Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Geometry ↓


Non-Euclidean Geometry


In the world of geometry, there exists a fascinating and complex universe known as non-Euclidean geometry. This field explores scenarios where the familiar rules and principles of Euclidean geometry do not apply. Non-Euclidean geometry emerged as a significant discovery in the history of mathematics, challenging classical notions and providing new insights into the shape and structure of space.

Understanding the basics

Before delving into non-Euclidean geometry, it's important to know what Euclidean geometry is. Named after the ancient Greek mathematician Euclid, this type of geometry deals with the properties and relationships of points, lines, surfaces, and solids in a flat, two-dimensional space. Euclid's most famous work, The Elements, stated five fundamental principles that form the basis of Euclidean geometry:

  1. A line can be drawn joining any two points.
  2. A finite line can be extended indefinitely in a straight line.
  3. A circle can be drawn with any center and radius.
  4. All right angles are congruent.
  5. If two lines are drawn in such a way that they intersect a third line such that the sum of the interior angles on one side is less than two right angles, then if the two lines are extended to a sufficient distance they will necessarily intersect each other on that side. (Parallel Postulate)

Parallel theory

The fifth principle, known as the parallel principle, may seem a little complicated, but plays a key role in distinguishing Euclidean geometry from non-Euclidean geometry. Essentially, the principle states that through any given point that is not on a line, there is a line parallel to that line. This idea is so ingrained in Euclidean geometry that mathematicians for centuries have considered it a universal truth.

The birth of non-Euclidean geometry

Non-Euclidean geometry emerged when mathematicians began to question the necessity of the parallel principle. This quest led to the creation of alternative geometric structures where the parallel principle does not apply. These include:

  • Hyperbolic geometry: Here, through a given point that is not on a line, there are infinitely many lines that do not intersect the given line. This creates a universe with constant negative curvature.
  • Elliptic geometry: In this structure, through a point that is not on a line, there is no line that does not intersect the given line. Think of it as constant positive curvature.

Visual example: Euclidean vs. hyperbolic lines

Euclidean parallel Hyperbolic line Hyperbolic line

Exploring hyperbolic geometry

Hyperbolic geometry imagines a universe where the interior angles of triangles are always less than 180 degrees. You can visualize hyperbolic space as a saddle-shaped surface, where lines move away from each other as they expand. This model rose to prominence through the works of Nikolai Lobachevsky and János Bolyai.

Some interesting properties emerge in hyperbolic geometry:

  • Triangle Angle Sum: The sum of the angles in a triangle is less than 180 degrees.
  • Parallel Lines: From a given external point, more than one line can be drawn parallel to a given line.
Triangle Angle Sum: A + B + C < 180°

Visualization of elliptic geometry

Elliptic geometry, on the other hand, invokes a sphere where lines eventually converge. A classic way to see this is to imagine the surface of a sphere. Here every "line" curves back to meet itself, and the sum of the angles in a triangle is greater than 180 degrees. This ellipse was intensively developed by mathematicians such as Bernhard Riemann and occurs frequently in models such as positively curved space.

As key effects of elliptic geometry, we note:

  • Triangle Angle Sum: The sum of the angles in a triangle is more than 180 degrees.
  • Parallel Lines: There are no parallel lines as all lines eventually cross each other.
Triangle Angle Sum: A + B + C > 180°

Implications and applications

Non-Euclidean geometry has far-reaching implications beyond abstract mathematics. It forms the basis for relativistic physics, in particular the theory of general relativity developed by Albert Einstein. According to this theory, space itself is not Euclidean, and the presence of mass and energy warps space-time into non-Euclidean configurations, affecting the motion of objects in space.

Non-Euclidean geometry also has applications in various other fields:

  • Art and Architecture: Artists and architects leverage non-Euclidean concepts to create mind-blowing structures and visuals.
  • Navigation: Ellipsoidal geometry concepts are essential for understanding the curvature of the Earth in global navigation systems.
  • Computer science: Algorithms for processing large amounts of data often benefit from understanding hyperbolic spaces, where data tend to cluster more naturally.

Further exploration into models

In addition to the hyperbolic and elliptic models, other constructions allow the visualization and exploration of non-Euclidean geometry. These include the Poincare disk model and the Beltrami-Klein model. These models help guide our understanding by providing concrete ways to explore the infinite complexity and beauty of non-Euclidean spaces.

Mathematicians and scientists explore these models to better understand their properties. For example, the Poincare disk model represents the entire hyperbolic plane within a finite circle, which maintains angles but distorts distances. One can draw geodesics (the equivalent of straight lines in non-Euclidean spaces) as arcs that look very different from straight lines in Euclidean terms.

Conclusion

Today, non-Euclidean geometry remains an important area of mathematical science where imaginative theories meet practical applications, reshaping our understanding of the nature of the universe. Breaking out of traditional theories, non-Euclidean constructions challenge established views and continue to inspire unprecedented breakthroughs in a variety of disciplines. As we move into more complex areas of reality, the importance of non-Euclidean geometry in expanding the boundaries of human understanding cannot be underestimated.


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Non-Euclidean Geometry

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