The Triple Quad Formula in Universal Hyperbolic Geometry 
The Triple Quad Formula in Universal Hyperbolic Geometry
by UNSW / N.J. Wildberger
Video Lecture 25 of 42
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Date Added: January 20, 2015

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

  1. Introduction to Universal Hyperbolic Geometry
  2. Apollonius and Polarity
  3. Apollonius and Harmonic Conjugates
  4. Pappus' Theorem and the Cross Ratio
  5. First Steps in Hyperbolic Geometry
  6. The Circle and Cartesian Coordinates
  7. Duality, Quadrance and Spread in Cartesian Coordinates
  8. The Circle and Projective Homogeneous Coordinates
  9. The Circle and Projective Homogeneous Coordinates II
  10. Computations with Homogeneous Coordinates
  11. Duality and Perpendicularity
  12. Existence of Orthocenters
  13. Theorems using Perpendicularity
  14. Null Points and Null Lines
  15. Apollonius and Polarity Revisited
  16. Reflections in Hyperbolic Geometry
  17. Reflections and Projective Linear Algebra
  18. Midpoints and Bisectors
  19. Medians, Midlines, Centroids and Circumcenters
  20. Parallels and the Double Triangle
  21. The J function, sl(2) and the Jacobi identity
  22. Pure and Applied Geometry: understanding the continuum
  23. Quadrance and Spread
  24. Pythagoras' Theorem in Universal Hyperbolic Geometry
  25. The Triple Quad Formula in Universal Hyperbolic Geometry
  26. Visualizing Quadrance with Circles
  27. Geometer's Sketchpad and Circles in Universal Hyperbolic Geometry
  28. Trigonometric Laws in Hyperbolic Geometry using Geometer's Sketchpad
  29. The Spread Law in Universal Hyperbolic Geometry
  30. The Cross Law in Universal Hyperbolic Geometry
  31. Thales' Theorem, Right Triangles and Napier's Rules
  32. Isosceles Triangles in Hyperbolic Geometry
  33. Menelaus, Ceva and the Laws of Proportion
  34. Trigonometric Dual Laws and the Parallax Formula
  35. Introduction to Spherical and Elliptic Geometries
  36. Introduction to Spherical and Elliptic Geometries II
  37. Areas and Volumes for a Sphere
  38. Classical Spherical Trigonometry
  39. Perpendicularity, Polarity and Duality on a Sphere
  40. Parametrizing and Projecting a Sphere
  41. Rational Trigonometry: An Overview
  42. 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.

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