas of coverage, maximising efficiency and reducing overlap. This significantly enhances the quality of service for users.

Exploring further, the impact of discrete geometry on modern technology is vividly seen in the realm of robotics, particularly in motion planning and spatial reasoning. Robots operate in environments that are discretised into navigable and non-navigable spaces, allowing them to plot paths and make decisions based on geometric algorithms. This is crucial in autonomous vehicles, where discrete geometry algorithms process sensor data to avoid obstacles and navigate efficiently.For example, pathfinding algorithms utilising discrete geometry, such as A* or Dijkstra's, empower autonomous vehicles to compute the most efficient route between two points, considering real-world constraints. Here's a simple representation of the A* algorithm in Python:

def a_star_algorithm(start, stop):
# Open set is initially empty
# Closed set starts with the initial position
# Assume some functions defined elsewhere: heuristic(), neighbours(), move_cost()
open_set = set(start)
closed_set = set()
g = {}
parents = {}
g[start] = 0

while len(open_set) > 0:
n = None
for v in open_set:
if n == None or g[v] + heuristic(v) < g[n] + heuristic(n):
n = v;
if n == None:
print('Path does not exist!')
return None
if n == stop:
reconst_path = []

while parents[n] != n:
reconst_path.append(n)
n = parents[n]
reconst_path.append(start)
reconst_path.reverse()
return reconst_path

for (m, weight) in neighbours(n):
if m not in open_set and m not in closed_set:
parents[m] = n
g[m] = g[n] + weight
else:
if g[m] > g[n] + weight:
g[m] = g[n] + weight
parents[m] = n

if m in closed_set:
closed_set.remove(m)
open_set.remove(n)
print('Path does not exist!')
return None

## Discrete geometry - Key takeaways

• Discrete Geometry: a subfield of mathematics focused on distinct geometric shapes like points, lines, and polygons, used in fields including computer science and architecture.
• Examples of Discrete Geometry: graph theory (network of computers connected by cables as vertices and edges), game playing (chess, Tetris), architectural design (tiling a floor), and urban planning (city layout).
• Discrete and Computational Geometry: combine to create efficient algorithms for problem-solving, such as pathfinding in networks and 3D graphics rendering.
• Convex and Discrete Geometry: study of convex sets, important in optimisation problems and linear programming, and examples include circles and Voronoi diagrams in cell phone network design.
• Discrete Differential Geometry (DDG): adapts differential geometry principles to discrete structures, critical for computer graphics and algorithmic modelling such as cloth dynamics simulation.
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What are common problems solved by discrete geometry?
Common problems solved by discrete geometry include tiling and packing problems, determining the densest sphere packings, finding the shortest networks connecting points (Steiner trees), and exploring the combinatorial properties of geometric configurations such as arrangements of lines, circles, and polygons.
What is the difference between discrete and continuous geometry?
Discrete geometry deals with objects that can be counted and separated, like points, lines, and polygons, whereas continuous geometry involves smooth objects that extend without break, such as curves and surfaces. The former focuses on distinct and isolated elements, while the latter on unbroken and flowing entities.
What are the key principles of discrete geometry?
Discrete geometry focuses on the study of geometric shapes and structures that are distinct and separate, emphasising combinatorial and algorithmic aspects. Key principles include the study of configurations, such as points, lines, and polygons, their properties, and the spatial relationships between them, often employing discrete and combinatorial methods for problem-solving.
How is discrete geometry applied in computer science?
Discrete geometry is applied in computer science for modelling the shapes and structures within computer graphics, optimising network designs, and enhancing data compression techniques. It is also critical in the development of algorithms for solving geometric problems relevant to computer vision and robotics.
What careers utilise knowledge of discrete geometry?
Careers utilising knowledge of discrete geometry include software development, especially in computer graphics and virtual reality, quantitative finance, architectural design, and various engineering fields, particularly those involving network design and spatial analysis. Academic research and teaching in mathematics also significantly involve discrete geometry.

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