Insights into Mathematics Courses

Spherical and Elliptic Geometries - An Introduction

An introduction to spherical geometry using great circles, antipodal points, spherical triangles, and circles on the sphere, illustrated with physical models.

• YouTube
• 32 minutes
• Self-Paced
• Free Video

Decimal Numbers - Arithmetic and Geometry Math Foundations

A careful mathematical study of decimal notation, including how decimal expansions represent fractions and how to convert them systematically.

• YouTube
• 28 minutes
• Self-Paced
• Free Video

The Fundamental Group - Algebraic Topology

Learn how homotopic paths and based loops combine to define fundamental groups, with examples from the disk and circle.

• YouTube
• 43 minutes
• Self-Paced
• Free Video

Knots and Surfaces I - Algebraic Topology - NJ Wildberger

An introduction to knot theory using Reidemeister moves and basic invariants including crossing number, linking number, and the Alexander–Conway polynomial.

• YouTube
• 52 minutes
• Self-Paced
• Free Video

The Two-Holed Torus and 3-Crosscaps Surface - Algebraic Topology - NJ Wildberger

Explore hyperbolic structures on a two-holed torus and a 3-crosscaps surface through hexagon and octagon models and plane tessellations.

• YouTube
• 39 minutes
• Self-Paced
• Free Video

More Applications of Winding Numbers - Algebraic Topology - NJ Wildberger

Uses the degree of a map from the circle to prove results about disk retractions, fixed points, and antipodes.

• YouTube
• 27 minutes
• Self-Paced
• Free Video

Non-Orientable Surfaces - The Mobius Band - Algebraic Topology

Explore non-orientable surfaces through the Möbius band, its single edge, chirality, crosscaps, and deformation into related models.

• YouTube
• 42 minutes
• Self-Paced
• Free Video

Polyhedra and Euler's Formula - Algebraic Topology

Explore the five Platonic solids and prove Euler’s formula for polyhedra using triangulation and flow lines on a sphere.

• YouTube
• 46 minutes
• Self-Paced
• Free Video

Two-Dimensional Objects - The Torus and Genus - Algebraic Topology

Introduces cylinders, tori, and n-holed tori, defining surface genus and examining plane coverings and their associated translation groups.

• YouTube
• 50 minutes
• Self-Paced
• Free Video

Number Systems Throughout History - Elementary Mathematics Explained

Explore how Babylonian, Egyptian, Chinese, Greek, and Roman cultures developed number systems, from primitive counting to the Hindu-Arabic system.

• YouTube
• 31 minutes
• Self-Paced
• Free Video

The Triple Quad Formula in Universal Hyperbolic Geometry

Explores and proves the Triple quad formula in universal hyperbolic geometry, relating quadrances through a challenging polynomial identity.

• YouTube
• 39 minutes
• Self-Paced
• Free Video

Pythagoras' Theorem in Universal Hyperbolic Geometry

Explore the hyperbolic version of Pythagoras’ theorem through quadrance, planar coordinates, examples, exercises, and a proof.

• YouTube
• 36 minutes
• Self-Paced
• Free Video

Quadrance and Spread - Universal Hyperbolic Geometry

Explore quadrance and spread as algebraic metrics, then compare rational trigonometry in Euclidean affine geometry with universal hyperbolic geometry.

• YouTube
• 36 minutes
• Self-Paced
• Free Video

The J Function, SL(2) and the Jacobi Identity - Universal Hyperbolic Geometry

Connects the J function and 2×2 matrix Lie brackets to the Jacobi identity, yielding a projective proof of a triangle’s orthocenter.

• YouTube
• 42 minutes
• Self-Paced
• Free Video

Parallels and the Double Triangle - Universal Hyperbolic Geometry

Explores parallels in Universal Hyperbolic Geometry through double-triangle constructions and associated perspective centers.

• YouTube
• 30 minutes
• Self-Paced
• Free Video