Archive for September 19th, 2005

Rational Trigonometry

A number of news outlets are reporting that an Australian professor has just rewritten the laws of trigonometry. Rather than using tools designed for circles, he uses squares and “spread”. Apparently the current system of trigonomtry is error prone, which doesn’t suprise me, but I’ve always been fascinated about what we (as a society and schools) would do if there was a major change in the way we carry out mathematical procedures. I’ve always felt that there was a time when we were very open to change in Science, Maths and English, but lately I feel we have become stubborn and these rules which may or may not be correct are bound into some kind of inflicted concrete. Perhaps it is just that there haven’t really been any major theoretical changes in my lifetime, sure computers have entered mainstream use, but it never seemed like a difficult thing to implement into classrooms, it happened gradually, helped in some ways by computer companies like Apple giving computers to schools for their own purposes, and in effect helping to bring a new generation into a new world.

Allow me to deviate slightly. Apple were very generous, particularly in parts of Australia where they gave schools a bunch of Apple Macintosh computers. Apple believed that students growing up with, and learning to use, Macintosh computers would grow up and continue to use Apple Macintosh. Although this wasn’t entirely unreasonable thinking, it wasn’t exactly correct. Microsoft, who Apple contracted to help write the first version of Macintosh (incredibly buggy by the way), then went and wrote Windows based on similar code for a different architecture. IBM and Microsoft had a bit of a partnership going with Windows and OS/2, however Microsoft did some evil things to IBM and released Windows 95, based quite firmly on OS/2 code, whilst retaining backward compatibility with older versions of Windows. Windows 3.11 (and previous) applications were able to run on OS/2, but Microsoft weasled out of their obligations with IBM and manged to give them a solid push into a downward spiral. By locking OS/2 out of running Windows 95 applications, they effectively drove nails into the coffin of a very good Operating System. Windows 95 was also the first Windows to look like Macintosh, and with the combination of code from Mac and OS/2, Microsoft obviously thought they had won the war, unfortunately (or fortunately I suppose) they were in such a rush that they botched the OS and left it filled with bugs and security issues that they have never been able to fix. The current incarnation of Windows continues the line of work that started with these rushed combinations, and it has become quite apparent that the people at Microsoft really don’t know how to write a decent Kernel or Operating System. It may be dominant, but it is far from perfect, and fixing it would break all the existing Windows applications, so they can’t do that…catch 22.

Anyway, back to Trigonometry and educational change. It seems that teachers and textbooks all rely on the existing rules, and even simplifactions and clarifications to these rules take time to be implemented. Apparently the new rules of trigonometry will make calculations easier and more accurate, so I do hope that they are understood and implemented soon. Due to the inherently simpler nature of the newer rules, I wouldn’t expect much retraining will be required for people who use and teach the existing rules, the thing that will take time, however, will be the implementation of the new rules in people’s heads.

After a while, the notion of how to utilise trigonometry becomes second nature, and that is the tricky bit. Whilst you can learn a new way to do something, it is not always easy to actually do it the new way when it comes to the time that you need to do it. In fact, that takes a lot longer, and that is what will slow down the implementation of the new rules.

The same thing happened with the conversion from imperial meaurements to metric measurements. People still talk in yards, pounds and inches because those are the measurements that are second nature to them. Mind you, I think there would be greater resistance to a measurement system change now than there was when we changed from imperial to metric, mainly due to that stubborness I mentioned earlier.

ABC Online explains the new Trigonometry pretty well:

The mathematical study of triangles has just got a whole lot simpler, according to a researcher who says his new theory of trigonometry is easier to use and more accurate.

Associate Professor Norman Wildberger, of the University of New South Wales in Sydney, Australia, says his theory of “rational trigonometry” is more like algebra as you can plug numbers into an equation and get an accurate result.

“We’re going to look at trigonometry in a new way,” says Wildberger.

“We’re going to leave sines and cosines to the circular motion part of mathematics and not force it on triangles.”

Wildberger says the trigonometry we know and love (or hate) today has its historical roots in the work of ancient astronomers, like Ptolemy, who studied the motion of planets.

In this case, the angle between two points in the sky, as seen from Earth, was a reasonable way of calculating the distance between them, he says.

But, says Wildberger, the problem came when others applied these theories, developed for spherical geometry, to the study of flat triangles.

“It’s all very well to do if you’re working on a sphere or a circle but when we actually study triangles there aren’t any circles there,” he says.

New concepts

The key purpose of trigonometry is to understand the relationships between the corners and sides of triangles.

It is used in areas like surveying, engineering and construction today.

Classical trigonometry calls the separation between two lines an “angle”, which is the length of a circular arc between two lines.

An angle can be calculated using an equation that relates the corners of a triangle (using the concepts of sine, cosine or tangent) to the length (distance) of the side opposite it.

But Wildberger says that distance is not the best way to measure the separation of two points and angle is not the best way to measure the separation of two lines.

“It’s not the concept that leads to a mathematics that is the most pleasant and the most useful and the most accurate,” he says.

Instead of distance, Wildberger’s trigonometry uses a concept called “quandrance”, the square of distance.

Instead of angle, he uses the concept of “spread”, calculated by dividing one quadrance by another.

The spread between two lines is a number between 0 (representing parallel lines) and 1 (representing lines at right angles).

Wildberger says it would be possible to make a new protractor that measures spread instead of angle.

You would then plug the values for the quadrance and spread into his set of equations.

More accurate

What’s better about the system, says Wildberger, is that all the terms in the equations can be calculated exactly, or are “rational”, hence the term for his new theory, “rational trigonometry”.

But sine, cosine and tangent, are usually only approximated, he says, making them “transcendental functions”.

This means that any complex calculation using classical trigonometry could result in a significant accumulation of errors.

Wildberger says he hopes that “rational trigonometry” will provide high school students with a simpler way of thinking about triangles that is both more accurate and easier to carry out.

And he says the improved accuracy will be important elsewhere such as in GPS surveying or when engineers design devices.

Wildberger says he developed the new theory while studying more complex maths.

“It wasn’t that I set out to do this. It was just a fluke, in a way, that I realised ‘Hey these ideas can change elementary trigonometry’.”

Wildberger’s book, Divine Proportions: Rational Trigonometry to Universal Geometry, published by Wild Egg books, is launched this week.

Samuel

September 19th, 2005 at 09:09pm

John Stanley’s Unlucky Caller

Regular readers might remember from the comments on this post, that I’m not a huge fan of John Mangos. I must say however, that in the last few days I have begun to like him. It does usually take me a few days to get to like him again, either that or it takes him a few days to settle in. In the last few days I think his presentation has changed slighty, possibly more towards the comments I made in that post about how I would run an afternoon show. Maybe TinMan passed on the message?

Anyway, you may also recall from that post that John Stanley had a caller make a bet with him. If John Stanley could avoid saying “good on you” for an entire show, the caller would dress up in a chicken suit and wave a Dragons flag by the side of a busy road near the St. George Leagues Club. John Stanley only just managed to survive that bet, and the caller followed through on his part of the deal, as can be seen in these photos from the 2UE website.
John Stanley\'s Chicken Man
John Stanley\'s Chicken Man
John Stanley\'s Chicken Man

Samuel

2 comments September 19th, 2005 at 07:18pm

2UE Continue To Have Fun

I’ve been noticing a few things over the last few days. 2UE are definetly still settling in to their new studios, and the presenters/panel operators have had a lot of fun. There have been plenty of incorrect sounds playing, callers that seem to invoke strange noises, and the best mistake of all last night.

2CC’s ads weren’t playing last night, which meant that John Kerr’s little accident was quite audible, he managed to leave his or one of his guests microphone on during an ad break. Normally this would be difficult to hear as the ads would have drowned out the noises in the studio, but because there were no ads, the minimal bits of noise prevented the emergency tape from kicking in, and we were able to hear John Kerr talking to his regular guests Simon & Dale (who review movies and other related things every 2nd Monday at 1am) about their thoughts of the new studios, he was also informing them of the details of a movie screening that he had invited them to. This was just before the 2am news and it was quite entertaining hearing the people moving around the studio as you could quite clearly hear them moving closer to, and further away from, the microphone. Eventually we heard the producer in the distance saying something, followed by running noises and then the microphone turned was turned off, leaving us with silence and then the emergency tape.

I have also noticed over the last few days that the pulse which starts the ad breaks is arriving before the audio, sometimes by as much as two seconds, which leads to ads talking over the top of the presenters. Also the 2UE promos are often audible in the background as they return from the ad breaks.

Interestingly John Laws went to take a call this morning and somehow set off the 2UE news theme, which not only confused and entertained John, it probably sparked mass confusion amongst relay station staff who would have looked at the clock to double check the time. I certainly checked the clock on the computer I was using at the time.

George Gibson was plagued with computer issues the other night, 2CC had their pre-news intros, followed by the time beeps and then no news theme, 2UE’s news started and then mysteriously switched to 3AW news, they didn’t send a pulse at the end of the news so there was 15 seconds of dead air and 15 seconds of backup tape, followed by something rather odd, a pulse and the 2CC news theme, with George Gibson talking because his intro didn’t play and his computer wasn’t behaving, which seemed to prevent him from taking phone calls. He took an ad break (which suprisingly played an ad break and not the missing weather on 2CC) and then at the end of the ad break George played his intro and was able to take phone calls again.

I’m sure these issues will be worked out eventually, but for now they produce quite a bit of entertainment for people like me.

Samuel

September 19th, 2005 at 12:01pm

Broadband Over Powerline Woes

Broadband Over Powerline (BPL) is an exciting new technology that utilises the existing powerlines to bring broadband Internet access to every home with a connection to the elctricity grid. BPL is really just a new adaption of existing Power Line Communication technology used in such things as home automation.

The alternating current that currently uses the power lines operates at 50-60 Hz depending on your location, whilst the newer BPL signals operate at around 1-30 MHz and can bring speeds of up to 2.7Mbit to homes. Whilst this is certainly a clever technology, it has certain issues.

Some people may have already worked this out simply from reading the above, but I’ll explain it anyway. Powerlines tend to be unshielded and untwisted, making them perfect antennas, and this also means that they are very good at creating radio frequency interference.

The main problem with BPL is the enourmous amount of interference that it does create, in some cases blocking out AM radio frequencies, as well as some amateur, government and defence frequencies. There have been reports of interference to the FM frequencies as well, although I find that slightly harder to believe.

Considering that virtually every suburban power line would be acting as a radio transmitter, the concerns of people in the radio industry (especially amatuer radio entusiasts) are quite understandable.

It’s not all bad news though, as the FCC and the ACMA (formerly known as the ABA) have released rules requiring “notching” of the frequencies used by BPL to avoid the interference. The majority of the BPL systems that are capable of notching frequencies operate at mainly higher frequencies and are therefore capable of higher speeds, it is expected that they will be able to provide speeds in excess of the 24Mbit potentially provided by ADSL2+

The important thing for now is making sure that the regulators are kept up to date on what interference is occuring, and to make sure that the electricity companies know that they can’t take shortcuts and create unacceptable interference.

BPL will be the way of the future, potentially providing much more than just Internet access, but it is important that it is setup in the correct manner, and the only way to ensure that is to keep a close eye on the regulators and the companies involved. I for one don’t want to lose AM radio, FM may be technologically superior, but I find AM has many benefits, including it’s incredible ability to “bounce” for incredibly long distances at night. People don’t seem to realise that AM radio is capable of stereo transmissions, or that digital radio will probably use frequencies currently used by AM radio. Certainly FM is better at reproducing music, but it is more suceptible to frequency drift.

I’m sure John B1_B5 understands the theory behind AM and FM better than I do and will hopefully be able to explain things a bit more clearly and correct any mistakes I may have made. I’ll admit that I don’t really understand the theory behind AM and FM radio very well, but I’ve done my best…

Anyway, I’m all for BPL, as long as it is implemented in a correct and cautious manner.

Samuel

12 comments September 19th, 2005 at 10:50am


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