The WildTrig series

Rational Trigonometry and Geometry

This is a series of YouTube videos on geometry and rational trigonometry meant for a wide general audience---high school teachers, mathematics students, general public with an interest in geometry.

Please work through these slowly and patiently, and don't hesitate to scribble notes and work out examples for yourself. The videos are aimed at an elementary level, but even professional mathematicians will learn something from them.

WildTrig0: An invitation to geometry: the WildTrig series --- Introduces the WildTrig series, inviting you to learn a new approach to geometry and trigonometry.

WildTrig1: Why trig is hard --- The usual trigonometry is overly complicated, inaccurate and logically dubious. This is the first of a series that shows you a better way---rational trigonometry!

WildTrig2: Quadrance via Pythagoras and Archimedes --- This video introduces the main notion of quadrance by going back to how the ancient Greeks thought about geometry. Pythagoras' theorem and a sister theorem called the Triple quad formula are the main results, and they are proven!

WildTrig3: Spread, angles and astronomy --- Angles have their origin in astronomy and spherical trigonometry. Here we introduce the rational alternative, called spread, and give examples from ISO paper sizes to the faces of a dodecahedron.

WildTrig4: Five main laws of rational trigonometry --- We derive from first principles the main laws of rational trigonometry, using the concepts of quadrance and spread to replace the usual distance and angle. Most everything works out much simpler.

WildTrig5: Applications of rational trigonometry --- Rational trigonometry is applied to solve four examples of practical problems, concerning a flagpole, a ladder, a kite and the distance from a point to a line.

WildTrig6: Heron's formula viewed rationally --- Heron's formula, originally due to Archimedes, is here recast in a simpler and more natural form. And we prove it, using one of the basic laws of rational trigonometry.

WildTrig7: Solving triangles with rational trigonometry --- Shows how to solve triangles using the framework of rational trigonometry with quadrances and spreads replacing distance and angle.

WildTrig8: Centers of triangles withrational trigonometry --- We study the circumcenter, orthocenter and centroid of a triangle using rational trigonometry.

WildTrig9: The laws of proportion for a triangle --- Using rational trigonometry we develop Menelaus' and Ceva's theorem and some related results, namely the Law of Proportions, and the Alternating spreads theorem.

WildTrig10: Geometry of circles with rational trigonometry --- We derive some of the most fundamental facts about a circle using rational trigonometry---the Subtended spread theorems and the Equal products theorem.

WildTrig11: Applications of rational trig to surveying (I) --- Rational trigonometry can be used to solve surveying problems, usually more simply than the current way. This video gives three examples: finding the height of a mountain, Regiomontanus' problem, and spreads over a right triangle.

WildTrig12: Cartesian coordinates and geometry --- Cartesian coordinates allow us to talk precisely about points and lines, parallel and perpendicular, and quadrance and spread---the two main concepts from rational trigonometry.

WildTrig13: Why spreads are better than angles --- This video reconciles two different definitions of the spread between two lines. It also shows why spreads are generally superior to angles in a Cartesian framework.

WildTrig14: Rational parameter for circles --- The usual unit circle is best described by rational parameters, not transcendental ones. This approach is much older, and connects with Pythagorean triples, along with rational trigonometry.

WildTrig15: Complex numbers and rotations --- Complex numbers are here explained using geometry and their intimate connection with dilations and rotations. Pure rotations are related to the parametrization of the unit circle.

WildTrig16: Rational trigonometry Quiz 1 --- This is a quiz on Rational Trigonometry, with seven interesting and challenging questions. You can also try to solve them using classical trigonometry---but remember: only correct answers are allowed.

WildTrig17: Rational Trigonometry: Solutions to Quiz 1 --- Five problems involving a triangle, a surveying problem, a kite, a piece of paper and a cyclic quadrilateral are solved using rational trigonometry.

WildTrig18: Medians, altitudes and vertex bisectors --- We discuss inversion in a circle, and medians, altitudes and vertex bisectors of a triangle. These are solutions to the final two questions of Quiz 1 in WildTrig16.

WildTrig19: Trigonometry with Finite Fields (I) --- An introduction to finite fields, based on first understanding rational numbers. This will be the basis of extending geometry and trigonometry to the finite setting.

WildTrig20: Trigonometry with Finite Fields (II) ---This video explains more about arithmetic in a finite field, shows how to visualize points in a finite plane, and how to compute quadrance and spread. Examples are taken from the field with seven elements.

WildTrig21: Trigonometry with Finite Fields (III) --- This video discusses lines in Euclidean geometry over finite fields, and shows why the Spread and Cross laws hold. The circumcenter and circumcircle of an example triangle are found over F_7.

WildTrig22: Highlights from triangle geometry (I) --- The Euler line, the nine-point circle, incenters and isogonal conjugation.

WildTrig23: Highlights from triangle geometry (II) --- Equilateral triangles, Napoleon's theorem and Morley's theorem are discussed from the view of rational trigonometry. We give an example over a finite field with 11 elements.

WildTrig24: Spread polynomials--- Spread polynomials are the rational trigonometry equivalent of Chebyshev polynomials. This video shows what they look like, how to compute them and how they link to geometry.

WildTrig25: Pentagons and five-fold symmetry --- Regular pentagons, pentagrams and five fold symmetry. Ruler and compass construction, golden ratio spreads, kite and dart from Penrose tiles.

WildTrig26:Applications of rational trig to surveying (II) --- How to cut a train track up a hill, how to compare two vertical spreads, how to measure a height from three vertical spread measurements.

WildTrig27:Stewart's theorem --- Stewart's theorem relates measurements on a triangle to an additional line through a vertex. Here we present the rational version, give a simple proof, some examples and an application.

WildTrig28: What size ladder fits round a corner? --- The famous ladder problem, here solved using rational trigonometry.

WildTrig29: Trisecting angles and Hadley's theorem --- Trisecting angles from a rational trigonometry point of view, featuring a new theorem of Hadley, and relations to the trisectrix.

WildTrig30: Polar coordinates and rational trigonometry --- After a review of the usual polar coordinates, we introduce the rational version, and illustrate via an ellipse, a lemniscate and the trisectrix

WildTrig31: Introduction to projective geometry --- A first look at Projective Geometry, starting with Pappus' theorem, Desargues theorem and a fundamental relation between quadrangles and quadrilaterals.

WildTrig32: Projective geometry and perspective --- Projective geometry has close links to artist's attempts to render accurately three dimensional objects in two dimensions. We introduce three basic rules for such perspectivities, concerning images of lines, images of parallel lines, and images of conic sections.

WildTrig33: Projective geometry and homogeneous coordinates --- One of the most important mathematical advances occurred in the 1800's with the introduction of homogeneous coordinates to describe the true nature of the projective plane.

WildTrig34: Lines and planes in projective geometry --- How to think about both projective points and projective lines via lines and planes in 3D geometry. Also we discuss some basic facts about 3D geometry, relating perpendicularity and quadrances.

WildTrig35: Affine geometry and barycentric coordinates --- Affine geometry is the geometry of parallel lines. Using parallelism, we show how to construct a ruled line, how to find the midpoint of a segment, and divide a segment into a given ratio. We connect this to Archimedes law of the lever, and then extend to barycentric coordinates with respect to a triangle.

WildTrig36: Affine geometry ane vectors --- Using vectors, we define parallelograms, discuss affine combinations, and show how to derive barycentric coordinates without any notion of weights.

WildTrig37: The cross ratio --- The cross ratio is the most important invariant in projective geometry, and plays a key role in hyperbolic geometry. We introduce it here using vectors, that is in the framework of affine geometry.

WildTrig38: More about the cross ratio --- We extend the cross ratio from four collinear points to four concurrent lines, and introduce the special cases of harmonic ranges and harmonic pencils.

WildTrig39: Harmonic ranges and pencils --- Four points on a line with a cross ratio of -1 form a harmonic range. Four lines through a point with a cross ratio of -1 form a harmonic pencil. These two notions are intimately linked, and related naturally to quadrangles and quadrilaterals.

WildTrig40: The fundamental theorem of projective geometry --- The fundamental theorem of projective geometry states that any four planar non-collinear points (a quadrangle) can be sent to any quadrangle via a projectivity, that is a sequence of perspectivities. We prove this by first establishing the simpler one-dimensional case of three points on a projective line.

WildTrig41: Conics via projective geometry --- Conics, such as circles, ellipses, hyperbolas and parabolas, can be defined purely within projective geometry, as realized by the nineteenth century German mathematician Steiner. This is done by using projectivities. There are essentially two dual constructions, one giving a line conic, the other a point conic. We illustrate using The Geometer's Sketchpad, a useful software program for students of geometry.

WildTrig42: An algebraic framework for rational trigonometry (I) --- Rational trigonometry can be developed purely algebraically, without any pictures. This video reminds you of the basic concepts of quadrance and spread and their definitions in terms of coordinates.

WildTrig43: An algebraic framework for rational trigonometry (II) --- The most powerful law in geometry is the Cross law, the rational analog of the Cosine law. It includes as special cases Pythagoras' theorem and the Triple quad formula. Here we sketch a purely algebraic derivation of the Cross law, and then how the other four main laws of rational trigonometry follow from it.

WildTrig44: How to learn mathematics--- A few thoughts on how to learn mathematics, that should also be relevant for this course in Rational Trigonometry and Geometry. The basic idea: `Mathematics is a landscape'. So learning mathematics is not that different from becoming knowledgable about an unfamiliar city or terrain. This is a short break from our usual thread of rational trigonometry!

WildTrig45: Einstein's special relativity: an introduction --- Einstein's special theory of relativity (1905) was recast by Minkowski in terms of the geometry of a four dimensional spacetime. This video gives an introduction to this idea, motivating our study of two dimensional relativistic (red) geometry.

WildTrig46: Red geometry (I) --- Red geometry is a two dimensional relativistic geometry in the spirit of rational trigonometry, using variants of the usual quadrance and spread. The usual grid plane is still the arena in which this geometry lives, but the notion of perpendicularity is new. A wonderful new world of geometry emerges from just a twist of the old definitions!

WildTrig47: Red geometry (II) --- Some examples of calculating red quadrances and red spreads in red geometry, and illustrating some of the usual laws of rational trigonometry. The concurrence of the red circumcenter, centroid and orthocenter on the `red Euler line' is shown in a special case.

The MathFoundations series

Foundations of mathematics, from the start

This is a series of YouTube videos on the foundations of mathematics, starting from elementary first principles, and then slowly building up the basics of arithmetic, geometry, algebra and analysis. This series also exposes many current weaknesses in modern mathematics. But it also shows how to resolve many--- but not all! --- of them.

The videos are aimed at an elementary level, and should be especially useful to teachers and to students of mathematics, and to the general public interested in mathematics.

MathFoundations1: What is a number? --- The first of a series that will discuss foundations of mathematics. Contains a general introduction to the series, and then the beginnings of arithmetic with natural numbers.

MathFoundations2: Arithmetic with numbers --- We introduce the two basic operations on natural numbers: addition and multiplication. Then we state the main laws that they satisfy.

MathFoundations3: Laws of arithmetic --- We explain why the basic laws for addition and multiplication hold, using a model of natural numbers as strings of ones.

MathFoundations4: Subtraction and division --- Subtraction and division are inverse operations to addition and multiplication. Part of a series on the foundations of mathematics

MathFoundations5:Arithmetic and maths education ---A one page summary of the contents of K-12 mathematics is followed by some basic principles that may be useful in mathematics education. For example---calculators are unnecessary. After that, some tips on how the foundations so far on arithmetic with natural numbers can guide primary school education.

MathFoundations6: The Hindu-Arabic number system --- This foundational talk introduces the most important development in the history of mathematics and science--the Hindu-Arabic number system. To motivate it, we start by reviewing natural numbers as strings of ones, then introduce the Roman numerals in a simplified form, then the Hindu-Arabic system.

MathFoundations7: Arithmetic with Hindu-Arabic numbers --- The Hindu-Arabic number system allows us to perform addition, subtraction and multiplication smoothly. We also connect these to primary school education.

MathFoundations8: Division --- The most challenging of the four basic operations, division is a source of confusion for millions of students. Here we explain why division is really repeated subtraction. Then we prove some basic division rules, and give a simplified form of long division---something every student should learn!

MathFoundations9: Fractions --- Fractions can be introduced in many different ways. We give a definition depending only on natural numbers, not geometry.

MathFoundations10:Arithmetic with fractions --- We define addition and multiplication of fraction to parallel the operations for natural number quotients. A crucial step is to check that these operations are actually well-defined, that is that they respect the notion of equality built into the definition of a fraction.

MathFoundations11:Laws of arithmetic for fractions --- Addition and multiplication for fractions and the laws they satisfy. All these reduce to the corresponding laws for natural numbers.

MathFoundations12:Introducing the integers --- The integers are introduced as pairs of natural numbers, representing differences. The standard arithmetical operations are also defined.

MathFoundations13: Rational numbers --- Rational numbers are obtained from the integers the same way fractions are obtained from natural numbers---by taking pairs of them. The main operations are defined. The rational numbers form a `field', an important technical term in mathematics whose definition we give precisely.

MathFoundations14: Rational numbers and Ford circles--- How to visualize rational numbers using lines in the plane through the origin and the rational number strip. We connect this with the lovely theory of Ford circles.

MathFoundations15: Primary school maths education --- What do foundational issues tell us about teaching mathematics at the primary school level? Here we give some insights into arithmetic with different kinds of numbers. We also introduce a two dimensional view of rational numbers.

MathFoundations16: Why infinite sets done't exist --- Historically mathematicians have been careful to avoid treating `infinite sets'. After G. Cantor's work in the late 1800's, the position changed dramatically. Here I start the uphill battle to convince you that talking about`infinite sets' is just that---talk, not mathematics. The paradoxes discovered a hundred years ago are still among us, even if we ignore them.

MathFoundations17: Extremely big numbers --- We look at extremely big numbers. This is the best way to get a feel for the immensity and complexity in the sequence of natural numbers. And why we have no right to talk about `all' of them as a completed `infinite set'. Our main tool is a cool inductive way of defining higher and higher operations, going beyond multiplication and exponentiation.

MathFoundations18: Geometry --- How to begin geometry? What is the correct framework? How to define point, line, circle etc etc? These are some of the issues we will be addressing in this first look at the logical foundations of geometry.

MathFoundations19: Euclid's Elements --- Euclid's book `The Elements' is the most famous and important mathematics book of all time. To begin to lay the foundations of geometry properly, we first have to make contact with Euclid's thinking. Here we look at the basic set-up of Definitions, Axioms and Postulates, and some of the highlights from Books I,II and III.

MathFoundations20: Euclid and proportions ---The ancient Greeks considered magnitudes independently of numbers, and they needed a way to compare proportions between magnitudes. Eudoxus developed such a theory, and it is the content of Book V of Euclid's Elements. This video describes this important idea.

MathFoundations21: Euclid's Books VI-XIII ---A very brief outline of the contents of the later books in Euclid's Elements dealing with geometry. This includes the work on three dimensional, or solid, geometry, culminating in the construction of the five Platonic solids.

MathFoundations22: Difficulties with Euclid --- There are logical ambiguities with Euclid's Elements, despite its being the most important mathematical work of all time. Here we discuss some of these, as well as Hilbert's attempt at an alternative formulation. We prepare the ground for a new and more modern approach to the foundations of geometry.

MathFoundations23: The basic framework for geometry (I) --- This video begins to lay out proper foundations for planar Euclidean geometry, based on arithmetic. We follow Descartes and Fermat in working in a coordinate plane, but a novel feature is that we use only rational numbers. Points and lines are the basic objects which need to be defined.

MathFoundations24: The basic framework for geometry (II)--- We discuss parallel and perpendicular lines, and basic notions relating to triangles, including the notion of a side and a vertex of a triangle.

MathFoundations25: The basic framework for geometry (III) --- Distance is not the best way to measure the separation of two points, as Euclid knew. The better way is using the square of the distance, called quadrance. Here we introduce this concept, and the two most important theorems in mathematics---with purely algebraic proofs.

MathFoundations26: The basic framework for geometry (IV) --- Angles don't make sense in the rational number system. The proper notion of the separation of two lines is the `spread' between them, which is a purely algebraic quantity and can be calculated easily over the rational numbers. This video highlights some of the advantages in replacing `angle' with `spread'. It also gives an explicit formula for the `inverse cosine' function, which rarely appears in trigonometry texts, despite the universal reliance on this function via calculators.

MathFoundations27: Trigonometry with rational numbers --- Rational trigonometry works over the rational numbers, and allows us a more elementary and logical approach to the basics of trigonometry. This video illustrates the Spread law, the Cross law and the Triple spread formula.

MathFoundations28: What exactly is a circle? --- Moving beyond points and lines, circles are the next geometrical objects we encounter. Here we address the question of how best to introduce this important notion, strictly in the setting of rational numbers, and without metaphysical waffling about `infinite sets.'

MathFoundations29: Parametrizing circles --- How to describe all the points on a circle, using a rational paratmetrization. This is a major improvement on the usual transcendental parametrization with circular functions. Also some interesting number theory arises when we ask which lines through the center of a circle meet that circle.

MathFoundations30: What exactly is a vector? --- The notion of vector is here made completely explicit. Points and vectors are different types of mathematical objects.

MathFoundations31: Parallelograms and affine combinations --- We use vectors to introduce parallelograms, the parametric representation of a line, and affine combinations, such as midpoints.

MathFoundations32: Geometry in primary school --- Some comments on the teaching of geometry in primary schools (K-6). I emphasize the importance of the grid plane, as well as constructions and drawing, and give examples of important topics.

MathFoundations33: What exactly is an area? --- While there is a naive idea of area in terms of number of unit squares that can fit inside a region, this is not the best definition. It is better to work with oriented triangles and maintain linearity.

MathFoundations34: Areas of polygons --- How to define the area of a polygon? The right way is to consider signed areas of oriented polygons. This leads to natural formulas that are important for calculus.

MathFoundations35: Translations, rotations and reflections (I) ---We introduce translations in the rational plane. However we do not assume the conventional understanding of functions and mappings, which actually has some logical difficulties and conceptual disadvantages. We prefer a cleaner and more flexible understanding emphasizing the use of expressions.