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.

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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!

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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 http://www.openlearning.com, 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:

https://www.openlearning.com/courses/mathsterminologyfornon-englishspeakingunistudents

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.

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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!

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

 

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Let H be a load of hogwash.

Let H be a load of hogwash. By which we mean, of course, that H is an unbounded category of fuzzy schemas, expressed in the first order language of obfuscation with only countably many incompleted disjunctions.

Now take the space L of all cohomological Aleph one completions of H, partially ordered by increasing complexity—the de-facto mathematical convention in the beginning twenty-first century, but we spell it out for grad students—and consider the set N of all normalized functors from L to its contragradient.

The model space of N clearly has an adelic inductive boundary, which we denote by N_infinity. Let M be the infinite unstable tensor product of Aleph squared many copies of N_infinity, and take G to be the stable homotopy group of the measure zero projection of the affine homological dual of the K theory retract of M upon its enveloping quantum C* algebra.

While there are many fascinating questions arising from the inverse scattering problem of the functorial pair (L,G), we are naturally interested in considering the projective Hom groups of M into the space of all transcendental harmonic twistings of G mod its radical.

 Assuming the Axiom of Unrestricted Freedom with NP dominance, the associated cardinality of all semi-stable injections of H into the perverse sheaf of pseudo-differential connections of the cotangent bundle T(L,G) ought to be wildly inaccessible, making the whole subject a bonanza for further investigations and grant applications. Which of course goes to show yet again that ZFC is indeed finger-licking good. 

Just some thoughts I had the other day, which i thought I might share with you.

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The real information revolution

Johannes Gutenberg’s introduction of the printing press around 1450 was one of the defining moments of the modern age, ushering in a new era where knowledge could be cheaply reproduced and widely distributed. Since then the printed word has come to dominate our understanding of what information is.

Whether it be a book, a pamphlet, a newspaper or magazine article, a letter, a legal document, or these days a pdf or ebook, we have been completely ingrained to understand that information is printed information, and that to learn something means more or less to read it and understand it. This is the bedrock of the educational system, including of course tertiary education; with its heavy reliance on textbooks, libraries and learned journals.

All true, until now.

In only the last five or perhaps ten years, a new paradigm is suddenly upon us, sweeping through the modern world like a wildfire fanned by the deep untapped desire of people to learn by watching and listening, not reading. We are talking about video, my friends: most notably YouTube videos, but also of course iTunesU, Vimeo, Coursera, OpenLearning etc. Young people increasingly go to YouTube as a default if they want to know something; now already the 2nd largest search engine in the world—next only to Google—and moving quickly to become one of the prime repositories of really useful knowledge on the planet.

For one-sentence knowledge, the printed word will remain king. What is the circumference of the earth? Who was the president after Lincoln? Where was the first mammoth discovered? For such tidbits of knowledge, the printed word is optimal. For large-scale knowledge, the printed word may also be harder to replace. But for everyday middle complexity information, which requires, or at least requests, something of an explanation, video will rule.

How do I fix my lawnmower? Who were the greatest conquerors in history and why? What is Rational Trigonometry, and why is it so superior? Where are the best surfing spots in Sydney? What is the best way of chatting up a girl? For this kind of important info, and much, much else besides, most of us would rather get the answer from a person, using a combination of audio and visual representations. Video cannot be beaten here in my opinion.

While MOOCS and all kinds of fancy e-learning systems are much the rage in tertiary education these days, it is useful to keep in mind that the key ingredients are almost always the videos themselves. We are returning to the rhythm and logic of an earlier vocal tradition, where knowledge was memorized and passed on from father to son, from mother to daughter, from leader to followers—by talking, explaining, showing. This is far closer to our biology than the current arcane system of letters and numbers that form our printed sentences, like this one. If I was reading this out loud on a video, then my emphasis, pauses, expressions and posture would convey just as much, maybe more, than the words themselves. As it is, you have only the words.

Video as information is an idea which may well prove to be more interesting and important than video as entertainment. It is happening now, as we speak. When I started posting math videos on YouTube in 2007, most of my colleagues thought it was a strange use of my time. Don’t academics spend all of their energy writing furiously to continuously augment their all-important list of printed publications? What’s the point of posting videos that you will get little academic credit for?

Some of my colleagues probably still feel this way, but I bet they are a lot less confident now. They are perhaps starting to acknowledge something that students have long known—that even interesting and pretty mathematics may be difficult or painful to learn from an article or book! And some of them are starting to realize that if you don’t join the video revolution, your work runs the risk of being left behind, forgotten and unused, no matter how good it looks officially on a CV.

A salutary story for me: when I was a graduate student at Yale, I had a desk in the annex of the library on 11 Hillhouse Avenue; a somewhat dark and hard-to-find room in the basement which was stacked to the rafters with ancient math journals (for which there was no more room in the main library upstairs). Late at night, bleary from too much mathematical pondering, I would pull down a volume from on high and have a look into journals from the 1800’s. Creakily the dusty tome would relinquish its grip on its neighbours, having been unmoved in at least half a century: then I would skim these lovely, elegant articles, thinking—why is no one reading this great stuff anymore?, and— is this what will happen to my work once I am gone?

This need not be the future of today’s mathematicians. Well-presented videos of interesting topics embodying deep understanding will be regarded like gems of classical music to future generations of students and scientists, is my guess. Maybe this is a tad poetical, but I really do believe in the huge potential for broadening understanding and interest in the general public towards mathematics—that most beautiful of disciplines!

So, young mathematicians, take my advice—by all means play the game of oft and repeated publication in learned journals, but also spend some time developing your skills at explaining and presenting your knowledge and work through videos, so that your ideas will be accessible, useful and engaging to a wide spectrum of listeners. It is the future of publication, as much as it is the future of knowledge distribution.

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Politics and the two great fears

Australia is two weeks away from another national election, featuring Prime Minister Kevin Rudd (Labour) versus Tony Abbott (Liberal). I won’t bother you with the details, which almost everyone agrees are not that compelling. But it does provide an opportunity to muse about the broad division in politics between the left and the right—a schism that seems pretty universal, across the western world at least.

These two positions are natural responses to the two great fears that have dominated political thought for thousands of years. The first is the fear of the rabble—the unwashed hordes—the lawless thugs—violently taking what we have struggled hard to reap, build and create. The second is the fear of the leaders—the aristocrats—the powers in charge— stealthily taking what we have struggled hard to reap, build and create.

There are good historical reasons for both fears.

In times gone by, the most efficient way to amass great wealth was simply to steal it. Get yourself an army, the more impoverished and violent the better, offer your men pillage, rape and looting, and off you go up and down the countryside, taking whatever you please. If you are extremely successful, you will eventually be hailed as a great conqueror (Alexander the Great, Cyrus the Great, Caesar, Attila, Genghis Khan, Timor the Lame, Pizarro, Cortes, Napolean, Queen Victoria, Hitler, etc etc)—at least for a while, until your fragile imperial structures collapse. Or not—if you are competent and perhaps lucky.

On a smaller scale, the peasantry can simply revolt without the coherence of an army. Sometimes people just get fed up with suffering inequalities and injustices at the hands of their betters. With these kind of insurrections a lack of clear direction might mean that countries descend into civil disobedience, lawlessness, and lack of respect for property that can last generations.

At the other extreme, there is an almost opposite fear. In the absence of social upheaval or external military conquest, there are good reasons to worry about the abuse of power by those in charge within the society. This can take the form of a feudal hierarchy, where a small  cohort live the life of luxury at the expense of the peasantry, which is almost everyone else, or in the more modern form of a dictatorship, where a strong and ruthless leader at the head of a vested minority seizes power, brutally eliminates any effective opposition, and sets about taxing and bleeding the citizenry. Both situations have been historically very common and are still with us today.

Another more subtle variant occurs in our modern democracies, where the rule of law forbids the more extreme forms of exploitation of the masses, but where the ruling classes have none-the-less figured out how to slowly and surely tighten their grip on power, and accumulate ever more wealth and influence at the expense of the proletariat. The techniques are well-known and indeed obvious: the wealthy and powerful go to the best schools, meet the right people, obtain the positions of decision-making, and then naturally steer the legal structure in directions which favour them and their class. This kind of insidious transfer from the poor to the rich seems almost to be a kind of natural law in stable economies. See the history of almost any western country since World War Two.

So which side of politics is someone likely to be on? Usually it’s a pretty good guess, if you can find out the income and assets of the person in question. Belong to the top 20%? Then you are most likely a die hard Republican (USA), or Conservative (Canada) or Liberal (Australia) etc. Belong to the bottom 30%? Then you are most likely a die hard Democrat (US), Liberal or NDP (Can) or Labour (Aus) supporter etc.

And if you are in the middle somewhere? Then you probably and rightly fear both the poor and the rich taking your fair share. In that case you will be pulled and pushed by both groups—the right wingers trying to convert you for patriotic reasons or for fear of outside groups, and the left wingers trying to convert you for moral reasons. And there will be other parties, like the Greens here in Australia, that try to occupy more of that middle ground, but who find it very difficult to actually gain power, without either the support of the rich or the working classes/peasantry.

Naturally this is all very simplistic, but sometimes simple explanations have something to tell.

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The essentials of effective tertiary mathematics education

I am in Austria right now; having spent a week in Innsbruck visiting Hans-Peter Schroeker and Manfred Husty, I am now in Graz at the Technical University, and I am talking with Anton Gfrerrer, Sybille Mick, Johannes Wallner and Johann Lang of the Institute for Geometry, and giving some talks. Central Europe has a long and distinguished history of excellence in research and teaching of geometry. Here in Austria students learn Descriptive Geometry, Hyperbolic Geometry, Projective Geometry and CAD systems for visualization: if only Australian students were exposed to half as much!!

Its a pleasure to be in this part of the world, both personally and academically. My father is from Austria, born in Linz, and my mother from Liechtenstein, and so I have been travelling to this part of the world on and off for many years visiting relatives. Here is a picture from Triesenberg in Liechtenstein, where I stayed for a few days visiting my aunt Maria, who lives a few kilometers down the mountainside.

What a beautiful area of the world! Photo 15-05-13 8 00 55 PM

Here is a shot from a town in Austria called Landeck where I overnighted on the way to Innsbruck. Some lovely walks around the countryside there.Photo 19-05-13 2 28 59 PM And I suppose I better show you a mountain, of which there are many around, but sadly the weather has not been entirely cooperative for sunny photos. This taken from the train on the way to Graz from Innsbruck. Photo 21-05-13 11 42 30 AM

Today I want to tackle the challenge of succinctly summarizing some essential features, in my opinion, of effective tertiary level mathematics education. This follows naturally from my last blog on MOOCS. Of course there is much to be said here, but suppose we had to just write a paragraph or two: an excellent exercise for focusing one’s thoughts.

Putting together an effective university level mathematics course requires:

1. A prior solid understanding of the mathematical content of the subject, and its connections and application to other areas within and outside of mathematics.

2. A carefully chosen syllabus that lays out a logical sequence of topics: not too many, not too few, pitched at the right level for the intended audience.

3. A written text, either notes or a book, which covers in detail the syllabus of the course, including a wide variety of examples. This can of course be online.

4. A series of lectures, given either live or via video, which explain the course content but perhaps do not go into quite as much detail as the written notes. These lectures should be obviously accessible, interesting and useful to the students for learning the material.

5. A complete and comprehensive collection of exercises for students to attempt. These should help students gain familiarity and mastery of the course content, to develop problem solving ability, and to spark further interest in other aspects of the subject. The exercises may possibly be organized into various levels of difficulty if appropriate.

6. A carefully prepared set of worked solutions to many of the exercises, and summary answers to the rest. This could be either in written or video form, or both.

7. A mechanism for grading student work at solving exercises and writing up solutions, and providing a reasonable level of feedback on their written work.

8. Effective and fair tests and final exam, that motivate students to study, review and ultimately absorb the material.

So if you can manage to incorporate all these aspects in a coherent way, you will for sure have an effective mathematics course.

And how much of this could be done on-line? Almost all of it, with the important exceptions of 7, and possibly 8. This is the key challenge in setting up online courses in mathematics—(in fact also of regular courses at university level!)—how to provide good feedback to students on the exercises that they ought to tackle.

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