Graphing Polynomials - Arithmetic and Geometry Math Foundations
Learn to graph polynomials using integer and rational points, explore quartic polynomials, and understand the foundations of mathematical graphing. Gain insights into Cartesian coordinates and tangent lines.
Laws of Arithmetic Using Geometry - Elementary Mathematics Explained - Lecture 5
Explore geometric interpretations of arithmetic laws using translations and reflections in the grid plane, enhancing understanding of commutativity in addition and multiplication.
Fundamental Group of Surfaces - Multiplication and Properties
Explores fundamental group of surfaces, proving loop multiplication forms a group. Covers torus and projective plane, with examples and a problem on describing fundamental groups.
Knots and Surfaces - Seifert Surfaces in Algebraic Topology - Lecture 2
Explore Seifert surfaces, their construction for knots like the trefoil, and learn about Euler numbers for surfaces with boundaries in this advanced algebraic topology lecture.
Sphere Geometry and Complex Analysis - Algebraic Topology Lecture 4
Explore advanced concepts in sphere geometry, including inversive geometry, Riemann sphere, complex analysis, and projective lines, enhancing understanding of algebraic topology.
The Geometry of Surfaces - Algebraic Topology Lecture 20
Explore the connection between 2D surfaces and classical geometries, focusing on isometries and introducing hyperbolic geometry through circle inversions in the Beltrami Poincare disk model.
Classification of Combinatorial Surfaces - Lecture 2
Explores the classification of two-dimensional surfaces using polygon reduction techniques. Covers cutting, pasting, and creating crosscaps and handles to achieve standard forms in algebraic topology.
Rational Curvature of a Polytope - Algebraic Topology Lecture 15
Introduces a new rational definition of polytope curvature, using turn-angle to connect curvature with Euler number. Explores opposite cones and applies Harriot's theorem on spherical triangles.
Duality for Polygons and the Fundamental Theorem of Algebra - Algebraic Topology Lecture 12
Explore duality in polygons and a rational proof of the Fundamental Theorem of Algebra using winding numbers. Gain insights into algebraic topology concepts and their applications.
Rational Curvature, Winding, and Turning in Algebraic Topology - Lecture 11
Explores rational curvature, turn angles, and winding numbers in algebraic topology. Introduces new geometric concepts for describing polygons and planar curves, emphasizing practical applications in mathematics.
Rational Curvature of Polytopes and the Euler Number - Algebraic Topology Lecture 16
Explore the relationship between polyhedron curvature and Euler number using rational formulation. Learn about Harriot's theorem and analyze various polyhedra, including tetrahedron, cube, and dodecahedron.
The Ham Sandwich Theorem and the Continuum - Algebraic Topology Lecture 14
Explore the Borsuk-Ulam theorem, Ham Sandwich theorem, and their applications in algebraic topology. Learn how these concepts relate to continuous maps and homeomorphisms in affine spaces.
Applications of Euler's Formula and Graphs in Algebraic Topology
Explore Euler's formula applications to Platonic solids, polyhedra, and higher dimensions. Learn about planar graphs and delve into advanced geometric concepts in algebraic topology.
Counting Using the Grid Plane - Elementary Mathematics Explained - Lecture 1
Explore geometry using graph paper, introducing grid points, segments, and rectangles. Learn to measure, count, and tile shapes while practicing arithmetic in a geometric context.
Trigonometric Laws in Hyperbolic Geometry Using Geometer's Sketchpad - Universal Hyperbolic Geometry 26
Explore trigonometric laws in hyperbolic geometry using Geometer's Sketchpad. Learn about Pythagoras' theorem, Triple quad formula, Spread law, and Cross law with practical examples and custom tools.
Geometer's Sketchpad and Circles in Universal Hyperbolic Geometry - Lecture 25
Explore Geometer's Sketchpad for dynamic illustrations in universal hyperbolic geometry. Learn custom tools, construct point duals, and discover various circles, including equidistant curves, in hyperbolic space.
Pure and Applied Geometry - Understanding the Continuum - Universal Hyperbolic Geometry 20
Explores the distinction between pure and applied geometry, challenging traditional views on real numbers and the continuum. Offers a rational number-based approach to understanding geometric concepts and continuums.
Reflections in Hyperbolic Geometry - Universal Hyperbolic Geometry Lecture 14
Explore reflections in hyperbolic geometry, comparing with Euclidean counterparts. Discover unique properties and surprising connections to circle geometry in this advanced mathematical exploration.
Canonical Structures Inside the Platonic Solids - Tetrahedron Symmetry - Lecture 49
Explore the hidden symmetries within the tetrahedron, uncovering a surprising three-fold structure and its connection to group theory in this insightful mathematical journey.
Teaching Connections Between Algebra and Geometry - Part 2
Exploring connections between algebra and geometry, including Cartesian geometry, conics, projective geometry, triangle geometry, and rational trigonometry for a fresh approach to math education.
Applications of Rational Spherical Trigonometry - Universal Hyperbolic Geometry Lecture 43
Exploring rational spherical trigonometry through a specific spherical triangle example, demonstrating its connection to hyperbolic geometry and three-dimensional space.