Rational Trigonometry and
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
An invitation to geometry: the WildTrig series
--- Introduces the WildTrig series, inviting
you to learn a new approach to geometry and trigonometry.
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!
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!
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.
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.
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.
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
Solving triangles with rational trigonometry
--- Shows how to solve triangles using the framework
of rational trigonometry with quadrances and spreads
replacing distance and angle.
Centers of triangles withrational trigonometry
--- We study the circumcenter, orthocenter and
centroid of a triangle using rational trigonometry.
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
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
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.
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
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
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.
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.
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.
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.
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.
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.
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.
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.
Highlights from triangle geometry (I) ---
The Euler line, the nine-point circle, incenters
and isogonal conjugation.
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.
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.
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.
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.
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.
What size ladder fits round a corner? ---
The famous ladder problem, here solved using rational
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.
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
Introduction to projective geometry --- A
first look at Projective Geometry, starting with
Pappus' theorem, Desargues theorem and a fundamental
relation between quadrangles and quadrilaterals.
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
Projective geometry and homogeneous coordinates
of the most important mathematical advances occurred
in the 1800's with the introduction of homogeneous
coordinates to describe the true nature of the
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.
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.
Affine geometry ane vectors --- Using vectors,
we define parallelograms, discuss affine combinations,
and show how to derive barycentric coordinates
without any notion of weights.
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.
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.
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.
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
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.
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.
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.
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!
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.
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!
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.
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.
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
Arithmetic with numbers --- We introduce the
two basic operations on natural numbers: addition
and multiplication. Then we state the main laws
that they satisfy.
Laws of arithmetic
--- We explain why the basic laws for addition
and multiplication hold, using a model of natural
numbers as strings of ones.
Subtraction and division
--- Subtraction and division are inverse operations
to addition and multiplication. Part of a series
on the foundations of mathematics
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
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.
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
--- 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
--- Fractions can be introduced in many different
ways. We give a definition depending only on natural
numbers, not geometry.
--- 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.
of arithmetic for fractions
--- Addition and multiplication for fractions
and the laws they satisfy. All these reduce to
the corresponding laws for natural numbers.
--- The integers are introduced as pairs of natural
numbers, representing differences. The standard
arithmetical operations are also defined.
--- 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.
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.
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.
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.
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
--- 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
--- 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.
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.
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.
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
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.
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.
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.
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.
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.
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.'
--- 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.
What exactly is a vector?
--- The notion of vector is here made completely
explicit. Points and vectors are different types
of mathematical objects.
Parallelograms and affine combinations
--- We use vectors to introduce parallelograms,
the parametric representation of a line, and affine
combinations, such as midpoints.
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.
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
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
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.