Some fundamental formulas from metrical algebraic geometry (seminar)

This coming Tuesday in the Pure Maths Seminar at UNSW I will be giving a talk: Here are the details, in case you are in Sydney and are interested. The talk is at the School of Mathematics and Statistics, UNSW, Kensington campus, building the Red Centre, which is up the main walkway from Anzac Parade.

Speaker:   Norman Wildberger (UNSW)
Title:         Some fundamental formulas from metrical algebraic geometry
When:       12:00 Tuesday, 9 June 2015
Where:      RC-4082, Red Centre, UNSW

Abstract:  At the heart of metrical algebraic geometry there are hierarchies of beautiful algebraic formulas, starting in one dimension and then working their way up both in dimension and complexity. These  can be viewed as generalizations of formulas of Archimedes, Ptolemy, Brahmagupta, Bretschneider, von Staudt and others, as well as classical results from Euclidean geometry.

One of the interesting aspects of our approach to these formulas, based on Rational Trigonometry, is the pursuit of generality: using general fields and arbitrary quadratic forms. Interesting connections emerge with the theory of special functions, differential geometry and operator analysis.

Most of these formulas would be challenging to find without the use of a computer: this is one reason why they have laid hidden for so long. We’ll see that there are many unanswered questions that invite exploration.

The talk will be accessible to undergraduates.

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Online tutorials for first year mathematics

The first teaching semester of 2015 has just finished here at UNSW. It has been very busy for me, which is a convenient excuse for not having blogged much lately. What’s been happening?

Teaching-wise I got a Learning and Teaching Grant from the University to put together new Online Tutorials for our Higher First Year Maths course MATH1141, which combines Algebra and Calculus. With the fantastic help of Daniel Mansfield, and also Jonathan Kress, we have been busily designing interesting questions that our best first year students can attempt on-line using Maple TA. That is a system devised by the clever people at Maple, based in Waterloo, Canada, which allows mathematical computation in the context of online delivery of course questions.

This means we can coerce, I mean motivate, students to actually do homework! How good is that? Up till now we, along with fellow educators around the world, plead and cajole our students to work on their own going over suggested exercises to practise the skills we are teaching them. But not altogether surprisingly, self-interest intrudes, and students often find themselves with something better to do, in other words something for which marks are assigned.

Our Online tutorials are based on videos to problem solutions that we made that show students how to solve questions from our Problem Notes.

Hopefully they watch the videos, and then answer a suite of five questions, ranging in difficulty from easy to more challenging, on that general topic. With the possibility of adding images, diagrams, lots of good looking and well-laid out mathematics, we can make our questions rather nifty. Here is an example of a typical question, which prompts students to discover the rational parametrization of the hyperbola x^2-y^2=1.


One of the exciting aspects of this is that we have realized we can get our students reading through proofs, and filling in blanks to complete them. This way they get the benefit of walking through a well presented argument and following that logical development—surely a key preliminary to getting them to create coherent proofs themselves.

And of course, there is the flexibility of the online platform that works in their favour, so they can choose the optimal time to do the assignments, which curiously seems to be an hour before Sunday midnight, just before the tutorials close.:)

Next session we are motoring on with more of the same in MATH1241, and next year we hope to roll this out for our big first year course (over 2000 students). Exciting, brave new times for teaching.


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My future glorious life!

I am so excited! Here drinking my morning green tea, i have just been visited by an image of St. Peter, who proclaimed to me that all my personal dreams are about to come true, provided i can write them down on this blog in the next half an hour!! Wow. So please excuse any spelling mistakes: I am about to restructure my life, and I need to do it fast, since i can’t type that quickly, but I need be careful, because I am not allowed to revise anything I write!

First of all, I am going to live a healthy life till I’m 100. Unfortunately he said I could only dictate what happens to me, not anyone else, so I can only hope the rest of my family is okay. And why not beyond a hundred? He said I couldn’t actually change my physical characteristics, so I reckon that beyond 100, frailty is too much of an issue. But I’m wasting time explaining these things to you!

Clearly I want a nice house for us, one that we own. Real estate prices are totally ridiculous here in Sydney, what with…oh I don’t need to explain why, it’s too long a story, suffice to say that for only $2 million I can get a beautiful three bedroom house in a good neighbourhood not far from the Uni of NSW. So that’s on the list. A new car, of course, perhaps a luxury vehicle, a Lexus say, since I am very happy with our old Camry. Another $2 million in the bank for a comfortable retirement ahead (not too soon, since I don’t want to get bored) would also be sweet. I can picture it now, a lovely new house, luxury car, money stashed away for retirement travelling and golfing.

Speaking of golf, how about a life membership at one of the nice golf courses close to where I live here. And please St. Peter, throw in a good set of clubs and some nice golf attire. Speaking of clothes, I haven’t redone my wardrobe in a serious fashion for many years: how about a gift voucher for a few thousand bucks at a local department store. I`ll want my theory of Rational Trigonometry to suddenly be understood by the academy, ensuring a promotion at work to full Professor; heh why not one of those Scientia Professorships that pay twice as much with only half the work? That should about wrap it up. But I still have another ten minutes to go. Am I being too modest here? What will my wife think when she finds out I had a blank cheque and came back with only a modest haul. And in a few years that Lexus will be a used car… Shouldn’t I be more ambitious?

And what if some friends come to visit? Let’s upgrade to a 5 bedroom house in Double Bay for $5 million. Let’s add a top of the range BMW, and–you only live once—a spanking new red Ferrari. Actually I don’t care much about cars, but surely I can get interested once these are parked in my triple garage. The rich and famous who live there are probably a bit stand-offish, so I`ll ask that I become an internationally famous mathematician: Rational Trigonometry will be such an educational breakthrough that I will be in hot demand on the lecture circuit, girls will ask for my autograph in the street, and pollies will invite me to their lavish dinner parties. Russell Crowe and James Packer will become my good buddies, and drag me to those tedious football games, where I can sit in the high class enclosures sipping champers. Speaking of, I need a wine cellar, and say 2000 bottles of the good stuff. Probably I want a library too, and a tennis court—let’s make that house a $25 million mansion at Piper’s Point instead.

But then all my new buddies will have holiday weekenders and villas in Tuscany. I want a holiday weekender too, say in Leura (I think I only have about 5 minutes left!!), and a villa in Tuscany. And a helicopter, and a horse riding ranch, say out near wherever James has his. And a private jet. Wait! I will need a lot more dough to pay for all the staff, maintenance and taxes that all this stuff requires. I want $100 million in cash. No, I want $10 billion in stock. Why limit myself? Why should Bill Gates outdo me in my future philanthropy? I want $100 billion in cash, bonds and stock. And–

What? It appears that St. Peter has been timing me, and I just ran over my thirty minutes. [Insert sad face here]. The deal is void. Whoops. Better not tell my wife about this.

But to tell the truth, it was all starting to sound like a major headache, if you know what I mean. Now I am back to the essential mystery of my life, and I better get off to work; I have a 9 am meeting.



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The infinitely real delusion, and my recent debate with James Franklin

In the last fifteen years or so, I have become increasingly disenchanted with the way modern mathematics deals with, or rather doesn’t deal with, the serious logical problems which beset the subject. These difficulties arise from a misunderstanding of the nature of `infinite sets’ and `the continuum’, and then extend further in many directions.

`Infinite sets’ are propped up, according to the standard dogma, by certain axiomatics, which lift the burden of having to actually define properly what we are talking about, and prove the various theorems that we would like to have true. What a joke these ZFC axiomatics are. The entire situation is ironic to the extreme: in fact Cantor’s Set Theory was vigorously opposed by most prominent mathematicians during his day, and then collapsed in a catastrophic heap at the beginning of the 20th century due to the discovery of irrefutable paradoxes. And now, fast forward a hundred years later: not only has Set Theory been resurrected, essentially with no new ideas—most of the key concepts go back to Cantor or Turing, and are just endlessly recycled—but now most of us believe that this befuddled and imprecisely laid out subject is actually the correct foundation for the rest of mathematics! This is little short of incredible. I feel I have woken from a dream, while most of my colleagues are still blissfully dozing.

And our notion of the continuum is currently modelled by the so-called ‘real numbers’, which in fact are far removed from most sensible people’s notions of reality. These phoney real numbers that most of my colleagues pretend to deal with on a daily basis are in fact hazy and undefined creations that frolic and shimmer in a fantasy underworld deep beneath the computational precisions of our computers, ready to alleviate us from the dull chore of striving for precise computations, and incorporating correct error bounds when we can obtain only approximations.

We are talking about irrational numbers here; numbers whose names even lay people are familiar with, such as sqrt(2), and pi, and Euler’s number e.

Supposedly there are myriads of other ones, given by various arcane procedures, formulas and properties. The actual theory and arithmetic of such real numbers is never laid out completely correctly; rather we find brief ‘summaries’ of the wished-for properties that these creatures have, properties that ensure that theoretically many standard computational problems have solutions, even if our computers can in fact not find them.

Ask a modern pure mathematician to make the computation pi+e for you, and see what kind of bemused look you get. Is not the answer the same as the question? Is this not how we all do `real number arithmetic’??

The belief in `real numbers’ supports a false mathematical dream-world where almost everything has a solution; a Polyanna fantasy land which can be conjured up by words but not written down on paper. (Of course the computer scientist or applied mathematician or scientist knows that in reality all meaningful computations occur with rational numbers or floating point decimals).

What a boon it is to live in the `infinitely real’ dreamscape of the modern pure mathematician! To conjure up `constructions’ and ` computations’ these days we need only scribble words, phrases and descriptions together. This is why so many of the ‘best’ journals are filled with page after page of what might be generously called `mathematical prose’. See my submission `Let H be a load of hogwash’ to get a feeling for this language of modern mathematics that the journals encourage.

Most pure mathematicians feel little obligation to address the claims of logical weakness. Objections such as mine may be safely ignored. Unlike scientists, we don’t feel the obligation to step up to the plate and respond rationally to criticism, as it clearly cannot be correct: since the majority rules! As long as we all play along, and ignore the increasingly obvious gaps between what our computers can do and what we are claiming, everyone can pretend that things are merry.

But could the tide be turning? A little while ago, James Franklin and I had a public debate (quite civilized and friendly I would add) in the Pure Maths Seminar in the School of Mathematics and Statistics UNSW, and lo and behold– the room was filled to capacity, people were huddled at the doors from outside trying to hear what was said, and my heresies were not met with a barrage of hoots, tomatoes and derision. Check out the debate either at the YouTube channel MathsStatsUNSW, or at my channel Insights into Mathematics (user njwildberger):

Judging from the many comments, it is no longer such a one-sided debate as it was a few decades ago. I reckon that young people’s comfort and trust in computers has a lot to do with it. What is it really, if you can’t get your computer to model it?? Only a fantasy.

You can join the revolution, too. Don’t be so accepting of everything you are told. Ask for explicit examples and concrete computations. Be suspicious of appeals to authority, or the well worn method of swamping with jargon. And of course, watch as many of my videos as you can, for a slow but steady introduction to: a more sensible world of pure mathematics.

Perhaps the forces of confusion and orthodoxy will soon be on the back foot.


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Austria, geometry and mountains

I have just attended the 16th International Conference on Geometry and Graphics, hosted this year in Innsbruck by Manfred Husty, Hans-Peter Schroeker and their team. It was a resounding success, and I had a great time, meeting new friends, from Mexico, Columbia, Russia, Serbia, Germany and elsewhere, and also old friends from here in Austria, Canada, Germany and Croatia.

Photo 2-08-2014 11 35 57 am

I gave a talk on work with my former student Ali Alkhaldi on the parabola in hyperbolic geometry, and canonical points associated to it, including our discovery of the fascinating Y-conic. Also on my mind is a paper on Incenter circles with my student Nguyen Le that I need to finish correcting for the illustrious journal KoG. I might tell you about that paper next time: in the meantime here are a few more pics of Innsbruck, whose German name means Bridge over the Inn (river). The Inn valley hugs the city from both sides, with outdoor activities, winter and summer, in the mountains directly accessible. Austrians who live here definitely stay fit!

Photo 2-08-2014 11 26 23 am

The old town has charm and character, and of course lots of tourists!

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Coming from Canada, and from Australia, it is interesting to imagine the pyschology of living in a city with such an august and established heritage; with the works of the ancestors constantly in view, and tradition playing much more of a role than where I come from. While the majority no doubt are strengthened and supported by the solidity and presence of that history, perhaps others feel confined by it?

Since my father is from Austria, I feel very comfortable in this country, and always enjoy my time here. The mountains are great, and on a nice summer day walking in the alpine countryside and forests, with grand vistas around, can’t be beat.

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Math Terminology for incoming Uni students

It’s been a while since I posted, I have been busy with the end of term, and our new video room in the School has kept me busy, putting together videos of solutions to first year tutorial problems (you can see some of this fine work at the School’s YouTube channel at mathsstatsUNSW) and getting ready to go overseas to Austria and Croatia for August and the first half of Sept.

But another interesting development is that I have dipped my toes into the world of MOOCs. If you’ve been following this blog, you know I have been musing about this topic, with mixed feelings. But better to get some experience directly, and since I have posted on Insights into Mathematics a series of videos on Maths Terminology, I thought I would put a mini-MOOC together. It launched this week, to great fanfare of course. :)

Seriously, you can check it out at, which is a very nice platform developed here in Australia for hosting courses (like Coursera and EdX I suppose). It’s orientation is towards student interaction, and I’ve had fun making crosswords, puzzles and quizzes to complement the YouTube lectures. It’s aimed primarily at students entering Uni or College, and planning on taking mathematics there, and the idea is to briefly review notation and terminology that they ought to know. Actually probably students from non-English speaking backgrounds might benefit most, but perhaps others will too.

The course only has 7 Modules, so you could finish the whole thing in a day if you were really dedicated. Here is the link in case you want to have a look:

Openlearning is headed by Adam Brimo, who has been very helpful in giving advice and information. The other guru behind the project is the famous Richard Buckland, from the School of Computer Science and Engineering at UNSW, who has also been helpful answering my dumb questions.

It’s early days, let’s see if interest develops. I am thinking that a platform like openlearning might be a good place to host discussions about Rational Trigonometry or the Foundations of Mathematics, allowing people to post, blog, chat etc.

We are also putting together at UNSW (we being Bruce Henry, Peter Brown, Chris Tisdell and Daniel Mansfield, with me) a PD course for high school maths teachers. That is coming along well, with the expert help of Iman Irannejad; who is a wizard with all things to do with filming and editing.

In a couple of days I head off to Austria for two Geometry conferences, one in Innsbruck and one in Supetar, Croatia. Should be fun, and hope to keep you posted.






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Bats, echolocation and Einstein’s Special Relativity

Lately I have been pondering Einstein’s theory of Special Relativity (SR). This has long seemed a fertile area to employ ideas of rational trigonometry, as the associated geometry, called sometimes Lorentzian geometry, rests on a symmetric bilinear form, and rational trig is all about setting up the machinery to study geometry starting from such a form. Quadrance and spread, the basic two measurements between points and lines, are simple rational functions of the dot product between vectors.

Perhaps surprisingly, I have slowly come to realize that SR actually can be derived not only from Einstein’s two basic postulates (that the laws of physics are the same in any two inertial frames, and that the speed of light is constant independent of the inertial frame) but rather from simple Newtonian mechanics, once we let go of the idea of an inertial frame and replace it with the simpler, more fundamental idea of an inertial observer. We replace a grid of equally spaced observers armed with coordinated clocks with just a single observer, armed with a single clock, and with a particular method of propogating signals, be it light, sound, water waves, or something else.

The whole story can be well described using the world of bats, who employ sonar echolocation to do their hunting at night. Turns out that many of the mathematical aspects of SR are already apparent in this humble setting. Sound, not light, is the basis of measurements. It is all rather surprising to me, and really only involves some elementary first year linear algebra.

I will be giving a talk about this subject in a few weeks here at UNSW: here are the details in case any reader is in the area and would like to come along.


Speaker:    A/Prof Norman Wildberger (UNSW)
Title:          Bats, echolocation, and a Newtonian view of Einstein’s Special Relativity
When:        12:00 Tuesday, 24 June 2014
Where:      RC-4082, Red Centre, UNSW, (Kensington campus, Sydney)

Abstract:   Einstein’s 1905 Special Relativity (SR) is a foundational theory of 20th century physics. While perhaps unintuitive and certainly surprising initially, it has a beauty and elegance which connects to a rich and interesting variant of Euclidean geometry. In this talk we present a simple but novel introduction to SR and the associated geometry, showing that the mathematical framework actually resides already in Newtonian mechanics, and could possibly have been discovered any time after 1700 if physicists had asked themselves the question: how would two (mathematically inclined) bats compare time and position measurements??

The unique abilities of bats to hunt their prey using (sonor) echolocation is one of the more remarkable aspects of the world of mammals. We will show that by adopting a `bat-centric’ point of view, and thinking about sound–not light!–as the source of physical measurement information, many of the standard pillars of SR, including Lorentz transformations, length contraction, time dilation, Einstein’s interval, and the twin paradox arise simply and naturally. Mathematically only some first year linear algebra is required. Holy Albert, Batman!


If there is any interest, we can have a Q&A session afterwards. Hope to see some of you!


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