Defending tau

The Daily Mail today has a piece about the proposed alternative to the number pi, tau (τ), equal to 2 x π.*

Basically, the reason we might want to change to tau is simple:

Pi is defined as the circumference of a circle divided by its diameter. We came up with this definition thousands of years ago, before modern geometry had taken off, and it’s a perfectly good number.

However, we now know that the diameter of a circle – the distance all the way across – isn’t really the best way to measure a circle. Instead, it’s better to measure the distance the radius of the circle – the distance from the centre to the edge. For example, if you’re working out a planet’s orbit, the distance from the planet to the Sun (radius) is a nice logical thing to measure, but the distance from the planet to the other side (diameter) of its orbit doesn’t really mean anything important.

Since the radius is half the diameter, circumference divided by radius is twice as big as circumference divided by diameter, which is why tau is twice as big as pi.

The definition of 1 radian. The red line (radius) and blue line (arc length) are the same length. By Stannered, CC-SA-BY-3.0

Tau is also nice because it makes working with angles a bit easier. The most natural way of measuring angles – the way you have to do it if you’re doing maths or physics – is to measure them in radians. The radian is defined in terms of the circle’s radius – one radian is the angle you get if you walk around the circle for a distance of one radius. If you travel 2π radians, you travel a distance which is 2 times pi times the radius, or pi times the diameter; in other words, you travel the length of a full circle, 360°.

However, that 2 in there is a bit of a pain. It basically means that, in physics or maths, whenever you’re dealing with a circle or a wave you get annoying factors of 2 popping up in your equations, and unless you’re very careful, it’s easy to forget a factor here or put an extra one in there, making your sums completely wrong. For example, to switch between ordinary frequency and angular frequency (effectively, switching from revolutions per second to radians per second), you multiply the frequency by 2π. This is a change we need to make a lot when working with waves, and it’s so easy to lose a factor of 2 when you’re working with a bulky equation.

That factor of 2 is only in there in the first place because we made the mistake of basing pi on the diameter instead of the radius. If we replaced pi with tau, 1 circle would be τ radians, and that factor of two would disappear. It wouldn’t be a groundbreaking change, but it would still be quite nice, and it would make the mathematics of circles a bit easier to understand. Of course, the hassle of teaching people to use tau instead of pi is probably greater than the benefits, so its unlikely we’ll ever give up pi.

Unfortunately, the Mail is quoting from a (paywalled) interview in The Times with University of Leeds lecturer Kevin Houston, whose and they seem to cut down what he’s said to just:

‘Mathematicians don’t measure angles in degrees, we measure them in radians, and there are 2pi radians in a circle,’ Dr Houston said.

“That leads to all sorts of unnecessary confusion. If you take a quarter of a circle, it has a quarter of 2pi radians, or half pi. For the number of radians in three quarters of a circle, you have to think about it. It doesn’t come naturally.

‘How much simpler it would be if we just used tau instead of pi,’ Dr Houston added. ‘The circle would have tau radians, a semicircle would have half tau, a quarter of a circle a quarter tau, and so on. You don’t have to think.’

Trimmed like that, it sounds like Houston is only promoting tau because he’s too stupid to work out three quarters of two (his Youtube video, linked by the Mail, explaining the mathematics of tau, shows this is not the case). So perhaps naturally, Daily Mail commenters have leapt at the chance to prove that they’re smarter than a mathematician, because really, what do mathematicians know about mathematics?!

If an ‘A’ level Maths student has trouble with the difference between Tau and Pi, then they should be on another course. Any excuse to dumb down the kids and stop them thinking for themselves!

Alan, Frankfurt, Germany (ex pat)

Right on! How dare we dumb maths down by making it slightly clearer!

‘The circle would have tau radians, a semicircle would have half tau, a quarter of a circle a quarter tau, and so on. You don’t have to think.’ …..It may be appropriate for DM readers who like not to think, but I’m afraid mathematicians do think, and aren’t interested in tau, thank you very much.

rupert, Yors4hire, UK

Except for all the mathematicians and scientists who support tau I daresay because they think.

This tau thing is clearly aimed at mindless rote learners but to those of us who actually understand maths, pi expresses something meaningful which is precisely why we refer to it so much.

Vincent, Glasgow

Pi and tau express more or less the exact same thing – the shape of a circle – tau just expresses it in a slightly more logical way.

As a retired Maths teacher this idea lacks intuitive sense. The whole point of the exercise is to relate the circumference of a circle to its diameter – and the ratio is the irrational number 3.14159… The ratio is NOT 6.whatever! That’s the basic theory. Then there are various formulae needed at basic school level: for example, the area of a circle is pi time radious squared. The new formula would be tau times radius squared divided by four: an extra calculation step. Or the volume of a cone pi r squared h…. and so on. And in times of austerity how many schools could afford a new set of textbooks filled with tau? (And all authored by Leeds people, no doubt!) And then there are the millions of calculators with the pi button built in…. No! This idea is like trying to say that from 2012 we’ll all drive on the right – indeed the idea of tau is worse than that because there are regions of the world already driving on the right….. This is a big UM and No No

Andrew, Cwmbran

No, but the ratio of the circumference of a circle to its radius IS 6.whatever (6.28318… in fact). The fact that the area of the circle becomes a half times tau r squared (0.5 τr² – there’s no dividing by four involved) is one of the downsides to tau, but as Kevin explains in the video, there is a deeper mathematical reason why we would expect that half to be there, so while it’s annoying it does make sense.

This is the way you can waste a lot of time while avoiding doing anything of real interest or value to anyone. What a waste of brain cells!! Is this supported by public funds? There are real problems in the world today that are worthy of serious consideration, but when academics waste their time on things like this, they prove that they have no value to society at large and should be dismissed.

Samuel, Dubuque, Iowa

One mathematician making a 5 minute Youtube video in his spare time? How dare he! He should devote every second of his life to curing cancer!

Big Wow… Is “2pi” the only mathematical innovation Leeds University can come up with, it’s best 21st Century contribution to the advancement of mathematics? Talented Maths kids doing their 5 science A levels must be crossing Leeds off as even a 5th choice.

Russ H, Bucks

Again, this is one guy working in his spare time. This is not the only thing the University of Leeds Maths department does! (Incidentally, 55% of Leeds pure maths research is “world leading” or “internationally excellent”, and a further 40% is “internationally recognised”.)

BUT….e^([pi]i)]=-1 and this does not work for tau. There are uses for Pi beyond circles!

Miles, Australia

(This refers to Euler’s identity, e^πi = -1, which gives us a nice way of representing imaginary numbers using angles. And despite what Miles says, it’s all about circles.)

That’s true, it doesn’t. Instead, e^τi = 1. This is even better than Euler’s original version, since we no longer have that minus sign (and all that minus sign told us is that pi radians = half a circle, something we already know). Quite a few comments are like this – “OH NO EULER’S FORMULA IS BEAUTIFUL, WE CAN’T CHANGE PI COS THAT WILL BREAK IT” – and yet no-one actually bothers working out what it would look like with tau.

Professional jealousy. Eienstein acceepted Pi and I can assure you he was more intelligent than this egghead. This man just wants to make a name for himself. Change all the books indeed. He is “daft” !

Ruckus, myrtle Beach SC (ex pat)

And why do we bother speaking English? If German was good enough for Einstein, it’s good enough for us!

All those comments have been upvoted, by the way, unlike this comment, which currently has a score of -1:

I am in full favour of this proposition. Unfortunately, a large number of these comments seem be be from people with only a basic understanding of mathematics. Using Tau in place of Pi would reduce the need for a constant in a plethora of standard calculations involving circles and spheres, the like of which children will be schooled in. Furthermore, the simplicity of the equations using Tau may increase understanding and encourage children to pursue an interest into further mathematics. From my experience of mathematics lessons, the majority of students didn’t dislike mathematics, but rather found it too complicated to enjoy. Once provided with a topic they were able to grasp, students began to enjoy working the problems. Once children have achieved a satisfactory grasp of simple circle equations, the transition to understanding the calculus is a much easier one. I think that should the readers have been taught using Tau the comments here would be better informed.

Pep, Manchester

Good old Daily Mail comments, eh?

* They claim the idea was invented by Kevin Houston at the University of Leeds. In fact, it’s much older than that – Bob Palais first came up with it in 2001 – and Houston is just promoting it. Also, as a conflict of interest thing, I guess I should point out that Kevin taught me a few years ago, and is a thoroughly nice guy.

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The Supermoon did NOT cause the earthquake

Normally, scientists are meant to be circumspect and reserved about cause and effect. We’re supposed to deal in probabilities, statistics and uncertainties. However, on this occasion, I have absolutely no qualms in saying this: The Daily Mail is wrong. The “Supermoon” had nothing to do with the terrible earthquake off the coast of Japan today.

Ben Goldacre and Phil Plait have already produced some admirable rebuttals to the Mail‘s exploitative scaremongering; these are my thoughts, but I recommend reading the linked posts.

The powerful tsunami that today slammed into Japan’s eastern coast comes just two days after warnings that the movement of the moon could trigger unpredictable events on Earth.

Except none of these “warnings” come from scientists. The only people who took the “Supermoon” seriously were internet conspiracy theorists and the tabloids. And…

Astrologers predicted that on March 19 – a week tomorrow – the so-called ‘supermoon’ will be closer to Earth than at any time since 1992, just 221,567 miles away, and that its gravitational pull will bring chaos to Earth.

An astrologer is the not same thing as an astronomer. A warning from an astronomer should (generally) be taken seriously. A warning from an astrologer isn’t worth the paper it’s printed on.

Others on the Internet have predicted it will cause further catastrophes such as volcanic eruptions and earthquakes.

“Others on the Internet”? Of course, everything that was ever written on the internet MUST be true.*

* Fully aware of the hypocrisy, thank you very much.

However the ‘supermoon’ date is still eight days away. But those that adhere to this particular belief could claim that this was still close enough for there to be some kind of effect.

Ok, as everyone knows, the moon orbits once a month. Once every two weeks, it comes close to the Earth – this is the “perigee” of its orbit – and in between these perigees we have the furthest points: the “apogee”. This means that eight days before the perigee, it will be as far away as possible from the Earth; the exact opposite of a “supermoon”. Its gravity would be weaker, not stronger.

The energy needed to produce an earthquake builds slowly through stress and strain in the rocks where tectonic plates meet, over years or decades. A tiny change in the position of the moon – a change which, remember, happens twice a month – will have virtually no effect compared the constant force of billions of tons of rock pressing against each other.

Two days ago, in an interview with ABC radio discussing the potential impact of the March 19 supermoon, astrologer Richard Nolle, who first coined the term in 1979, said he was convinced that lunar perigees cause natural disasters on Earth.

‘Supermoons have a historical association with strong storms, very high tides, extreme tides and also earthquakes,’ he said.

The only one of those that is true is that “Supermoons” can cause high tides (and even then, it only changes the tidal range by a couple of percent). There is no association between “Supermoons” and storms or major earthquakes. And again, an astrologer is NOT an astronomer.

Natural disasters are unfortunately common. Earthquakes, storms and floods happen with depressing regularity – look hard enough and you can find disasters happening around any supermoon. What’s important is to the look at the disasters that weren’t linked to a supermoon. No scientific study has shown an increase in quakes or storms around supermoons.

What do the actual scientists say?

Dr David Harland, space historian and author, said: ‘It’s possible that the moon may be a kilometre or two closer to Earth than normal at a perigee, but it’s an utterly insignificant event.’

Professor George Helffrich, a seismologist at the University of Bristol was equally dismissive.

‘Complete nonsense. The moon has no significant effect on earthquake triggering.

‘If the moon triggers “big” earthquakes, it would trigger the many of millions of times more “small” earthquakes that happen daily. There is no time dependence of those; hence no moon effect.’

[…]

John S Whalley, geoscience programme manager at the University of Portsmouth, agreed there was no correlation.

‘There is no established correlation between variations in the orbit of the moon and either the number or magnitude of earthquakes.

‘It is all too easy, with hindsight, to link major earthquakes to variations in all sort of parameters.

‘The real test is to look at the vast numbers of earthquakes of all magnitudes that occur on a daily basis worldwide.

‘Any correlation with the lunar orbit would have to be established on the basis of this population of earthquakes, not on individual high magnitude events. In need hardly add that no such correlation has been established.’

I would end this post with another glib link to the SMBC comic, but frankly I don’t feel like this the right time for daft jokes. Nor is it the appropriate time for a newspaper with a readership of millions to be giving a platform to charlatans and conspiracy theorists, who seize on a terrible tragedy just to get a little bit of publicity for their pseudoscientific claims.