Universal Hyperbolic Geometry : 1 42
Universal Hyperbolic Geometry (UHG) is a dramatically new approach to the classical subject initiated by Gauss, Lobachevsky and Bolyai. While classical hyperbolic geometry is set in the interior of the unit disk, or the upper half plane, UHG involves the entire projective plane, together with a distinguished conic. In this sense it is parallel to Cayley Klein geometry, which calls the distinguished conic the absolute.
But UHG introduces metrical structure in a purely algebraic way, without use of transcendental functions such as the log or sin, and exactly parallel to rational trigonometry in the plane. That means that projective notions of quadrance between points and spread between lines are the main measurements. Everything becomes much simpler, more beautiful and more general, extending to arbitrary fields (including finite fields!) and also to other quadratic forms.
And there are many new theorems and insights. Welcome to a new hyperbolic world!
A big THANKS to EmptySpaceEnterprise for putting together the Content Indexes for quite a few from the series.
But UHG introduces metrical structure in a purely algebraic way, without use of transcendental functions such as the log or sin, and exactly parallel to rational trigonometry in the plane. That means that projective notions of quadrance between points and spread between lines are the main measurements. Everything becomes much simpler, more beautiful and more general, extending to arbitrary fields (including finite fields!) and also to other quadratic forms.
And there are many new theorems and insights. Welcome to a new hyperbolic world!
A big THANKS to EmptySpaceEnterprise for putting together the Content Indexes for quite a few from the series.
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 3way 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) 

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 3way 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) 

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 

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 9:09 

This video outlines the basic framework of universal hyperbolic geometryas the projective study of a circle, or later on the projective study of relativistic geometry. Perpendicularity is defined in terms of duality, the polepolar 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 BeltramiKlein 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 BeltramiKlein model @26:45 Pythagoras' dual theorem @28:43 Main formulas for triangles that involve both quadrances and preads @31:13 

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 nonparallel 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) 

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 bookkeeping 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 coordinates theorem. @30:50 Quadrance in coordinates theorem @34:24 Spread in coordinates theorem @46:30 Remark on challenge of parallel lines @49:03 

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 onedimensional 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 

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 onedimensional 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 WT31WT41 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 

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 

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 

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 ortholinealso 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 orthoaxis of a triangle. The orthoaxis 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 orthoaxis; examples in Geometers Sketchpad @42:56 

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 orthoaxis 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 orthoaxis theorem; importance of orthoaxis 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, orthoaxis, 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 

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; 

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 explicitbut 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:u1t2:u2), f(t1:u1t2: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 

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 

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 

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 162) 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 163; 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) 

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 orthoaxis, 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 zpoint whether or not it has midpoints pg 9: @29:07 zpoint orthoaxis theorem; zbhs harmonic range theorem; remark exercise request; geometer_sketchpad illustrations @32:18 

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 orthoaxis, 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 181; xpoint orthoaxis theorem; shxb crossratio theorem. pg 9: @22:42 Second double triangle perspective theorem; ypoint/second double point defined pg 10: @24:44 Double dual triangle perspective theorem; zpoint 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 

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 

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 Geometryrational numbers :: Applied Geometrydecimal 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 

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 

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 3dim 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) 

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 

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 

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 

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 

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 27.13 

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 282 @31:24 pg 6: @31:48 exercises 28.35 

This video establishes important results for right triangles in universal hyperbolic geometrythese 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 

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 302; exercise 303 @30:27 

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 31.46 

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. 

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.


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. Classical 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. 

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 anglesa 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. 

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. 

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 11 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. 

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 MathFoundations29.
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. 

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'. You can find easy links to all my YouTube videos, and also access screenshot pdfs for my videos at http://wildegg.com . 

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. 

Rational trigonometry can be developed purely algebraically, without any pictures. This video reminds you of the basic concepts of quadrance and spread and introduce new laws for spherical and elliptic trigonometry. These are natural consequences of applying Rational Trigonometry to the three dimensional projective setting. Remarkably, the main laws end up being exactly the same as those in Universal Hyperbolic Geometry! It means there is a deep symmetry between spherical and hyperbolic geometries, which in fact extends to other geometries too.
Because of the fundamental nature of these topics, the video is designed to be viewed without any prior familiarity with either Rational Trigonometry or the other videos in this series. The main object is an elliptic triangle: a triple of lines through the origin. In the projective setting we refer to a line through the origin, or onedimensional subspace, as a projective points. The (projective) quadrance between any two such projective points is nothing but the Euclidean spread between the two lines, which is naturally defined in terms of the dot product of two direction vectors for the lines, suitably renormalized so it is independent of scaling. Any elliptic triangle has a polar triangle. There is an intimate connection between polar triangles and inverses of matrices. The (projective) spread between two lines of an elliptic triangle is defined as the (projective) quadrance of the associated points of the polar triangle. This way quadrances of a triangle and spreads of the polar triangle are naturally the same. We show that the projective quadrance is alternately described elegantly using a bit of linear algebra: essentially a two by two determinant suitably renormalized. To study an elliptic triangle, we need not just the three quadrances formed by pairs of points, but also the three spreads formed by the lines of the triangle. Equivalently, these are the quadrances formed by the polar triangle. The main laws relate the three quadrances and spreads of an elliptic triangle: they are the Spread law, the Cross law, and the Dual cross law. These are among the most important formulas in all of mathematics. This video should be of considerable interest to physicists, chemists, video game creators, astronomers, engineers working with 3D dynamics or graphics, and computer scientists. It should also be the driver of a new way of thinking about elementary geometry in linear algebra courses, and it naturally leads to a whole new approach to the metrical structure of algebraic geometrycurrently a largely invisible subject. In short, this video and the next are indispensible for all students of geometry and related disciplines. 

We explain how to prove the main laws of rational elliptic/spherical trigonometry, using basic linear algebra involving dot products, 3x3 matrices and their inverses.
The main object of interest is an elliptic triangle, and in the previous video, UHG41, we stated the basic laws: the Spread law, the Cross law and the Dual cross law. Here we prove them. These are among the most important formulas in geometry, and hence in mathematics. As easy consequences of these laws, we prove Thales' theorem, Pythagoras' theorem, the Dual Pythagoras' theorem, and the Triple Quad Formula in this elliptic or projective setting. These are complementary, and closely related to, corresponding results in (planar) affine trigonometry. Remarkably, all these formulas have exactly the same form as they do in Universal Hyperbolic Geometry. 

We review the basics of rational spherical/elliptic trigonometry, a cleaner more logical view of classical spherical trigonometry which is intimately linked with hyperbolic geometry.
We illustrate the basic laws by having an indepth look at a specific example of a spherical triangle, formed by three vectors positioned at the origin in three dimensional space. This will give us an excellent basis for investigating more deeply many aspects of the geometry of three dimensional space. 

We look at a famous motivating problem from spherical trigonometry: how to figure out the separation between two points on the sphere with given latitude and longitude. In particular we look at Sydney Australia and New Delhi India, and calculate both how far apart they are, and what bearing from North a pilot should aim at to fly from Sydney to Delhi.
We use both a classical derivation, using the standard formulas of classical spherical trigonometry, as well a modern derivation, using rational spherical trigonometry. Hopefully it will become quite clear why the modern method is superior. 

We look at the geometry of the regular tetrahedron, from the point of view of rational trigonometry. In particular we reevaluate an important angle for chemists formed by the bonds in a methane molecule, and obtain an interesting rational spread instead.


The geometry of the regular tetrahedron centres around the spread of 8/9. Here we look at the A series of paper sizes, including the A4 letter paper that is the ISO standard used in almost all of the world outside North America (where 8.5" x 11" is the standard).
Curiously the shape of the A4 paper has some rather interesting mathematical properties, and the spread of 8/9 plays also a central role. By looking at a cube in the right way, we will explain this seeming coincidence. Throughout we adopt the modern powerful point of view of Rational Trigonometry: giving us a lot more control and insight into the geometry of the world around us! 

The Platonic solids have fascinated mankind for thousands of years. These regular solids embody some kind of fundamental symmetry and their analogues in the hyperbolic setting will open up a whole new domain of discourse. Here we give an introduction to these fascinating objects: the tetraheron, cube, octahedron, icosahedron and dodecahedron. In particular we go back to the ancient Greeks to see the importance that Euclid attached to these objects.


We look at the symmetries of the Platonic solids, starting here with rigid motions, which are essentially rotations about fixed axes.
We use the normalization of angle whereby one full turn has the value one, and also connect the number of rigid motions with the number of directed edges. Duality also plays a role, in that the symmetries of the cube and octahedron are essentially the same, as are the symmetries of the dodecahedron and icosahedron. 

Each of the Platonic solids contains somewhat surprising addition structures that shed light on the symmetries of the object. Here we look at the tetrahedron, and investigate a remarkable threefold symmetry which is contained inside the obvious fourfold symmetry of the object. We connect here with some group theory, and discuss aspects of the symmetric group S_3.


The cube and the octahedron are dual solids. Each has contained within it both 2fold, 3fold and 4fold symmetry. In this video we look at how these symmetries are generated in the cube via canonical structures. Along the way we discuss bipartite graphs.
This gives us more insight into the rigid motions of the cube, and in particular introduces the important notion of even and odd symmetries. 

The dodecahedron is surely one of the truly great mathematical objectsrevered by the ancient Greeks, Kepler, and many mathematicians since. Its symmetries are particularly rich, and in this video we look at how to see the fivefold and sixfold symmetries of this object via internal structures.
We will look at handle sets, inscribed cubes, tetrahedra and face diameters and how they related to the rotational symmetries of the dodecahedron. 

John Flinders Petrie was the son of the famed Egyptologist Flinders Petrie, and a good friend of Donald Coxeter. He discovered lovely polygonal paths on polytopes or polyhedra, that in the case of the Platonic solids have remarkable properties when we project these solids orthogonally onto particular planes.
We also expand our view of the Platonic solids by considering more general tesselations of a surface, including the three famous regular tesselations of the plane. The Schlafli symbol of a tesselation is also introduced. These will have important consequences when we extend this thinking to the hyperbolic situation. 

Euclid showed in the last Book XIII of the Elements that there were exactly 5 Platonic solids. Here we go through the argument, but using some modern innovations of notation: in particular instead of talking about angles that sum to 360 degrees around the circle, or perhaps 2 pi radians, we normalize our "turn angle" so that all the way around is exactly 1: the natural unit here. Using turn angles, we can use the Schlafli symbol {p,q} notation to discuss the various possibilities.
First there is the planar situation, where we find the usual three regular tesselations, and then in the spherical case we find exactly five possibilities for {p,q} corresponding to the spherical regular polytopes, i.e. Platonic solids. However this is really only part of the argument: the mathematical constructions of the Platonic solids still remaina quite separate question from the physical constructions of these objects. It turns out the the tetrahedron, cube and octahedron have pleasant constructions that generalize very nicely to higher dimensions. However the icosahedron and dodecahedron are quite a different story! The reality is that the complete argument is much more subtle than is usually conceeded. Euclid's proof, while quite attractive, does not completely work mathematically, convincing though it may be physically. 
