Modern Geometry: A Visual and Logical Approach

A Journey Through Axioms, Models, and Non-Euclidean Worlds

What you'll learn
- Understand the Axiomatic Method and how mathematical systems are built from foundational assumptions
- Explore and compare models of axiomatic systems, including isomorphic structures and finite geometries
- Analyze Fano, Young, and Incidence Geometries, and evaluate the independence of their axioms
- Investigate parallel postulates and their alternatives, leading to Affine and Non-Euclidean geometries
- Study the structure and properties of Spherical, Hyperbolic, Projective, and Elliptic geometries
- Learn the Betweenness Axioms, Plane Separation, and the logic of Ordered Geometry
- Apply theorems involving angles, rays, and triangles within ordered and congruence geometries
- Master the Congruence Axioms for segments and angles, including triangle congruence criteria
- Explore advanced geometric constructs like Saccheri and Lambert quadrilaterals, angle of parallelism, and defect
- Understand how area and distance are definExamine Continuity Axioms, metric axioms, and the measurement of segmened and behave in hyperbolic and elliptic spaces
- Discover the principles of Projective Geometry, including duality and polar relationships
- nalyze elliptic polygons, lunes, and the obtuse angle hypothesis in elliptic geometry

Requirements
- A basic understanding of high school geometry (points, lines, angles, triangles)
- Some exposure to algebraic thinking and set notation is helpful but not required
- Curiosity about how geometry works beyond Euclid

Description
Explore the Geometry That Goes Beyond Triangles and Circles—Into the Logical Foundations of Space Itself

Most students encounter geometry as a set of rules for measuring angles, calculating areas, and proving theorems about triangles and circles. ButModern Geometryis something entirely different. It’s not just about shapes—it’s about thestructure of space, thelogic of axioms, and themathematical systemsthat define how we understand the world.

This course is a deep and intellectually rich journey into theaxiomatic foundations of geometry, where we don’t just accept the rules—we question them, reconstruct them, and explore what happens when we change them. You’ll begin by learning theAxiomatic Method, the formal process by which entire mathematical worlds are built from a handful of assumptions. From there, you’ll exploremodels of geometry, including finite systems likeFano and Young geometries, and discover how different sets of axioms lead to radically different geometric realities.

You’ll move beyond the familiar Euclidean framework intoIncidence Geometry,Affine Geometry, and the vast landscape ofNon-Euclidean Geometry—includingSpherical,Hyperbolic,Projective, andElliptic geometries. These aren’t just theoretical curiosities—they’re essential to understanding modern physics, computer graphics, and the very nature of space and time.

Along the way, you’ll study theBetweenness Axioms,Ordered Geometry, and the logic behindcongruence,angle relationships, andtriangle theorems—not as static facts, but as consequences of deeper structural rules. You’ll see howparallel postulatesshape entire geometries, howdefect and areabehave in curved spaces, and howduality and polarityredefine relationships in elliptic systems.

Why Take This Course?

Rigorous yet accessibleexplanations of abstract concepts

Visual and logical walkthroughsof geometric models and theorems

Historical and philosophical contextfor modern geometric developments

Real mathematical reasoning—not just memorization

Lifetime access, downloadable resources, and a certificate of completion

Who this course is for:
- University students studying mathematics, physics, or philosophy
- Educators seeking a deeper understanding of geometric foundations
- Anyone curious about how geometry evolves beyond Euclid
- Learners preparing for graduate-level geometry or mathematical logic
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