Math seminars
This series consists of mathematical seminars presented by N J Wildberger and perhaps also seminars by colleagues. It will be aimed at undergraduate math majors, PhD students, mathematicians and physicists. Most but not all of these will have been presented to the School of Mathematics and Statistics, UNSW Sydney.
The Wild Linear Algebra playlist gives a geometric course on Linear Algebra.
We present a very brief survey of a few classical results in Euclidean triangle geometry, and then give an introduction to triangle geometry in the new setting of universal hyperbolic geometry. This occurs in a projective Beltrami Klein model involving both the inside and outside of a distinguished circle.
This lecture assumes no familiarity with either classical or universal hyperbolic geometry, and introduces only the concepts of perpendicularity, midpoints and bilines (analogs of angle bisectors) via simple projective constructions using the fixed circle. With this framework we present a nice cross section of interesting results involving orthocenters and the orthoaxis, the Double triangle, the x,z,b,h and s points, Incenters, Apollonian points and Incircles, Circumcenters and Circumcircles, Medians, Centroids etc. A good high school student can easily appreciate the results, which are illustrated by lots of picturesthere are no formulas in this talk! This talk was presented to the AMSI Summer School at UNSW in January 2012 by N J Wildberger. Thanks to Ali Alkhaldi for videoing. 

This is the full lecture of a seminar on a new way of thinking about Hyperbolic Geometry, basically viewing it as relativistic geometry projectivized, that I gave a few years ago at UNSW. We discuss three dimensional relativistic space and its quadratic/bilinear form, particularly the upper sheet of the two sheeted hyperbola, which is a Riemannian manifold under the induced metric from the Lorentzian quadratic form x^2+y^2z^2. Two views give the Poincare and Klein models (both due to Beltrami!) and it is the latter that we extend to include the entire projective plane.
We discuss some related projective geometry, harmonic ranges and cross ratio, introduce the main metrical notions of quadrance and spread, illustrate with diagrams of circles, connect with classical notions of distance and angle, and state the main laws of trigonometry as well as some important secondary results, such as the Parallax theorem. We explain the Null perspective theorem and related null trigonometric results, the 48/64 theorem and finish with the remarkable Jumping Jack theoremmy personal favourite theorem in mathematics. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... . 

Lectures notes at http://www.maths.unsw.edu.au/seminars/201207/threedimensionalgeometryzomeandelusivetetrahedron.
The geometry of three dimensional space, despite its obvious importance, is a sadly neglected topic in modern mathematics. One of the reasons is that the topic is rather awkward and difficult when approached with the standard tools of metrical affine and spherical geometry. In this talk we show that ideas of rational trigonometry, both in the planar and spherical (elliptic) setting, open new doors to our understanding, supported by fascinating algebraic relations. We will start with a quick linear algebra presentation of rational trigonometry, introduce the beautiful and remarkable ZOME construction system, and then tackle the trigonometry of a tetrahedronthe fundamental object in three dimensional geometry. We introduce the basics of rational trigonometry, first in the affine and then in the projective setting, using the notions of quadrance and spread instead of distance and angle. (Quadrance can be thought of as the square of the distance, and spread the square of the sin of an angle; both notions come in an affine and a projective flavour.) The main laws of affine rational trigonometry, namely the Cross law, Spread law and Triple spread formula are proved using only a bit of first year linear algebra and familiarity with the dot product. All maths teachers/lecturers who currently teach trigonometry might like to have a think about the educational implications of this first fifteen minutes. The projective formulas will be familiar to those following the UnivHypGeom series. The main object is a projective triangle, or tripod, consisting of three concurrent lines in 3D space, and the corresponding three planes formed by pairs of those lines. The spreads between the lines are called the projective quadrances of the tripod, while the spreads between the normals to the planes are the projective spreads of the tripod. The projective relations between the three projective quadrances and three projective spreads are deformations of the planar laws. The Zome construction system is introduced and we exhibit the primitive Zome triangles, and a particular tetrahedron derived from a dodecahedron and one of its faces. The heart of the talk is the derivation of new rational formulas for three dimensional trigonometry, involving side quadrances, face spreads, dihedral edge spreads, solid spreads at the vertices, and 144 times the square of the volume, which we call the quadrume. These laws are not at all complete, in the sense that there are other relations and further secondary quantities too which we do not discuss. We illustrate the main laws with explicit concrete formulas for not only our example tetrahedron, but also the right isosceles tetrahedron (analog of the 90/45/45 triangle), and then the regular tetrahedron. It seems remarkable that the metrical structure of this fundamental object is laid out for the first time here, on YouTube, in 2012! This talk was given to the Pure Mathematics Seminar of the School of Mathematics and Statistics at UNSW on July 31, 2012. Thanks to Michael Cowling, David Hunt, Michael Reynolds and Thomas Britz for the interesting questions, and thanks to Nguyen Le for help with the filming. 

This is a talk I (Norman Wildberger of the School of Mathematics and Statistics, UNSW) gave to the Independent Schools Heads of Departments of Mathematics meeting at Knox Grammar School, Sydney on August 13, 2012. The aim was to try to address the issue of a new national mathematics curriculum for Australia that ACARA, the Australian Curriculum Assessment and Reporting Authority, have proposed for advanced mathematics for Years 11 and 12. The two courses are called Mathematical Methods and Specialist Mathematics. For those teaching in NSW, these are roughly at the level of Twounit and Fourunit mathematics.
In the talk I summarize the main points of a detailed report prepared by the School of Mathematics and Statistics at UNSW by Mr Peter Brown (Director of First Year Studies), Dr. Daniel Chan (Dept. of Pure mathematics), Assoc. Prof. David Warton (Dept. of Statistics) and myself. We are highly critical of the proposed drafts. I explain our reasons briefly, and refer interested viewers to the report itself, available at http://www.maths.unsw.edu.au/news/201207/schoolsresponsedraftseniormathematicscurriculumacara . To quote: The proposed national curriculum will be, in our opinion, a setback for mathematics education in NSW, and we would support the Federal government having a fresh look at the project. The second part of the talk is an attempt to reposition geometry in the high school curriculum. The construction of a national curriculum is an important opportunity to ask key questions about how we can encourage participation and learning in mathematics, and strengthen school leavers understanding, competence and confidence in the subject. Geometry, I will argue, is a historically vital subject which we need more of today rather than less (ACARA proposes to banish geometry almost entirely from the main advanced mathematics course: Mathematical Methods!!) To make the case, I will briefly introduce a number of topics that are potentially appealing and illuminating for high school students. These include conics and Cartesian geometry, the rational parametrization of a circle, Pappus' and Pascal's theorems in projective geometry, triangle centres and the Euler line, Archimedes' law of the lever and convex combinations, centers of mass and means of probability distributions, Olympic rankings and even a bit of rational trigonometry! This is a smorgasbord of geometric delights that hopefully will encourage the realization that geometry can have broad appeal, is important to the modern world, and underpins other areas of mathematics as well as of course science and engineering. Afternote: We are happy to report that ACARA ended up with a much more palatable final senior curriculum! Many different kinds of input and feedback were listened to, including a report from UNSW, and the result was a much more balanced and positive curriculum! 

This is a seminar talk given to the School of Mathematics and Statistcs, UNSW in April 2013. It describes joint work with Nguyen Le on a generalized triangle geometry.
We begin with an introduction to rational trigonometryan approach to the subject that is almost purely algebraic, replacing distances and angles with quadrances and spreads and allowing extension to general fields, and indeed to arbitrary bilinear forms. In particular the formulas of rational trigonometry apply to relativistic geometry, including the red and green planar geometries that complement the familiar Euclidean blue geometry. Together these three geometries combine in a lovely way, leading to a subject called chromogeometry. Triangle geometry is seeing a great revival of interest due to the power of dynamic geometry packages such as GSP, C.a.R and Geogebra, and the online Encyclopedia of Triangle Centers maintained by Clark Kimberling. Here we present a very general approach to this subject, using rational trigonometry and linear transformations to reduce to a standard triangle. This becomes particularly interesting when studying the incenter hierarchy and the natural fourfold symmetry it supports. I report on some interesting concurrences associated to the incenters, Gergonne and Nagel points, Mittenpunkts, Spieker and Bevan points. Finally I mention a remarkable result on what happens when we consider the blue, red and green incenters of a given triangleall at the same time! 

This is a seminar given at the University of Newcastle in April 2013. We explore the possibilities for a new more computational approach to mathematics which replaces the current dubious reliance on `real numbers' with the much more solid and natural rational numbers. The key idea is that of rational trigonometry (RT), whose basic laws are here introduced in a simple way using only very elementary linear algebra. The five main laws of RT are described, and proofs of the Cross law, the Spread law and the Triple spread formula are given.
Paul Miller's simple and elegant spread protractor is described. Some examples from the Zome construction system are illustrated. We discuss the beautiful new spread polynomials that arise from considering composites of spreads. The quadruple quad and quadruple spread formulas are described, and the relation with cyclic quadrilaterals is described. Then we move to three dimensional applications, introducing projective rational trigonometry, and projective versions of the planar theory described earlier. 

aA brief introductory lecture on the Global Positioning System (GPS), and how both Einstein's special and general theories of relativity need to be considered to ensure that it works properly.
We describe the satellite setup of GPS, the use of cesium atomic clocks to transmit time and position information, and the geometry of why four satellite positions determine where we are. Precursors to this system include LORAN (Long Range Navigation system) of beacons used at sea, and earlier audio systems used in WW1 to try to determine enemy gun positions by measuring differences of distances from observers. We discuss the accuracy of the clocks, and why Einstein's Special theory (SR) and General theory (GR) of relativity both are affecting this situation. We present simplified views of SR and GR, starting with the MichaelsonMorley experiment which showed that the speed of light was constant in different uniform motion coordinate systems, then Einstein's remarkable conclusion that there is no fixed reference coordinate system for the world: the laws of physics are the same for different observers moving at uniform motion with respect to each other. A bizarre consequence is that simultaneity of events is a relative notion. GR is explained in terms of the Equivalence Principle relating observers in accelerated frames and in gravitational fields: this allows us to apply the (relativistic) Doppler effect to conclude that clocks higher in a gravitational field appear to run faster. Both of these effects have effects on the running of the atomic clocks in the GPS satellitesin fact they work in opposite directions, and their cumulative effect must be taken into account by the engineers who manage the system.The HinduArabic number system allows us to perform addition, subtraction and multiplication smoothly. We also connect these to primary school education. It is hard to over emphasize how important this numerical system is for the history of science, technology and the modern world. 

This is a seminar recorded on June 24 2014 in the School of Mathematics and Statistics, UNSW. A/Prof N J Wildberger shows how to derive the main kinematic formulas of Special Relativity by taking a novel Newtonian point of view, emphasizing the role of inertial observers rather than inertial frames, and by framing the discussion in terms of sound (not light!) and how two bats might set up coordinates to measure their spacetimes.
This way the relativity of simultaneity, time dilation, length contraction and the curious Einstein addition of velocities become simple consequences of high school algebra applied to a scenario not far removed from every day life. At about 25:00, the board is a bit hard to read (sorry about that) but here are the two main equations there: t_1=(xt)/(rs) and t_2=(x+t)/(r+s). 

Infinity has long been a contentious issue in mathematics, and in philosophy. Does it exist? How can we know? What about our computers, that only work with finite objects and procedures? Doesn't mathematics require infinite sets to establish analysis? What about different approaches to the philosophy of mathematicscan they guide us?
In this friendly debate, Prof James Franklin and A/Prof Norman Wildberger of the School of Mathematics and Statistics, Faculty of Science, UNSW, debate the question of `infinity' in mathematics. Along the way you'll hear about Jim's new book: `An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure', published this year by Palgrave MacMillan. Unfortunately, the microphone could not pick up audience questions and responses very well. The correct answer to Norman's question at the end of the game he described was given by Roberto Riedig: `any number you want'! As for this interesting game itself, Norman seems to remember getting the idea from Wolfgang Mueckenheim, who also ventures into heretical waters: see for example his paper "Physical Constraints of Numbers", Proceedings of the First International Symposium of Mathematics and its Connections to the Arts and Sciences, A. Beckmann, C. Michelsen, B. Sriraman (eds.), Franzbecker, Berlin 2005, p. 134  141. For those interested in this kind of nonstandard position, they can also look for Norman's paper: `Set Theory: Should you Believe?' You can also check out my blog entry at http://njwildberger.com/2015/11/26/infinityreligionforpuremathematicians/. Thanks to Nguyen Le for videoing. 

This video tries to expose some of the very serious logical weaknesses that we currently have in pure mathematics. These center around historical problems with the continuum, analysis and set theory, but increasingly are coming to view with the growing power and relevance of computational thinking.


I discuss my book Divine Proportions: Rational Trigonometry to Universal Geometry, which gives a novel way of thinking not only about trigonometry, but also Euclidean geometry. It also lays the ground work for a more rational and logical approach to other geometries, including hyperbolic geometries.


This is a talk I gave to maths teachers at the 2015 annual meeting of MANSW. In it I explain some aspects of the intimate correspondence between algebra and geometry that might contribute to a resurgence of interest in geometry as a core subject for school students.
In particular in this lecture I look at geometry and basic arithmetic, the Euclidean algorithm and continued fractions, along with some projective geometry: Pappus' theorem and Pascal's theorem. 

This is the second half of a talk I gave to maths teachers at the 2015 annual meeting of MANSW. In it I explain some aspects of the intimate correspondence between algebra and geometry that might contribute to a resurgence of interest in geometry as a core subject for school students.
In particular in this lecture I look at Descartes' Cartesian geometry and conics, two basic theorems of projective geometry and a nonstandard view of the parabola, some classical triangle geometry and the Euler line, and give a quick introduction to Rational Trigonometry. We need some new ideas for mathematics education! 

Streamed live on 8 Nov 2016
This lecture, which begins at 2:45, shows how Big Number theory, together with an understanding of prime numbers and their distribution, resolves the Goldbach Conjecture, which states that every even number greater than two is the sum of two primes. Notions of complexity and computation, along with the distribution of prime numbers, play an important role. Along the way we will meet some seriously big numbers. The talk will be given by A/Prof N J Wildberger (UNSW). Norman would like to especially thank Daniel Mansfield and Joshua Capel for overcoming technical problems to set up the live stream and video it. Thanks guys! Note: These views on mathematics put forth here by A/Prof N J Wildberger are his own and do not represent that views of the School of Mathematics and Statistics at UNSW Australia. 
