Lecture Description
The Triple quad formula is the second most important theorem in hyperbolic geometry (just as it is in Euclidean geometry!) It gives the relation between the three quadrances formed by three collinear points. It is a quite challenging theorem to prove: relying on a remarkable polynomial identity. It is a deformation of the Euclidean Triple quad formula, and happens to agree in form with the Euclidean Triple spread formula. We sketch an argument for this seeming coincidence. This is one of the more algebraically challenging of the videos in this series. CONTENT SUMMARY: pg 1: @00:11 Triple quad formula; example; suggested exercise @07:12 pg 2: @07:31 Understanding the Triple quad formula; comparing the corresponding formulas in affine/projective geometry; notice the direction of the arrows @11:10 pg 3: @11:39 Triple spread formula from affine RT; spread in vector notation ; spread in vector notation @13:50 pg 4: @15:35 Euclidean dot products; Relativistic dot products pg 5: @19:40 Why the Triple quad formula holds; note on 4 main laws of hyperbolic trigonometry @22:06 pg 6: @23:38 Triple quad formula; proof pg 7: @27:44 Triple quad formula; proof continued; a small miracle @30:40 ; remark about proof @32:25 ; encouragement to do algebra @33:18 pg 8: @35:28 The Triple spread function is defined; exercises 21.1,2 pg 9: @36:21 exercises 23.3,4 ; Complimentary quadrances theorem; Equal quadrances theorem (THANKS to EmptySpaceEnterprise)
Course Index
- Introduction to Universal Hyperbolic Geometry
- Apollonius and Polarity
- Apollonius and Harmonic Conjugates
- Pappus' Theorem and the Cross Ratio
- First Steps in Hyperbolic Geometry
- The Circle and Cartesian Coordinates
- Duality, Quadrance and Spread in Cartesian Coordinates
- The Circle and Projective Homogeneous Coordinates
- The Circle and Projective Homogeneous Coordinates II
- Computations with Homogeneous Coordinates
- Duality and Perpendicularity
- Existence of Orthocenters
- Theorems using Perpendicularity
- Null Points and Null Lines
- Apollonius and Polarity Revisited
- Reflections in Hyperbolic Geometry
- Reflections and Projective Linear Algebra
- Midpoints and Bisectors
- Medians, Midlines, Centroids and Circumcenters
- Parallels and the Double Triangle
- The J function, sl(2) and the Jacobi identity
- Pure and Applied Geometry: understanding the continuum
- Quadrance and Spread
- Pythagoras' Theorem in Universal Hyperbolic Geometry
- The Triple Quad Formula in Universal Hyperbolic Geometry
- Visualizing Quadrance with Circles
- Geometer's Sketchpad and Circles in Universal Hyperbolic Geometry
- Trigonometric Laws in Hyperbolic Geometry using Geometer's Sketchpad
- The Spread Law in Universal Hyperbolic Geometry
- The Cross Law in Universal Hyperbolic Geometry
- Thales' Theorem, Right Triangles and Napier's Rules
- Isosceles Triangles in Hyperbolic Geometry
- Menelaus, Ceva and the Laws of Proportion
- Trigonometric Dual Laws and the Parallax Formula
- Introduction to Spherical and Elliptic Geometries
- Introduction to Spherical and Elliptic Geometries II
- Areas and Volumes for a Sphere
- Classical Spherical Trigonometry
- Perpendicularity, Polarity and Duality on a Sphere
- Parametrizing and Projecting a Sphere
- Rational Trigonometry: An Overview
- Rational Trigonometry in Three Dimensions
Course Description
This is a complete and relatively elementary course explaining a new, simpler and more elegant theory of non-Euclidean geometry; in particular hyperbolic geometry. It is a purely algebraic approach which avoids transcendental functions like log, sin, tanh etc, relying instead on high school algebra and quadratic equations. The theory is more general, extending beyond the null circle, and connects naturally to Einstein's special theory of relativity. This course is meant for mathematics majors, bright high school students, high school teachers, engineers, scientists, and others with an interest in mathematics and some basic algebraic skills. NJ Wildberger is also the developer of Rational Trigonometry: a new and better way of learning and using trigonometry. Look up for his course in order to familiarize with this new development. He also has recorded very organized and detailed lecture series on Algebraic Topology, History of Mathematic, Linear Algebra, and more.