Universal Hyperbolic Geometry

Video Lectures

Displaying all 42 video lectures.
I. Introduction to Universal Hyperbolic Geometry
Lecture 1
Introduction to Universal Hyperbolic Geometry
Play Video		 Introduction to Universal Hyperbolic Geometry
This is the introductory lecture to a series on hyperbolic geometry which introduces a radically new and improved way of treating the subject, making it more algebraic and logical, with improved computational power and many new theorems. In this lecture we summarize the differences between this UNIVERSAL HYPERBOLIC GEOMETRY and traditional courses taught at universities. We briefly review some of the framework introduced by Lobachevsky, Bolyai and Gauss.This course will open up a new area of investigation into geometry, accessible to a broad audience. Let's return to those days where high school teachers and ordinary lay people could contribute to the discovery of new and beautiful facts.Norman Wildberger is also the discoverer of Rational Trigonometry, an important new direction for classical trigonometry (and which really ought to be revolutionizing mathematics education!!) His YouTube site Insights into Mathematics at user: njwildberger also contains series on MathFoundations, History of Mathematics, LinearAlgebra, Rational Trigonometry and even one called Elementary Mathematics (K-6) Explained.
Lecture 2
Apollonius and Polarity
Play Video		 Apollonius and Polarity
This is the start of a new course on hyperbolic geometry that features a revolutionary simplifed approach to the subject, framing it in terms of classical projective geometry and the study of a distinguished circle. This subject will be called Universal Hyperbolic Geometry, as it extends the subject to arbitrary fields, as well as to the outside of the light cone/null circle.We begin by going back to Apollonius of Perga (in present day Turkey, not Italy!) and his understanding of the crucial role of polarity in studying conics, in particular the circle. Given a fixed circle, to each point in the plane we associate a line called the polar, and conversely to a line we associated a point called its pole. This duality is all important for hyperbolic geometry.CONTENT SUMMARY: conics @00:00 the circle @5:00 Thales thm @05:20 Greek m'ment @06:20 polarity, pole of A and polar of a @09:06 def' of polarity (projective def') @14:40 Polar independence Theorem @19:13 projective def' of polarity @23:34 polar of a point inside the circle @25:10 3-way symetry @26:40 hands_on experience @27:52 pole, polar starting with four points on circle @29:12 Quadrangle and quadralateral @29:26 Polar duality thm @31:30 The distinguished circle @37:40 Pole of a line thm @38:00 (THANKS to EmptySpaceEnterprise)
Lecture 3
Apollonius and Harmonic Conjugates
Play Video		 Apollonius and Harmonic Conjugates
Apollonius introduced the important idea of harmonic conjugates, concerning four points on a line. He showed that the pole polar duality associated with a circle produces a family of such harmonic ranges, one for every line through the pole of a line. Harmonic ranges also occur in the context of vertex bisectors, as combinations of vectors, and associated with the sides of a quadrangle. CONTENT SUMMARY: polarity holds for general conics @03:58 geometers sketchpad in use @05:02 How to find the polar of a null point @05:50 Harmonic conjugates @08:58 discussion of various types of geometry @12:17 More on harmonic conjugates @16:37 examples of harmonic ranges and harmonic ranges theorem @24:00 Harmonic pencils and Harmonic bisectors theorem @28:34 Harmonic vector combinations theorem @32:37 Harmonic quadrangle theorem @34:34
Lecture 4
Pappus' Theorem and the Cross Ratio
Play Video		 Pappus' Theorem and the Cross Ratio
Pappus' theorem is the first and foremost result in projective geometry. Another of his significant contributions was the notion of cross ratio of four points on a line, or of four lines through a point. We discuss various important results: such as the Cross ratio theorem, asserting the invariance of the cross ratio under a projection, and Chasles theorem for four points on a conic. We show that the notion of cross ratio also works for four concurrent lines. CONTENT SUMMARY: Pappus' theorem @00:52 cross ratio @02:46 cross ratio transformation theorem @11:08 cross ratio theorem @13:54 Chasles theorem @16:19 The cross ratio is the most important invariant in projective geometry
Lecture 5
First Steps in Hyperbolic Geometry
Play Video		 First Steps in Hyperbolic Geometry
This video outlines the basic framework of universal hyperbolic geometry---as the projective study of a circle, or later on the projective study of relativistic geometry. Perpendicularity is defined in terms of duality, the pole-polar correspondence introduced by Apollonius, and we explain that the three altitudes of a triangle meet in a point- the orthocenter H. The basic measurements of quadrance and spread in this geometry arise from the cross ratio of suitable points and lines. We state the main formulas: Pythagoras' theorem, the Triple quad formula, Pythagoras' dual theorem, the Triple spread formula, the Spread law and the Cross law and its dual. These are closely related to, but different from the corresponding laws in Rational Trigonometry. CONTENT SUMMARY: notion of perpendicularity @04:48 Perpendicularity via duality @05:33 Do the altitudes of a triangle meet in a point? @10:54 Quadrance: m'ment beween points @15:14 exercise @18:41 remark on Beltrami-Klein model @19:11 Pythagoras' theorem and Triple quad formula @20:30 Spread: m'ment between lines and quadrance spread duality theorem @23:29 Remark on Beltrami-Klein model @26:45 Pythagoras' dual theorem @28:43 Main formulas for triangles that involve both quadrances and preads @31:13 (THANKS to EmptySpaceEnterprise)
Lecture 6
The Circle and Cartesian Coordinates
Play Video		 The Circle and Cartesian Coordinates
This video introduces basic facts about points, lines and the unit circle in terms of Cartesian coordinates. A point is an ordered pair of (rational) numbers, a line is a proportion (a:b:c) representing the equation ax+by=c, and the unit circle is x^2+y^2=1. With this notation we determine the line joining two points, the condition for colinearity of three points (using the determinant), the point where two non-parallel lines meet and the the condition for concurrency of three lines. We state the rational parametrization of the circle and show that a line meets a circle in either 1,2 or 0 points. These theorems are fundamental in applying Cartesian coordinates to Euclidean geometry and also, as we shall see, to hyperbolic geometry. CONTENT SUMMARY: Line through two points theorem @04:31 Collinear points theorem @6:56 Determinants @08:09 Number system we will use @11:28 Concurrent lines theorem @15:10 Affinely parallel lines and Point on two lines theorem @16:42 Parameterization of unit circle theorem @19:33 experience with parameterization of unit circle @24:23 Meets of line and circle theorem @25:52 (THANKS to EmptySpaceEnterprise)
Lecture 7
Duality, Quadrance and Spread in Cartesian Coordinates
Play Video		 Duality, Quadrance and Spread in Cartesian Coordinates
In this video we connect the notions of duality, quadrance and spread to the Cartesian coordinate framework, giving explicit formulas for the dual of a point, the quadrance between points, and the spread between lines in terms of coordinates.The proofs involve some useful preliminary results on lines formed by two points on the unit circle, and the meets of two such lines. Some careful algebraic book-keeping is required, along with some pleasant identities. This is a challenging lecture, so take it slowly. CONTENT SUMMARY: Duality in coords theorem @06:00 Line through two null points theorem. @09:21 Meet of interior lines theorem @15:34 Return to Duality in coords theorem @20:00 Perpendicularity in co-ordinates theorem. @30:50 Quadrance in co-ordinates theorem @34:24 Spread in coordinates theorem @46:30 Remark on challenge of parallel lines @49:03 (THANKS to EmptySpaceEnterprise)
Lecture 8
The Circle and Projective Homogeneous Coordinates
Play Video		 The Circle and Projective Homogeneous Coordinates
Universal hyperbolic geometry is based on projective geometry. This video introduces this important subject, which these days is sadly absent from most undergrad/college curriculums. We adopt the 19th century view of a projective space as the space of one-dimensional subspaces of an affine space of one higher dimension. Thus the projective line is viewed as the space of lines through the origin in two dimensional space, while the projective plane deals with one dimensional and two dimensional subspaces of a 3 dimensional affine xyz space (called respectively projective points and projective lines). This relates to Renassiance artists attempts to render perspectives correctly; we illustrate by looking at a parabola in a somewhat novel way.The usual two dimensional view of the projective plane emerges by intersecting with the plane z=1 in the ambient x,y,z space. This way the circle is the two dimensional representation of a cone: a view relating back to the ancient Greeks. CONTENT SUMMARY: Projective Geometry: Affine and projective geometry @00:23 Perspective and points at infinity @08:00 example of affine vs projective view @13:11 One dimensional geometry as starting point @17:48 affine space and vector space @22:30 page change @22:28 One dimensional projective geometry @26:27 page change @31:04 (THANKS to EmptySpaceEnterprise)
Lecture 9
The Circle and Projective Homogeneous Coordinates II
Play Video		 The Circle and Projective Homogeneous Coordinates II
Universal hyperbolic geometry is based on projective geometry. This video introduces this important subject, which these days is sadly absent from most undergrad/college curriculums. We adopt the 19th century view of a projective space as the space of one-dimensional subspaces of an affine space of one higher dimension. Thus the projective line is viewed as the space of lines through the origin in two dimensional space, while the projective plane deals with one dimensional and two dimensional subspaces of a 3 dimensional affine xyz space (called respectively projective points and projective lines). This relates to Renassiance artists attempts to render perspectives correctly; we illustrate by looking at a parabola in a somewhat novel way.The usual two dimensional view of the projective plane emerges by intersecting with the plane z=1 in the ambient x,y,z space. This way the circle is the two dimensional representation of a cone: a view relating back to the ancient Greeks. For more information on projective geometry, see WT31-WT41 in my WildTrig series. CONTENT SUMMARY: 2 dimensional geometry the arena of hyperbolic geometry @00:12 page change: adopting a viewing plane @08:07 The circle in projective homogeneous coords.(very important picture in mathematics and physics: special relativity) at the heart of hyperbolic geometry @16:14
Lecture 10
Computations with Homogeneous Coordinates
Play Video		 Computations with Homogeneous Coordinates
We discuss the two main objects in hyperbolic geometry: points and lines. In this video we give the official definitions of these two concepts: both defined purely algebraically using proportions of three numbers. This brings out the duality between points and lines, and connects with our 3 dimensional picture of lines and planes in the space, or our 3 dimensional picture of the projective plane. We derive several important theorems: the formulas for the lines joining two points, and dually the point where two lines meet. We introduce the J function for making such computations. CONTENT SUMMARY: Lines and planes through the origin as points and lines on the viewing plane @00:01 A projected line on the viewing plane @05:32 Official definitions: hyperbolic point, hyperbolic line @08:48 examples: plot points, plot lines @14:29 find a line given 2 points @21:55 A graphical llustration: @25:15 page change: solution to prob. on previous page @26:28 Join of two points theorem @28:50 Meet of two lines theorem @31:51 Duality rinciple @34:46 formulas have application to cartesian geometry 37:38 meet of lines app. to cartesian geom. @40:03 hyperbolic Geometry is a computational subject memorize j function @42:18 (THANKS to EmptySpaceEnterprise)
Lecture 11
Duality and Perpendicularity
Play Video		 Duality and Perpendicularity
Perpendicularity in universal hyperbolic geometry is defined in terms of duality. One big difference with classical HG is that points can also be perpendicular, not just lines. Once we have perpendicularity, we can define altitudes. We also state the collinear points theorem and concurrent lines theorem, using homogeneous coordinates and determinants. CONTENT SUMMARY: pg1: @00:04 pg2: examples of viewing plane (good for study) @02:44 pg3: exercises: point duality theorem line duality theorem @09:03 pg4: proof (of point duality theorem) @10:14 pg5: Perpendicularity: lines and points @11:48 pg6: examples of perpendicular points and perpendicular lines @13:48 pg7: Altitudes of triangles @20:22 pg8: Concurrent lines theorem and proof @27:48 pg9: Collinear points theorem @30:09 (THANKS to EmptySpaceEnterprise)
Lecture 12
Existence of Orthocenters
Play Video		 Existence of Orthocenters
In classical hyperbolic geometry, orthocenters of triangles do not in general exist. Here in universal hyperbolic geometry, they do. This is a crucial building block for triangle geometry in this subject. The dual of an orthocenter is called an ortholine---also not seen in classical hyperbolic geometry!This lecture also introduces a number of basic important definitions: that of side, vertex, couple, triangle, trilateral. We also introduce Desargues theorem and use it to define the polar of a point with respect to a triangle. The lecture culminates in the definition of the orthic line, orthostar and ortho-axis of a triangle. The ortho-axis will prove to be the most important line in hyperbolic triangle geometry. CONTENT SUMMARY: pg 1: @00:15 Orthocenter of a triangle Basic definitions: a side, a vertex pg 2: @03:49 more definitions: a couple, a triangle, a trilateral pg 3: @06:37 A triangle has points, lines and vertices; a dual triangle; pg 4: @10:45 A dual couple; Altitude line theorem; pg 5: @12:52 Altitude point theorem; pg 6: @15:02 Orthocenter theorem; examples of no orthocenter in Poincare model @17:37 pg 7: @18:24 Proof of Orthocenter theorem pg 8: @29:43 The Ortholine theorem; the dual to the orthocenter theorem;examples from geometers sketchpad diagrams @31:40 pg 9: @32:28 Desargues theorem (a foundational theorem of projective geometry) pg 10: @34:35 Establishing the polar of a point with respect to a triangle; cevian lines; the Desargue polar pg 11: @39:07 Orthic axis, orthostar and ortho-axis; examples in Geometers Sketchpad @42:56 (THANKS to EmptySpaceEnterprise)
Lecture 13
Theorems using Perpendicularity
Play Video		 Theorems using Perpendicularity
We review how perpendicularity in hyperbolic geometry comes from duality, and then introduce duality for triangles and trilaterals. Then we discuss the orthic triangle and its dual, defining the important Base center point, which lies on the ortho-axis of a triangle, and is also somewhat remarkably the orthocenter of a triangle of orthocenters formed from the bases of the triangle. We introduce a general strategy for approaching theorems in the subject, and introduce our Standard Triangle 1: which will be used to algebraically illustrate many concepts and as an arena for numerical investigations. CONTENT SUMMARY: pg 1: @00:09 Review of basic algebraic framework; pg 2: @04:38 Dual triangle; triangle, associated trilateral, dual trilateral, associated dual triangle; constructing altitudes to find orthocenter pg 3: @09:30 orthocenter and orthic triangle; dual of orthic triangle; Base center theorem @11:46 ; point of perspectivity; base center of triangle pg 4: @13:39 Base ortho-axis theorem; importance of ortho-axis pg 5: @15:17 Three steps to understanding theorems; GPS pictures illustrating base center theorem @18:39 pg 6: @19:30 standard triangle #1 pg 7: @23:46 computing altitudes with standard triangle #1 pg 8: @26:58 computing orthic lines, orthic axis, ortho-axis, base center, using standard triangle #1 pg 9: @31:34 Base triple orthocenter theorem; GSP pictures of base triple orthocenter theorem @33:02 pg 10: @33:28 more (st#1) base triple orthocenter (THANKS to EmptySpaceEnterprise)
Lecture 14
Null Points and Null Lines
Play Video		 Null Points and Null Lines
Null points and null lines are central in universal hyperbolic geometry. By definition a null point is just a point which lies on its dual line, and dually a null line is just a line which passes through its dual point. We extend the rational parametrization of the unit circle to the projective parametrization of null points and null lines. And we determine the joins of null points and meets of null lines using these coordinates. CONTENT SUMMARY: pg 1: @00:09 Null points and null lines; definitions; pg 2: @5:37 Rationalparametrization of unit (null) circle; fix exceptional point; moving to projective parameterization; pg 3: @10:21 Projective parametrization of the unit circle; dual statement (projective parametrization of null lines); pg 4: @14:42 remarks to connect parametrization with linear algebra; mention of chromogeometry; pg 5: @18:12 Join of null points theorem; proof 1; pg 6: @23:31 Join of null points theorem; proof 2; pg 7: @27:40 Meet of null lines theorem; pg 8: @32:31 Introduction of Standard Triangle #2 (st2); A triply nill triangle; (THANKS to EmptySpaceEnterprise)
Lecture 15
Apollonius and Polarity Revisited
Play Video		 Apollonius and Polarity Revisited
Armed with explicit formulas for null points and null lines, along with their meets and joins, we return to the polarity of Apollonius with which we began this series. Our aim is to establish a fundamental fact that was previously stated without proof: that the dual or polar of a point can be found by two auxiliary (interior) lines and an associated quadrangle of null points. The key point is that the diagonal line formed by the (other) diagonal points of this quadrangle depend only on the original point. Our main tool is an explicit---but lengthy!---- formula for the meet of two interior lines formed by two pairs of null points. As usual we illustrate with a concrete explicit example. CONTENT SUMMARY: pg 1: @00:10 null point, null line, join of null points, meet of null lines F(t1:u1|t2:u2), f(t1:u1|t2:u2); interior line, exterior point pg 2: @04:45 drawing an interior line and exterior point pg 3: @07:09 quadrangle, quadrilateral, Apollonius, polarity pg 4: @08:36 a quadrangle of 4 null points; g function for joins and a meet; formula for the meet of 2 interior lines pg 5: @12:49 quadrangle computation example; important observation about 3 diagonal points; statement of polarity of Appolonius pg 6: @18:40 Nil quadrangle diagonals theorem, proof pg 7: @21:33 calculation showing 3 diagonal points are mutually perpendicular; pg 8: @23:51 homogeneous coordinates to affine coordinates, pole/polar corollary (THANKS to EmptySpaceEnterprise)
Lecture 16
Reflections in Hyperbolic Geometry
Play Video		 Reflections in Hyperbolic Geometry
Symmetries are crucial in studying geometry. In Euclidean geometry we have translations, rotations, reflections, dilations and also projections and perspectivities. This lecture introduces reflections into universal hyperbolic geometry. First we discuss the two different kinds of reflections (in a point or in a line) in Euclidean geometry. The hyperbolic version rests on some remarkable facts, also directly connected to the geometry of a circle. Somewhat surprisingly, reflection in a point is the same as reflection in the dual line, so the two notions agree in hyperbolic geometry. CONTENT SUMMARY: pg 1: @00:10 Reflections in hyperbolic geometry, symmetries of a geometry, types of symmetries; pg 2: @04:57 Reflections in Euclidean geometry, multiplication of transformations pg 3: @08:37 Euclidean reflections in points and in lines pg 4: @13:54 exercises pg 5: @16:35 Reflections in Universal Hyperbolic Geometry, simple and elegant generators pg 6: @19:15 Reflections in a hyperbolic setting defined; reflection in a point a; pg 7: @22:10 reflection is well defined pg 8: @23:58 a reflection also acts on lines pg 9: @26:29 example of reflections in point_a outside and its dual pg 10: @29:48 example of reflections in point_a inside the circle (THANKS to EmptySpaceEnterprise)
Lecture 17
Reflections and Projective Linear Algebra
Play Video		 Reflections and Projective Linear Algebra
Reflections are the fundamental symmetries in hyperbolic geometry. The reflection in a point interchanges any two null points on any line through the point. Using the projective parametrization of the circle, we associate to the reflecting point a 2x2 projective matrix. So we need to develop some basics about projective linear algebra: where we consider vectors and matrices but only up to scalars. CONTENT SUMMARY: pg 1: @00:10 Reflection in a is determined by its action on null points pg 2: @04:01 Reflections on null points; null point, join of null points (red, green, blue bilinear forms), a lies on L; null point reflection formula (the star formula) pg 3: @08:01 Linear algebra (in 2 dim's) in a nutshell; projective rather than affine linear algebra; pg 4: @15:22 Projective linear algebra (in 2 dim's) pg 5: @19:56 (the star formula rewrite); The projective matrix of the point a; trace and determinant; a is a null point when determinant of its projective matrix is zero; trace zero matrix pg 6: @26:37 Reflection matrix theorem; example pg 7: @30:35 Point/matrix correspondence; sl(2) Lie algebra; null point zero determinant exercise 15.1a pg 8: @33:36 exercise 15.2; Reflection matrix conjugation theorem; pg 9: 36:32 example of Reflection matrix conjugation theorem pg 10: @41:43 proof of Reflection matrix conjugation theorem pg 11: @47:20 2 Corollaries (THANKS to EmptySpaceEnterprise)
Lecture 18
Midpoints and Bisectors
Play Video		 Midpoints and Bisectors
Midpoints of sides may be defined in terms of reflections in points in hyperbolic geometry. Reflections are defined by 2x2 trace zero matrices associated to points. The case of a reflection in a null point is somewhat special. The crucial property of reflection is that it preserves perpendicularity, which then implies that reflections send lines to lines. Midpoints of a side bc can be constructed with a straightedge when they exist, and in general there are two of them! This is a big difference with Euclidean geometry. Bisectors of vertices are defined by duality. CONTENT SUMMARY: pg 1: @00:10 point/matrix correspondence; reflection matrix conjugation theorem; exercise 16.1 pg 2: @03:28 Definition of reflection of a general point pg 3: @07:32 another example; Null reflection theorem; proof (exercise 16-2) pg 4: @09:29 Matrix perpendicularity theorem; reflections as generators of isometries in hyperbolic geometry pg 5: @14:30 Reflection (preserves) perpendicularity theorem; remark about trace; proof pg 6: @18:11 reflection (preserves) lines theorem; proof; Line/point reflection notation pg 7: @21:24 exercise 16-3; Concept of Midpoint between 2 points pg 8: @24:26 Geometrical construction concerning midpoints pg 9: @28:59 Another geometrical construction concerning midpoints; Harmonic quadrangle and harmonic conjugates UHG2 revisited pg 10: @31:42 another midpoints construction pg 11: @33:34 Not all sides have midpoints; side/vertex midpoints/bisectors (THANKS to EmptySpaceEnterprise)
Lecture 19
Medians, Midlines, Centroids and Circumcenters
Play Video		 Medians, Midlines, Centroids and Circumcenters
Here we introduce basic aspects of triangle geometry into the superior framework of universal hyperbolic geometry, a purely algebraic setting valid over the rational numbers. We begin by reviewing the centroid and circumcenter in the Euclidean setting. In the hyperbolic plane, midpoints of a side don't always exist. If we consider a triangle in which each side has midpoints, there are then 6 medians, and their dual lines, called midlines here, although they play the role of perpendicular bisectors. The medians meet in 4 centroids. The midlines meet in 4 circumcenters. There are some remarkable connections between centroids and circumcenters, culminating in the z point of the triangle. Remarkably it lies on the ortho-axis, and together with the base center, orthocenter and orthostar, forms a harmonic range of points. CONTENT SUMMARY: pg 1: @00:11 Euclidean triangle centers; midpoints of sides; perpendicular bisectors; centroid and circumcenter; centroid as balancing point; midlines; circumcenter (C), centroid (G), orthocenter (H) pg 2: @04:25 Euler line (C,H,G: colinear); No analog of Euler line in hyperbolic geometry pg 3: @06:07 Midlines of a side; midlines as perps of midpoints; pg 4: @09:47 # of midpoints of a triangle; duals of midpoints; median defined; pg 5: @12:16 example of meets of medians of a triangle; circumlines; geometers sketchpad illustrations @17:50 pg 6: @18:28 Meets of medians theorem; Joins of midpoints theorem; Meets of midlines theorem; circumlines/circumcenters duality; pg 7: @20:14 construction of meets of midlines (circumcenters); prior to metrical constructions remark; remark on classical hypergeometry pg 8: @25:03 Centroid circumcenter correspondence theorem; The z_point of the triangle; remark - every triangle has a z-point whether or not it has midpoints pg 9: @29:07 z-point ortho-axis theorem; zbhs harmonic range theorem; remark- exercise request; geometer_sketchpad illustrations @32:18 (THANKS to EmptySpaceEnterprise)
Lecture 20
Parallels and the Double Triangle
Play Video		 Parallels and the Double Triangle
We discuss Euclid's parallel postulate and the confusion it led to in the history of hyperbolic geometry. In Universal Hyperbolic Geometry we define the parallel to a line through a point, NOT the notion of parallel lines. This leads us to the useful construction of the double triangle of a triangle, and various perspective centers associated to it, the x, y and z points of a triangle. The x and z point lie on the ortho-axis, the y point generally does not. CONTENT SUMMARY: pg 1: @00:11 parallel's in hyperbolic geometry pg 2: @05:55 Better definitions of parallel lines pg 3: @09:29 Construction of the parallel P to L through a; no "P is parallel to L" pg 4: @13:01 Applying parallel's to a triangle; Double triangle in Euclidean geometry; pg 5: @14:51 Example of the double trilateral and double triangle pg 6: @16:33 Construction of double triangle algebraically using st#1 pg 7: @18:43 Double triangle midpoint theorem; Double triangle perspective theorem; The center of perspectivity x_point/double_point defined pg 8: @21:08 Exercise 18-1; x-point ortho-axis theorem; shxb cross-ratio theorem. pg 9: @22:42 Second double triangle perspective theorem; y-point/second double point defined pg 10: @24:44 Double dual triangle perspective theorem; z-point revisited also called the double dual point pg 11: @26:58 zbhs harmonic range theorem; zbxh harmonic range theorem; cg illustrations @28:15; UHG18 closing remarks @28:41 (THANKS to EmptySpaceEnterprise)
Lecture 21
The J function, sl(2) and the Jacobi identity
Play Video		 The J function, sl(2) and the Jacobi identity
We review the basic connection between hyperbolic points and matrices, and connect the J function, which computes the joins of points or the meets of lines, with the Lie bracket of 2x2 matrices. This connects with the Lie algebra called sl(2) in the projective setting. The Jacobi identity then gives a new proof of the concurrence of the altitudes of a triangle, in other words the existence of the orthocenter. CONTENT SUMMARY: pg 1: @00:11 Introduction; the J function, sl(2), the Jacobi identity pg 2: @05:41 sl(2) Lie algebra in a nutshell pg 3: @10:07 Jacobi Identity; proof; simpler identity pg 4: @13:52 Projective algebra of matrices; pg 5: @20:20 Review of connection between matrices and points and lines; Projective parametrization of null circle; Important - hyperbolic points are associated to projective trace zero matrices pg 6: @24:38 Continued review; General formula for reflection; Bracket theorem; the bracket computes the J function pg 7: @28:46 proof of Bracket theorem pg 8: @34:42 The meaning of the Jacobi identity (THANKS to EmptySpaceEnterprise)
Lecture 22
Pure and Applied Geometry: understanding the continuum
Play Video		 Pure and Applied Geometry: understanding the continuum
The distinction between pure and applied geometry is closely related to the difference between rational numbers and decimal numbers. Especially when we treat decimal numbers in an approximate way: specifying rather an interval or range rather than a particular value. This gives us a way of explaining the distinction between a line meeting a circle exactly or only roughly. This video addresses a very big confusion in mathematics: the idea that `real numbers' are a proper model for the `continuum'. THEY ARE NOT!! The true foundation for mathematics rests in the rational numbers and concrete constructions made from them. So we point out some of the logical deficiencies in the usual chat about the square root of 2, or pi, or e. And show the way towards a much more sensible approach to one of the most important problems in mathematics: how to understand the hierarchy of continuums. CONTENT SUMMARY: pg 1: @00:11 Circles, lines, rational numbers, real numbers pg 2: @04:00 Errett Bishop quote; Pure Geometry and Applied Geometry compared pg 3: @05:58 Pure Geometry|rational numbers :: Applied Geometry|decimal numbers; rational number framework pg 4: @07:32 Decimal numbers pg 5: @11:40 infinite decimals; pg 6: @22:31 Applied mathematicians; rough decimal pg 7: @26:06 example; look at pixels pg 8: @30:58 rough or exact solutions of a polynomial curve, Fermat curve pg 9: @32:52 unit circle pg 10: @34:53 Continuum Problem: To understand the hierarchy of continuums (THANKS to EmptySpaceEnterprise)
Lecture 23
Quadrance and Spread
Play Video		 Quadrance and Spread
This is the first video in the second part of this series on Universal Hyperbolic Geometry (UHG), introducing algebraic definitions of the main metrical concepts: quadrance between points and spread between lines. We first review the basics of Rational Trigonometry (RT) in the Euclidean affine setting, motivating the move to the hyperbolic projective setting. The five main laws of RT are laid out and compared with the four main laws of UHG. CONTENT SUMMARY: pg 1: @00:11 Metrical notions (over rational numbers!); measurements pg 2: @03:40 Affine geometry/Projective geometry compared pg 3: @07:50 Preliminary: Rational Trigonometry in Euclidean Geometry; WildTrig series mentioned pg 4: @13:31 Further development in the Euclidean affine case; Main laws of Rational Trigonometry; 1st and 2nd most important results in mathematics @16:37 ; the most powerful law among the 5 @18:30 ; pg 5: @20:02 Trigonometry in Universal Hyperbolic Geometry; In principle one could start the series here; the main definitions pg 6: @25:45 Main laws of Hyperbolic trigonometry; njwildberger opinion @30:08 pg 7: @31:53 exercises 21-(1:5) pg 8: @33:19 exercises 21-(6:9); right triangle, dual laws; closing motivational remarks @34:28 (THANKS to EmptySpaceEnterprise!)
Lecture 24
Pythagoras' Theorem in Universal Hyperbolic Geometry
Play Video		 Pythagoras' Theorem in Universal Hyperbolic Geometry
Pythagoras' theorem in the Euclidean plane is easily the most important theorem in geometry, and indeed in all of mathematics. The hyperbolic version, stated in terms of hyperbolic quadrances, is a deformation of the Euclidean result, and is also the most important theorem of hyperbolic geometry. We review the basic measurement of quadrance (not distance!) between points. CONTENT SUMMARY: pg 1: @00:11 Pythagoras' theorem in UHG; points, point/line incidence, quadrance/cross ratio pg 2: @05:25 projecting 3-dim onto 'viewing plane' pg 3: @11:03 quadrance in planar coordinates; GSP illustrations of different quadrances in the plane @13:22 pg 4: @13:58 quadrance planar formula; note - null point restriction; zero denominator convention; example pg 5: @17:22 Pythagoras' theorem (hyperbolic version); the importance of the theorem @18:04 ; example pg 6: @22:42 exercises 22.1,2 pg 7: @24:22 The proof of Pythagoras' theorem; a small miracle @27:04 ; suggested exercise @28:21 pg 8: @29:03 The proof of Pythagoras' theorem continued from (pg 7); "That's a proof" @33:06 pg 9: @34:31 exercises 22-(3:5) (THANKS to EmptySpaceEnterprise)
Lecture 25
The Triple Quad Formula in Universal Hyperbolic Geometry
Play Video		 The Triple Quad Formula in Universal Hyperbolic Geometry
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)
Lecture 26
Visualizing Quadrance with Circles
Play Video		 Visualizing Quadrance with Circles
To really understand the fundamental concept of quadrance between points in universal hyperbolic geometry, which replaces the more familiar notion of distance, it is useful to think about circles. Circles are conics, defined in terms of quadrance, and in our usual two dimensional picture they can appear as ellipses, parabolas or hyperbolas. We illustrate three different families, with three different centers. A careful study of these examples will give the student a good understanding of this crucial concept in geometry.This is part of the UnivHypGeom series, a new treatment of hyperbolic geometry, purely algebraic, and much prettier. CONTENT SUMMARY: pg 1: @00:11 Visualizing quadrance with circles pg 2: @03:58 circles in the hyperbolic plane; note - remark on letter c @05:18 dletter x @08:17 ; conics introduced; choice of center @09:52 pg 3: @10:08 example 1; of pictures of circles centered at 0; recap @15:35; pg 4: @16:21 example 2; c=[1:0:2] ; Exercise 24.1 ; remark of no quadrance between zero and one @22:07 pg 5: @24:11 example 3; circles with centers outside the null circle; c=[2:0:1];Exercise 24.2 ; How these curves appear in classical hyperbolic geometry @32:02 (THANKS to EmptySpaceEnterprise)
Lecture 27
Geometer's Sketchpad and Circles in Universal Hyperbolic Geometry
Play Video		 Geometer's Sketchpad and Circles in Universal Hyperbolic Geometry
We describe Geometer's Sketchpad (GSP): a dynamic software package that we use to illustrate constructions and measurements in universal hyperbolic geometry. Starting with basic properties of GSP, we then explain custom tools and how to make them. In particular we show how to construct the dual of a point, with respect to the standard null circle. For our main application, we illustrate various circles in hyperbolic geometry, both with centers inside the null circle, as well as center outside, in which case we get circles usually called equidistant curves.CONTENT SUMMARY: Introduction to Geometers Sketchpad (GSP): @00:11 lecture start: @01:53; tools @2:39; menu_items @07:23; custom tools for hyperbolic geometry @10:55; constructing the dual of a point @12:49; circles in hyperbolic geometry @16:57; closing remarks @24:36 (THANKS to EmptySpaceEnterprise)
Lecture 28
Trigonometric Laws in Hyperbolic Geometry using Geometer's Sketchpad
Play Video		 Trigonometric Laws in Hyperbolic Geometry using Geometer's Sketchpad
We use Geometer's Sketchpad to illustrate the four main laws of trigonometry in Universal Hyperbolic Geometry. These are Pythagoras' theorem, the Triple quad formula, the Spread law and the Cross law. We use custom tools to calculate quadrance and spread. We illustrate each law with a variety of examples. Along the way we show dual triangles, and explain some other features of GSP. CONTENT SUMMARY: Introduction @00:11; Quadrance @02:09; Spreads 04:37; Dual triangle @07:10 ; perpendicularity; Pythagoras' theorem @09:26; Triple quad formula @12:12; The Spread law @15:16; The Cross law @16:17 (THANKS to EmptySpaceEnterprise)
Lecture 29
The Spread Law in Universal Hyperbolic Geometry
Play Video		 The Spread Law in Universal Hyperbolic Geometry
The spread between two lines in hyperbolic geometry is exactly dual to the notion of the quadrance between two points. The Spread law is the third of the four main laws of trigonometry in universal hyperbolic geometry. Its proof also relies on a remarkable polynomial identity, just as did the proofs of Pythagoras' theorem and the Triple quad formula. In this video we review the definition of spread, give an example relating it to the spread between lines in Euclidean geometry, and give a proof.CONTENT SUMMARY: pg 1: @00:11 ; spread; quadrance spread duality; pg 2: @03:04 ; example pg 3: @04:36 ; Spread law (hyperbolic version); proof pg 4: @06:49 ; proof continued; big expression resolution @08:52; observation on how to remember factors @11:41 ; the heart of the proof @12:49 ; formula(*); pg 5: @13:15 ; proof continued; formula(***); "And that's a proof of the spread law." @17:05 pg 6: @17:29 ; Harvesting consequences of proof of spread law; quadrea of the triangle introduced pg 7: @22:33 ; Exercises 27.1-3 (THANKS to EmptySpaceEnterprise)
Lecture 30
The Cross Law in Universal Hyperbolic Geometry
Play Video		 The Cross Law in Universal Hyperbolic Geometry
The Cross law is the fourth of the four main laws of trigonometry in the hyperbolic setting. It is also the most complicated, and the most powerful law. This video shows how we can prove it with the help of a remarkable polynomial identity. We also give an application to the relation between the quadrance and spread of an equilateral triangle. CONTENT SUMMARY: pg 1: @00:11 Cross law in Euclidean R.T. review; Cross law in UHG; Cross Dual law pg 2: @06:43 Reluctant exposure to classical hyperbolic geometry; exercise 28.1 @09:25 ; observation - Euclidean cross law as a limiting case @11:02 pg 3: @12:46 Cross law (hyperbolic version); proof; heavenly assistance 16:06 pg 4: @16:21 proof continued; using a computer (Scientific Workplace) to verify an identity @21:41 ; proof complete @26:47 pg 5: @28:22 A pleasant consequence of the cross law; Equilateral triangle theorem; proof; exercise 28-2 @31:24 pg 6: @31:48 exercises 28.3-5 (THANKS to EmptySpaceEnterprise)
Lecture 31
Thales' Theorem, Right Triangles and Napier's Rules
Play Video		 Thales' Theorem, Right Triangles and Napier's Rules
This video establishes important results for right triangles in universal hyperbolic geometry--these are triangles where at least two sides are perpendicular. Besides Pythagoras' theorem, there is a simple result called Thales' theorem, giving a formula for a spread as a ratio of two quadrances. Together these allow us to state a very simple form for Napier's rules in this algebraic setting. CONTENT SUMMARY: pg 1: @00:11 Review of the 4 main laws of trigonometry; pg 2: @02:55 right triangles described; singly right, doubly right, triply right pg 3: @06:16 Thales theorem; A kind of 'similarity' for right triangles @09:05 ; In classical hyperbolic geometry this result is obscured @11:48 pg 4: @12:36 Thales theorem relationship to Euclidean RT; the spread as a crucial ratio @15:30 pg 5: @17:41 Quadrea as the single most important number associated to a triangle; reminder on how to obtain an altitude @18:41 ; Quadrea are @22:11 pg 6: @22:42 Napier's rules; suggested exercise @24:38 pg 7: @26:10 proof of Napier's rules; pg 8: @30:02 proof continued; suggested algebra exercise @35:46 pg 9: @36:49 When in doubt create some right triangles; exercise 29.1 @37:37; pg 10: @39:35 exercise 29.2 pg 11: @40:53 exercise 29.3,4 (THANKS to EmptySpaceEnterprise)
Lecture 32
Isosceles Triangles in Hyperbolic Geometry
Play Video		 Isosceles Triangles in Hyperbolic Geometry
Isosceles triangles have some special formulas associated to them, which are not obvious.They are also connected directly to the construction of the midpoint(s) of a side. CONTENT SUMMARY: pg 1: @00:11 Definition of isosceles triangle; theorem (Pons Asinorum); proof pg 2: @03:49 Notation for isosceles triangle; Isosceles triangle theorem @04:59 pg 3: @06:17 proof is an application of the Cross law pg 4: @11:20 connecting isosceles triangle formulas with formulas for equilateral triangles and right triangles; suggested exercise @ 13:26; importance of checking formulas against previous ones @14:04 pg 5: @14:34 definition of midpoint; definition of midline pg 6: @19:47 Midline theorem; proof pg 7: @22:26 Isosceles mid theorem; proof left as an exercise @24:55 pg 8: @26:20 Exercise 30.1 pg 9: @29:17 Exercise 30-2; exercise 30-3 @30:27 (THANKS to EmptySpaceEnterprise)
Lecture 33
Menelaus, Ceva and the Laws of Proportion
Play Video		 Menelaus, Ceva and the Laws of Proportion
The classical theorems of Menelaus and Ceva concern a triangle together with an additional line or point, and give relations between three ratios of distances (or quadrances). These results are also valid in universal hyperbolic geometry. However we also give some other lovely and simple results: the Triangle proportions theorem and the Alternate spreads theorem. CONTENT SUMMARY: pg 1: @00:11 classical results (main tool- spread law); Menelaus; remark on trigonometry @01:32 ; Menelaus' theorem @02:01 pg 2: @05:20 dividing a segment into a ratio; Menelaus' vector theorem @08:23 ; pg 3: @13:24 Ceva's theorem; cevian lines; the previous are classical results @17:55 ; relationship of results to UHG @18:15 pg 4: @18:38 Back to Universal Hyperbolic Geometry; Menelaus theorem; proof; pg 5: @24:23 remark on duality; menelaus' dual theorem @25:15 ; exercises 31-(1:3) pg 6: @27:45 Triangle proportions theorem; proof (the Spread law as the main ingredient) pg 7: @31:36 Alternate spreads theorem; proof pg 8: @34:14 Ceva's theorem; proof pg 9: @38:24 Ceva's dual theorem; exercises 31.4-6 (THANKS to EmptySpaceEnterprise)
Lecture 34
Trigonometric Dual Laws and the Parallax Formula
Play Video		 Trigonometric Dual Laws and the Parallax Formula
This video introduces a simple universal analog (called the Right parallax formula) to the Angle of parallelism formula found by N. Lobachevsky and J. Bolyai in classical hyperbolic geometry. First we establish the dual laws of the main trigonometric laws for Universal Hyperbolic Geometry. The Right parallax theorem is proven using the Cross dual law, and we also show how it is related to the classical result of Lobachevsky and Bolyai. Two further interesting variants are given as Exercises: the Isosceles parallax and General parallax formulas.
II. Spherical and Elliptic Geometries
Lecture 35
Introduction to Spherical and Elliptic Geometries
Play Video		 Introduction to Spherical and Elliptic Geometries
We introduce PART II of this course on universal hyperbolic geometry: Bringing geometries together. This lecture introduces the very basic definitions of spherical geometry; lines as great circles, antipodal points, spherical triangles, circles, and some related notions on points, lines and planes in three dimensional space. The ideas are illustrated with physical models.
Lecture 36
Introduction to Spherical and Elliptic Geometries II
Play Video		 Introduction to Spherical and Elliptic Geometries II
We continue our introduction to spherical and elliptic geometries, starting with a discussion of longitude and latitude on a sphere. We mention the close historical connections between spherical geometry and astronomy, going back to the ancient Greeks, to the Indians and to the Arabs. We explain the relationship of spherical geometry and Euclid's 5 postulates. Elliptic geometry is the result of identifying antipodal points on the sphere. Measurement on the surface of a sphere uses angles to define spherical distances, but additional functions are required. We describe Ptolemy's tables of chords and later Indian and Arab work on tables of sines. The final result is Menelaus' theorem, which first appears in the spherical setting, using chords.
Lecture 37
Areas and Volumes for a Sphere
Play Video		 Areas and Volumes for a Sphere
The beautiful formulas for the surface area and volume of a sphere go back to Archimedes, who also discovered some other remarkable facts relating spheres to circumscribing cylinders. We describe these results.Then we introduce rational turn angles---a renormalization of the notion of angle so that perpendicular lines are represented not by 90 degrees, or by pi/2 radians, but rather by 1/4 turn. This is mathematically the most natural parametrization of an angle, and we restate the sum of angles in a triangle and quadrilateral in terms of turn angles. We state a useful Proportionality Principle. A famous theorem of Harriot (or Girard) gives the ratio of the area of a spherical triangle to the area of the sphere in terms of the sum of turn angles.
Lecture 38
Classical Spherical Trigonometry
Play Video		 Classical Spherical Trigonometry
This video presents a summary of classical spherical trigonometry. First we define spherical distance between two points on a sphere, then the angle between two lines on a sphere (i.e. great circles). After a quick reminder of the circular functions cos,sin and tan, we present the main laws: the (spherical) sine law, two Cosine laws, Pythagoras' theorem, Thales theorem, Napier's Rules, the law of haversines and a few more, namely Napier's and Delambre's Analogies.We will not be using these laws in any substantial way: shortly in this course we will redevelop the framework of spherical trigonometry in a completely new, simplified algebra fashion. We will obtain analogs of the above laws, but analogs that are much more rigorous, powerful and accurate.
Lecture 39
Perpendicularity, Polarity and Duality on a Sphere
Play Video		 Perpendicularity, Polarity and Duality on a Sphere
This video discusses perpendicularity on a sphere, associating two poles to every great circle, and one polar line (great circle) to every point. This association is cleaner in elliptic geometry, where there is then a 1-1 correspondence between elliptic points (pairs of antipodal points on a sphere) and elliptic lines (great circles).We introduce the polar triangle of a triangle, and explain the supplementary relation between angles and sides in a triangle and sides and angles in the polar triangle.Then we extend the duality of Apollonius from the case of a circle to the case of a sphere; this associates to a point in three dimensional space (actually projective space) a plane. The dual of a line is another line.
Lecture 40
Parametrizing and Projecting a Sphere
Play Video		 Parametrizing and Projecting a Sphere
This video introduces stereographic and gnomonic projections of a sphere. We begin by reviewing three dimensional coordinate systems. A rational parametrization of a sphere is analogous to the rational parametrization of a circle found in Lecture 29 of "Math Foundations with Prof. Wildberger." Stereographic projection projects from the south pole of the sphere through the equatorial plane. Gnomonic projection projects from the center of the sphere through a tangent plane. Both are very important. Gnomonic projection works more naturally in the elliptic framework, where we identify antipodal points on a sphere.
III. Overview of Rational Trigonometry
Lecture 41
Rational Trigonometry: An Overview
Play Video		 Rational Trigonometry: An Overview
This is probably the most important video in this series; it introduces Rational Trigonometry from first principles using a vector approach. The main notions of quadrance and spread replace distance and angle, and are introduced purely algebraically. The scalar/inner/dot product plays an important role, and allows us to introduce perpendicularity algebraically, and to introduce the spread between two vectors. The main laws are Pythagoras' theorem, the Triple quad formula, the Cross law, the Spread law and the Triple spread formula. The proofs of the main laws are given as Exercises, but I give some hints. Please spend time to write out solutions carefully!For more information on this purely algebraic approach to trigonometry and geometry at an elementary level, see my YouTube playlist WildTrig, and my book 'Divine Proportions: Rational Trigonometry to Universal Geometry'.
Lecture 42
Rational Trigonometry in Three Dimensions
Play Video		 Rational Trigonometry in Three Dimensions
We extend rational trigonometry to three dimensions, using a vector approach and the dot product to define quadrance of a vector and spread between two vectors. The main laws of Pythagoras' theorem, the Triple quad formula, the Cross law, Spread law and the Triple Spread formula still apply.However there are some new developments. To motivate this, I recast the spread between two lines in terms of a ratio with a 2x2 determinant, and then introduce the solid spread made by three vectors also in terms of a ratio with a 3x3 determinant.