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 ItselfMost students encounter geometry as a set of rules for measuring angles, calculating areas, and proving theorems about triangles and circles. But Modern Geometry is something entirely different. It's not just about shapes—it's about the structure of space, the logic of axioms, and the mathematical systems that define how we understand the world.This course is a deep and intellectually rich journey into the axiomatic 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 the Axiomatic Method, the formal process by which entire mathematical worlds are built from a handful of assumptions. From there, you'll explore models of geometry, including finite systems like Fano and Young geometries, and discover how different sets of axioms lead to radically different geometric realities.You'll move beyond the familiar Euclidean framework into Incidence Geometry, Affine Geometry, and the vast landscape of Non-Euclidean Geometry—including Spherical, Hyperbolic, Projective, and Elliptic 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 the Betweenness Axioms, Ordered Geometry, and the logic behind congruence, angle relationships, and triangle theorems—not as static facts, but as consequences of deeper structural rules. You'll see how parallel postulates shape entire geometries, how defect and area behave in curved spaces, and how duality and polarity redefine relationships in elliptic systems.Why Take This Course?Rigorous yet accessible explanations of abstract conceptsVisual and logical walkthroughs of geometric models and theoremsHistorical and philosophical context for modern geometric developmentsReal mathematical reasoning—not just memorizationLifetime access, downloadable resources, and a certificate of completion
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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