PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD ↓


Geometry


Geometry is a branch of mathematics that studies the size, shape, properties, and dimensions of objects and spaces. It is one of the oldest fields of mathematical science and has applications in many fields, including art, architecture, engineering, physics, and computer science.

Introduction to geometry

The word "geometry" comes from two Greek words: "geo," meaning "earth," and "metron," meaning "measurement." Therefore, geometry was originally concerned with the measurement and relationships of land and space. Over the centuries, it has become a vast discipline involving abstract concepts and models.

Basic concepts

Geometry involves understanding and representing the characteristics, sizes, and shapes of figures and spaces. Mainly, it deals with points, lines, surfaces, angles, and solids.

Score

In geometry, a point is a place that has no size, shape or dimensions. It only represents a position in space.

Lines

A line is a one-dimensional figure that extends indefinitely in both directions. Here is a simple representation of a line:

A line can be defined by two points. For example, the line L can be described by two points ( P_1 ) and ( P_2 ).

Angles

The angle is formed by two rays that share a common end point. This end point is called the vertex of the angle. Angles are usually measured in degrees.

θ

In the graphic above, the angle θ represents the amount of rotation between two lines emanating from a common endpoint.

Types of geometry

Geometry can be broadly divided into several types:

Euclidean geometry

Named after the Greek mathematician Euclid, this form of geometry deals with plane (two-dimensional) and solid (three-dimensional) spaces. It involves the study of points, lines, and surfaces in various combinations.

Here, in Euclidean space, a circle and its radius are shown. The beauty of Euclidean geometry lies in its simplicity and the logical conclusions that can be drawn from the basic principles.

Non-Euclidean geometry

Non-Euclidean geometry emerges when the parallel theory of Euclidean geometry is changed. Types of non-Euclidean geometry include hyperbolic and elliptic geometry, which explore curved surfaces.

Analytic geometry

Analytical geometry, often called coordinate geometry, uses algebraic equations to describe geometric problems. The Cartesian coordinate system is an important part of analytic geometry, allowing geometry to be translated into algebra.

Plane geometry

Plane geometry includes shapes such as circles, triangles, and squares that can be drawn on a flat surface. In plane geometry, we mainly deal with two dimensions: length and width.

Triangle

A triangle is a polygon with three sides and three vertices. The sum of the interior angles of a triangle in Euclidean space is always equal to 180 degrees.

Triangles are classified based on their sides and angles. For example:

  • An equilateral triangle has all three sides equal and angles equal to 60 degrees.
  • An isosceles triangle has two equal sides.
  • A scalene triangle has all its sides of different lengths.

Solid geometry

Solid geometry is about three-dimensional objects such as cubes, prisms, pyramids, spheres, and cylinders. These shapes occupy a volume and have surfaces.

Cubes

A cube is a three-dimensional shape with six equal square faces, eight vertices, and twelve edges.

In the cube shown, each face forms a square and all angles between the faces are right angles.

Conclusion

Geometry is a fundamental area of mathematics that is not only about drawing and measuring but also about understanding the spatial relationship between different shapes and forms. It equips us with the tools to deduce properties and solve complex problems in various subjects.


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