Tag Archives: STEM

The Deceptiveness of Coincidence

A friend of mine recently posted about a chain of events – people sharing birthdays – that was so unlikely that a lottery ticket purchase was called for. Most people might make similar comments as the oddity of these events struck them. There followed some discussion about these problems and it made me think of trying to set out the issues more carefully.

I’ve done a lecture, in some variations (for school audiences), almost every year about the deceptiveness of common sense in mathematics. Indeed, it provided the title of this blog.

The  Birthday Problem

One of the problems I examined is the famous birthday problem. Let’s look at two aspects of this.

How many people must be in a room before we are certain that two of them share the same birthday?

This can be answered by using the so called pigeon hole principle. In this case, the pigeon holes are all the days of the calendar in a year, 365, or let’s say 366 to allow for the worst case scenario of a leap year. The people in the room are the pigeons. You can place 366 pigeons into the holes and keep them all separate, but any more ensures that at least two must share one of the holes. So, similarly, you need 367 people to be absolutely sure that one pair in the room shares a birthday.

Let’s look at a more subtle question.

How many people must be in a room so that it is more likely than not that two of them share the same birthday?

It turns out this number is deceptively low. We reach a 50% chance (more likely than not) with just 23 people in the room. Perhaps even more surprisingly we reach a 99.9% chance with 70 people in the room. The calculations for the probabilities are set out in the link to the lecture, and the wikipedia article as mentioned above, but I don’t want to focus on that. I want to focus on why the result is so shocking to most people.

The issue is the collision of relatively rare events (a birthday) with numerous combinations. For instance if you want to consider how many pairs one can form in the room we can see this.

If there are three people in the room, A, B and C, there are three possible pairs: AB, AC, and BC. If there are four people in the room, A, B, C and D, there are six possible pairs: AB, AC, AD, BC, BD, and CD. There’s actually a nice formula for this and you can work this out for more examples.

You don’t need to know the formula, but if you want it is  \frac{n!}{(n-2)!2!}.

People in Room Possible Pairs
3 3
4 6
10 45
15 105
20 190
23 253
30 435
50 1225
70 2415

You can see just how rapidly the number of pairs increases. It might seem un-obvious that 23 people is as likely as not to produce a duplicate birthday, but it’s perhaps less so when you realise that we are not predicting which pigeon hole has two pigeons in it to go back to our analogy. But given 253 possible pairings, it seems less unlikely even intuitively that there will be “collisions” within one of those pairs.

Coin Tosses

Let me talk about a similar problem, flipping a (fair) coin to get a Head (H) or a Tail (T). Let’s consider that we flip the coin three times. The possible outcomes are:

HHH, HHT, HTH, THH, TTH, THT, TTH, TTT.

These are all absolutely equally likely, and even this is sometime unintuitive. The coin has no memory of how it performed, and no desire to conform to some concept such as a “law of averages“. But we tend to categorise these 8 possible events into 4 categories.

  • Three Heads: (one way this can happen, so 1/8 probability)
  • Two Heads, One Tail: (three ways this can happen, so 3/8 probability)
  • One Head, Two Tails: (three ways this can happen, so 3/8 probability)
  • Three Tails: (one way this can happen so 1/8 probability).

In doing so, we can see why the HHH and TTT events look particularly exciting and rare. They are no more rare than any other individual outcome, but it’s the groupings of how one can get two heads that makes that collection of events more unexciting.

To take this to more of an extreme. If we flip a coin ten times, then these outcomes are all exactly equally likely.

  1. HHHHHHHHHH
  2. HTHTHTHTHT
  3. HHTHTTHTTH
  4. THTTHHHTHT
  5. TTTTTTTTTT

But some will be far more remarked upon that others. But even if we have an experiment that flips a coin 10 times over and over again enough times, we will expect to see all of these (more or less, eventually) equally often, but we will probably not consider these exciting except maybe outcomes 1,5 and maybe 2.

Genealogy

Another example, closer to home. Let’s assume that there have been 7,500 generations of Homo Sapiens since its emergence as a species.

The probability that in each of those 7,500 generations in both your paternal and maternal line that all the pairings took place between the right people at the right time in the right circumstances to give rise to you is astronomical. You have two parents, four grand parents, 8 great grand parents and so on. Just for that collection of people. If we add you into the mix, then the number of pairings involved over these 7500 generations is

2^{7501}-1

That number has 2259 digits, you can see them here on Wolfram Alpha.

That’s the number of people involved to get to you, including you. They all had to survive and meet at the right time, and um, do everything else at exactly the right time in the right way. What are the odds? And yet you are here… You might find that exceptional, and of course in a way it is, but the point is, you are not more likely or unlikely than most other human beings that have been born, or countless potential humans who might have been but were not born because their parents never even met. The fact that you specifically exist is unlikely, but the fact that if you did not someone else would exist instead is highly likely and predictable.

Naturally, however, we tend to think of our own existence rather more than some random person on the other side of the planet. But each one of us is an example of the coin flipping perfectly as needed every single time. Sometimes a Head, sometimes a Tail, but that exact perfect sequence essential to produce you. Naturally we are each quite taken with the outcome that produces us and also our nearest and dearest. But if it had not happened, we wouldn’t be here to wonder about it.

Availability and Psychology

As you can see there is more to this than mathematics and probability, there is also perception and psychology, specifically a concept called the Availability Heuristic. Some things stand out to us as being significant when they are no more significant than other events we ignore.

You might live in a city of a million people. You might personally know only a hundred. You might pass a thousand on your commute to work each day. You do that hundreds of times a year. The number of possible combinations of people you will see over a year is enormous, but you won’t remember it as a special event when you bump into all the people who aren’t dear to you.

The Anthropic Principle

A nice example of this thinking is the question that many people have asked which is essentially: why is the Universe, and our Earth, set up so nicely to allow humans to exist? If certain physical constants were just a bit different then matter as we know it couldn’t form. If we were just a bit closer to the Sun we couldn’t survive and so on.

These things are all true. But in a Universe with inappropriate physical constants there will be no matter by definition, and certainly no sentient matter like us to notice.

If the Earth was closer to the Sun, homo sapiens would not exist as it currently does, but like as not some other species would, and it would also think how perfectly the world was formed for them. This concept, that we tend to think the Universe is so special to have created us, despite the fact that if not we wouldn’t be here to think about it, is often called the Anthropic Principle.

We believe that the improbability of life itself or us as a species is an example of the collision between billions of individually improbable events with the billions of events than could have occurred, and the billions of years in which they might have done so.

We end up having to think about such things with great care because our evolution on the African savannah didn’t really equip us to consider them. So we shouldn’t be surprised that we are surprised, but we still are.

It’s the old cliché: you are unique and special; just like everyone else.

You are unique, just like everyone else

There are always dangers in imagining our special place as a species.

“This is rather as if you imagine a puddle waking up one morning and thinking, ‘This is an interesting world I find myself in — an interesting hole I find myself in — fits me rather neatly, doesn’t it? In fact it fits me staggeringly well, must have been made to have me in it!’ This is such a powerful idea that as the sun rises in the sky and the air heats up and as, gradually, the puddle gets smaller and smaller, frantically hanging on to the notion that everything’s going to be alright, because this world was meant to have him in it, was built to have him in it; so the moment he disappears catches him rather by surprise. I think this may be something we need to be on the watch out for.”

Douglas Adams, The Salmon of Doubt

 

Academic Family Tree, LaTeX and Tikz

A few years ago, I found out some information on my academic genealogy, going through my supervisor, Brian McMaster, back through others to G.H. Hardy, Newton and Galileo and a little further before the records run dry. Of course it is nice that mathematics is an old and well established discipline with great records. And also, the number of scholars back then was much lower, so it’s not surprising that famous names turn up quite quickly, if you just have the full records – just as on a less cerebral level – the cliché goes that Charlemagne got about quite a bit. A bit of work with my mathematical colleagues all show such interesting family trees.

Anyway, I slowly built an SVG file in Inkscape to reflect this family tree, but it was slow and rather a pain to maintain. So I’ve long wondered if it can be done better in LaTeX, with the Tikz graphics packages. Tikz is awesome, and while it has quite a learning curve as for LaTeX itself, I used it extensively in producing the slides for my recent inaugural lecture. This meant mixing the formulae with the graphics was pretty effortless. This is a slightly different problem.

A quick scan, showed that Tikz can be used to build flowcharts rather easily. So I began in this way, building nodes of people and then positioning them relative to each other. A bit tedious but much more pleasing that doing it in a vector graphics program, for me anyway. I rapidly ran into problems in building very large page styles. In particular odd things happened when I tried to include graphics (the image became the whole page, regardless of scaling).

I experimented briefly with the tikzposter package, but couldn’t really get anywhere with it – perhaps because I hadn’t started from the ground up with it. I think I looked at beamerposter but again ran into problems, possibly because I didn’t start that way. In the end it was the simple a0poster package that did it for me, anyway – although I had to force the font throughout to be “\tiny”.

I also discovered the hard way that the wrapfig package doesn’t seem to work within a Tikz node, so I’ve messily improvised by adding another node for the images of people and relating its position to the relevant biographical section.

I harvested some basic biographical information from Wikipedia and one other website.

I managed to get something basic working. The image is big so the version below is blurry. A link to the smaller PDF and the LaTeX source on GitHub are shown below.

PDF DownloadLaTeX File

Family Tree
The actual document is very large in terms of page size, so click on the PDF for that clearer (smaller) version, this image is just to show indicative layout.

FamilyTree

Necessary And Sufficient

This article contains links to materials and extra resources to my Inaugural Professorial Lecture, with the same name, delivered on 17th February 2016 at Ulster University.

 

Twitter

  • If you have any comments or questions, use the Twitter hashtag #nesssuff and I’ll pick them up later and try to address them. My Twitter ID is @ProfCTurner.
  • The Vote of Thanks will be given Sarah Flynn, whose Twitter ID is @sarahjaneflynn.

Synopsis

“Necessary and Sufficient: a look at elegance, efficiency and completeness in Engineering and its Mathematics”

Engineers and Pure Mathematicians have a surprising amount in common, despite working at opposite ends of many problems; one at the totally theoretical end and the other at that of practical realisation, sometimes centuries apart. They both use tools created or designed mainly by other members of their own profession; they both enjoy testing things to destruction in order to explore how they work; and they both enjoy finding solutions to problems that cover all the requirements but which tend to do so in an efficient and elegant way.

This lecture explores how basic concepts that began with natural numbers to count livestock in antiquity eventually gave rise to complex numbers, and how techniques to measure buildings and the movement of the stars evolved into techniques to analyse data in totally new ways.

Some modern applications, ranging from every day examples such as photographs taken by smart-phones through to research applications, will also be considered.

Finally, the lecture will examine the implications for how Engineers can be educated to bring the power of some of humanity’s most beautiful abstract ideas to bear on the practical problems that surround us in everyday life.

Lecture Slides

(“Director’s cut” and “Commentary/Video” to be uploaded at a later date).

PDF Download – Videos not embedded, no pauses (~3 MB)

PDF Download – Full Size Slides with pauses and embedded Video (~52 MB)

For those interested the slides were produced with PDFLaTeX, Beamer and Tikz. Diagrams with plots and positions of complex numbers are all calculated as the PDF is compiled. The presentation was stored in a git repository and a Makefile was used to produce the various versions.

A GitHub repository with some files missing (due to them being University property) is available here. But this does contain all Tikz diagram source code, cow images, and a LaTeX Beamer template aligned to the Faculty template that was produced. Faculty colleagues can request the required University images for their own presentations. The Makefile shows how to create different versions of the talk, with embedded or linked videos, and with or without pauses.

The content of the talk is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Extra reading

Introduction

The Evolution of Numbers

Circular Functions and Fourier Series

Event Brite

  • The lecture was advertised on EventBrite here.

Tweaking Newton’s Law of Universal Gravitation

I answered a question on Quora about the consequences of Newton’s law of Universal Gravitation being in inverse law and not an inverse square law for 60 seconds. The outcome is not so good, what with all matter acting as a black hole and all.

https://www.quora.com/What-would-happen-if-the-law-of-gravity-becomes-1-r-instead-of-1-r%C2%B2-for-60-seconds/answer/Colin-Turner-6

Chess Boards, Exponential Growth and the Ice Bucket Challenge

Yesterday I was finally nominated for the Ice Bucket Challenge, I had actually thought this inevitable for the reasons in this post, but then it all kind of passed by. As it happens, it was my Daughter Aimee’s fault in the end. :-).

So I thought I would use the opportunity to bore people about the arithmetic behind it, in the end, I suspect that wasn’t too coherent since I was being watered at the time by my older Daughter Aimee, while my younger Daughter Matilda generally screamed at both of us.

So here is what I was trying to say. I like to talk to my students about a new version of an old problem. There’s an old story about a chess game where the winner will take away a certain amount of rice. The amount is calculated by having one grain on the first square, two on the next, four on the next, and so on, doubling all the way to the 64th square. My modified version is to consider coins, a UK 10 pence piece to be precise, piling up on the squares as we go along. The question is, how high is the pile of coins on the last square? I encourage you to guess, approximately what sort of size that is.

We double each time, so the number of coins on the first eight squares are 1, 2, 4, 8, 16, 32, 64, 128.
Another way to write that is
20, 21, 22, 23, 24, 25, 26, 27.

So, we are going along 64 squares, and start at 0, so the number of coins on the last square will be
263. This is a big number, but just how big? Do we change our guess on this information?

Now 263 ≈ 9,223,000,000,000,000,000 = 9.223 × 1018
and a ten pence piece is 1.85 mm thick, or 0.00185 m, so when we multiply these we get

17,063,000,000,000,000 m = 1.7063 × 1016 m.

Wow. That seems like a lot. Just how big is that number as a distance?

To put it in perspective…

  • by the 19th square, the coins are higher than the radio mast on the Empire State building;
  • by the 29th square, the pile of coins would have reached the Moon;
  • by the 48th square, the pile of coins would have reached the Sun;
  • by the 63rd square, one from the end, the coins are a light year high.

This is an example of the staggering power of exponential growth, and just how unintuitive it is.

So what has this got to do with the Ice Bucket Challenge? Well suppose you start with a single individual, who then nominates three people, each of whom nominate three people. Now on the Chess Board we have 1, 3, 9, 27 and so on. This is, again, exponential growth with powers of three rather than two (actually somewhat higher growth).

Mathematically these are often called Geometric Progressions or just G.P.s for short. These are sequences of the type

a, ar, ar2, ar3, ar4, …

There is a formula that can be derived (it’s not hard, the derivation is on the above Wikipedia link) for the Sum of the first n terms. (In Mathematics, contrary to popular opinion, a Sum specifically means the result of an addition process).

S_n = \frac{a(1-r^n)}{(1-r)}

In the above case we can see that a = 1 because that’s the first number in 1, 3, 9, … and r = 3 because that is the number we are multiplying by each time. So here

S_n = \frac{a(1-r^n)}{(1-r)} = \frac{(1-3^n)}{-2}

In other words, every “generation” each person nominates a further three people, so the number of people added each generation ramps up exponentially, and the total number involved increased rapidly too.

Generation New People Cumulative Total
1 1 1
2 3 4
3 9 13
4 27 40
5 81 121
6 243 364
7 729 1,093
8 2,187 3,280
9 6,561 9,841
10 19,683 29,524
11 59,049 88,573
12 177,147 265,720
13 531,441 797,161
14 1,594,323 2,391,484
15 4,782,969 7,174,453
16 14,348,907 21,523,360
17 43,046,721 64,570,081
18 129,140,163 193,710,244
19 387,420,489 581,130,733
20 1,162,261,467 1,743,392,200
21 3,486,784,401 5,230,176,601
22 10,460,353,203 15,690,529,804

So you can see that after just 22 steps in the nomination process, you have literally more people than you have on the planet. Pyramid selling schemes become untenable for this reason too… you quickly run out of suckers on any given land mass.

So if this really ran its course unimpeded (and I know that the original challenge was money or dowsing, with a requirement to undertake the challenge in 48 hours (some say 24) then the whole planet would have done the challenge in at most six weeks. If each person had used say 4 litres of water then at the time of writing the 7,259,289,122 people alive today would have collectively used 29 billion litres of water. Since an Olympic swimming pool has 2.5 million litres of water, that represents over 11,614 swimming pools filled with water. While it is easy to criticise those who have been uneasy about the water wastage here, to be honest they have a good point. Also one organisation would have billions of charity income rather than it being spread.

Clearly this has not happened[citation needed] so people have run out on enthusiasm here and there. Anyway, have fun, and give some money to something you love, if you want, when you want to.

I am donating to

Since people I care about have been troubled by both, and

since it can’t be denied I live in extraordinary luxury to be able to waste perfectly good water on this kind of thing.

Aikido in Dead Straight Lines

There’s been an elephant in the room on my blog for quite a while now, and it has prevented me completing a number of articles that I have had in draft for some time.

A bit over a year ago, my friend and mentor, Alan Ruddock died. I’ve been trying to articulate what that meant for me, and what I thought about Alan, but I have repeatedly failed. This doesn’t fully resolve that issue, but at least I can put down some thoughts about Alan here now, or at least about Aikido

Disclaimer: this post probably badly needs some photos, and I’ll try to retro-fit that at some point.

I was kindly asked to present an hour at this year’s Galway Aikido Summer School, where Alan Ruddock and Henry Kono traditionally taught together for many years. This year Henry continued with his excellent classes in the morning and other instructors that knew Alan took an hour each in the afternoon. I confess I was a bit daunted by some of the others teaching in these slots, particularly the inestimable Lorcan Gogan of PSAC, with whom I shared a session. When I said to him I would have to follow that, he simply replied “Hey, I had to follow Henry!”. A fair point. But I did find myself more reticent than usual in my teaching style.

I chose to try and present some thoughts that have arisen from Alan talking about his “dead straight line”. That is that Aikido is often thought of as being circular in nature but Alan was keen to stress it was not at its heart.

In Aikido, we often see uke (the attacker) whirled around nage (the defender) in circles. There are beautiful diagrams about this, and allusion to circles everywhere. Our own club is called the Belfast Aikido Circle. So it’s impossible to deny circles don’t appear. Indeed, there are semi-physical, semi-mystical links to squares and triangles too.

But what Alan meant, in my opinion, (all disclaimers apply) is that you always behaved as if you were operating in a dead straight line.

Aristotle believed that objects, in a perfect (celestial) environment travel in perfect circles, but many centuries later, Newton thought otherwise.

A body will continue in its state of rest, or uniform motion in a straight line, unless acted upon by a resultant force.

is better known as Newton’s First Law of Motion. In Aikido terms its consequences are simple, if an attacker comes along a straight line, and is subsequently diverted off that straight line, Force, and Energy has been added by someone. Not in some kind of mystical sense of the use of these words you might see in other places, but in their elementary definitions in Physics. So the Force has been added, the big question then is by whom, followed up by why, and an analysis of the consequences.

Let’s look at the question of “whom” first. In a previous article, I wrote at some length about the spectator problem in Aikido. Sitting at the side lines you can never know for sure just who is doing what to whom. You can see the nage’s hands rise, move or turn, but you cannot know, from outside, whether this is nage initiating these things, or a reaction to uke’s movements and attack.

In fact, we can take it a step further than this and say that at best only the two involved can fully know, since it is entirely possible that neither of them will fully know either. In other words, the problem of “whom” is a really knotty one; it’s altogether possible that no-one knows.

In theory, at the beginning of a “technique” the nage first moves to a position of safety from the immediate attack and then “blends” with the attack. Even this is an over simplification since nage can be more proactive, but let’s set that side for a moment. This moment of initial blending is pivotal. Alan used to tell a story about O-Sensei coming to watch a class of aikido at the Hombu dojo, and after watching, smiling, for some time, he announced “you are all doing a wonderful job, after having your heads cut off.” The analysis of this could be that, especially when facing an armed attack, even if the blend if a bit out, there may be more than a bit of you missing by the time you start your technique.

Aiki means the harmony of ki, or energy, so your blend is the moment where you, and your attacked end in a position of aiki, where both your energies are pointed in the same direction. If the uke’s energy was directed at you in a dead straight line, as it often is, then in theory, you should both be pointing in the same dead straight line.

But this is often not the case. You can immediately see deviation from the straight line. Why is this? From Newton, the answer is obvious, one or both person(s) have put extra force into the situation.

This may have been uke, who immediately realising things are not going as planned, starts to react and often turns towards nage to strike them; this being the case nage has to move off the original straight line too to provide space for uke, and to keep the aiki principle, continuing to move with the uke. If that’s all it is then this is fine. The force and energy being contributed by nage are minimal, it is necessary and sufficient.

But all too often, the honest truth is that as nage, we anticipate the move from uke, or worse don’t even think about it, and just start whirling them around us. It may not be immediately clear why this is a Bad Thing. There are two reasons; one is that the aiki principle has been immediately broken, you are no longer in harmony with the attacker but attempting to direct them. The second is why the aiki principle itself matters; when you inject force and energy into the situation a skilled opponent can make use of it. In fact that’s practically the central principle of aikido. Probably 90% of the time, particularly in training, your opponent may not even notice, so we all get away with it, and we probably never learn.

So here are some thoughts on training to enforce honesty on this.

Try and do some techniques along a straight line on the mats. You’ll need your uke to be initially very well behaved. Don’t even think of it as an attack, imagine that you meet a friend at the gates of a park. You see them as they approach and you walk backwards, then sideways, then alongside them as you extend your hand to shake hands with them. They key here is not to make it a conflict. This is astoundingly easy to do when you aren’t worrying about having your head punched; just practice the naturalness of this first.

Then, as you walk along your straight line, just allow your hand to rise across the line, the analogy I used as if you are pointing at a squirrel in a tree. It’s a simple, gentle irimi-nage that requires a compliant sensible uke who just has to keep walking straight, even into the “throw”.

It’s a silly exercise, and you can play with greatly shortening it, but the big deal is, practising doing the throw along a straight line.

The next phase is to allow your uke to, once every so often (and without prior warning), lifting their free hand and turning to – well – lovingly caress your face shall we say? This changes the dynamic, the uke wants to move off the straight line, and so you have to as well. But the key of this exercise is:

  1. do you actually pull them off the line on occasions where they do not turn;
  2. can you really convince yourself you don’t move off the line with a little extra force?

It may be a useful exercise.

Another example worthy of note is Shomen Uchi, Ikkyo (tenkan). Or that when your opponent tries to strike your forehead, you turn, contact their arm and bring it, and the uke to the ground.

Traditionally this is often done by grabbing the arm as it descends and whirling your uke around you. There can be real consequences to this because their other fist is being whirled around you too. There are other problems too: you can have a tendency to pull the arm so closely into your space that uke can merely step behind you and topple you over their leg. It isn’t a nice fall. It also provides, ironically, a slow descent of uke into the ground, since you provide them with a lot of implicit support.

So try, from the moment of Shomen Uchi contact where your hands meet, to step to the side, and without gripping contact the arm with both hands (the other hand at the elbow). Let your uke continue on their straight line straight forward and downwards. Potentially this is a hard fall, you are providing no force or breaking but just allowing them to sail majestically (potentially teeth first) into the mat. Some care needs be taken with this initially! It is most important that you do not push them down, just act as a ratchet so as they descend they cannot rise.

Again, in real life uke will often start heading towards you early on, trying to recover their balance, and also because in aikido, the tendency for uke and nage to start running around in circles is ever present, and once again, that’s fine, the issue is not to interfere with this, but also not to seek to amplify it by hauling them around.

At an early stage you may find it useful not to grab their wrist; this can, in any case, cause all sorts of postural problems, and once you grab something the urge to pull it in is not far behind.

A suitable training for this can be found at your local Supermarket. Find yourself an empty trolley, and as you move it around, you will probably grip it with both hands and pull in one while you push out the other. If you are really mindful to your body you may notice muscles in your abdomen and spine taking the strain. But then fill the trolley to the brim, you will rapidly discover that this trick is not so easy. It’s also not so wise, your muscles will shriek in protest at you, and if you do this kind of thing with a large opponent you will likely both be unable to move them, and injure yourself trying.

These are some simple thoughts on the Dead Straight Line that Alan used to talk about. There should always be a delicate positive pressure in that straight line in front of you, so that as things move and the gap appears there you will be immediately slot yourself into it. It is the spike in O-sensei’s “Ki” calligraphy; it is the imaginary sword held in front of you that tells you always where you want to go.

Your curved movements are still a succession of Dead Straight Lines.

There is a nice, and relatively elementary parallel from Einstein‘s General Theory of Relativity. The curved paths that objects like planets make, as commented on by Aristotle appear circular and curved, but actually they are Dead Straight Lines (geodesics) through curved space-time. When the Moon travels around the Earth, it does so in a series of Dead Straight Lines so that it keeps missing the Earth and it perpetually falls towards it. If you do decide to have uke orbit around you, then this is the same principle you need to prevent them spiralling into you, complete with their body weaponry.

But anyway, try playing with Dead Straight Line. I hope you will find it rewarding.

Tau versus Pi

Today, two of my friends independently sent me a story about Tau Day which I had hitherto never heard of. One of them asked for me comment about whether this had any point to it. At first I thought the article was just mathematical trolling, thought about it a bit more, thought there might be a real point to it, thought some more and concluded it seemed rather silly.

The argument is about whether the mathematical constant pi, would be better being replaced throughout mathematics with an alternative tau, which is just twice pi (in other words, replacing pi everywhere with a half of this tau). It’s suggested that formulae with tau will be more simple.

Basic Geometry

So this is all about the fact that pi was defined historically as the ratio of the circumference to the diameter of the circle, a very old classical reference stemming back to Greek geometry (incidentally pi is also known as Archimedes’ constant since he attempted to calculate an approximation to it). Once upon a time, the formula used in schools would have been:

C = \pi D

related the circumference C to the diameter D. But generally now, we use the radius r rather than the diameter. And so that gives us (for circumference and area):

C = 2 \pi r \quad ; \quad A = \pi r^2

The argument for tau begins by observing the extra 2 in the first formula, and wouldn’t be nicer if we just defined tau to be twice pi so that these formula would be so much nicer. Would they?

C = \tau r ; \quad A = \frac{\tau}{2} r^2

Set aside for the moment the fact that pi is probably the most recognisable Greek letter in the world that speaks languages based on the Latin alphabet (aside from those that are, or appear to be the same). Set aside the fact that tau is used for other specific purposes in much of modern Mathematics, and in particular in the discipline of Topology. The first formula might be nicer, but the second one is probably worse, and by enough to make the improvement of the first rather parlous. OK. But the article talks about this being the problem behind radians, so maybe that’s where we get the big gain. Let’s explore that.

Radians instead of Degrees

There’s nothing particularly clever about using degrees. It’s an arbitrary choice (360 degrees in a circle) that probably owes a lot to do with historical factors in one civilisation. It is true that when you start to do some significant mathematics with degrees, it starts to look quite unwieldy. The classic two formulae to consider are the length of an arc and area of a sector.

Suppose we have a circle of radius r and we want to work out the length of an arc (a part of the circumference) where the angle subtending this arc is theta degrees (don’t panic, no more Greek to come). Then in degrees the formula will be:

s = \frac{\theta}{360} \times 2 \pi r = \frac{2 \pi r \theta}{360}

The reason why is that the fraction on the left is the fraction of the relevant angle out of all the angle available, multiplied by the total arc length available (the whole circumference). The formula is not beautiful, and the similar formula for sector area is also a big ugly.

A = \frac{\theta}{360} \times \pi r^2 = \frac{\pi r^2 \theta}{360}

You will note that in both cases there is a 360 on the bottom of the fraction and a 2 pi on the top. This looks like nature’s way of trying to tell us something. What would happen if we used an unit of angle so that, instead of having 360 of them in a circle, we had 2 pi of them in a circle (proponents of tau will just say tau of them in a circle)? The formula, derived using the same logic, become much nicer.

 s = \frac{\theta}{2 \pi} \times 2 \pi r = r \theta
 A = \frac{\theta}{2 \pi} \times \pi r^2 = \frac{1}{2} r^2 \theta

So we get

 s = r \theta \quad ; \quad A = \frac{1}{2} r^2 \theta

Now these are beautiful, elegant formulae, and the underpinning of why radians (the unit of angle we are talking about here) are used instead of degrees in much of higher mathematics, the formula are much simpler (particularly true when using calculus). Also, look at that first formula, it has all the resonance of F = ma. 1 unit of arc length is found in a circle of radius 1 unit with an angle of 1 radian. So beautiful is this that it used as the definition of the radian in many books. So far, so good. Did we really need tau to produce these? Does it matter that it it tau and not pi that cancels out? I can’t see why.

Fourier Series

Another example owes to the work of Fourier, who showed that repeating patterns can be broken into sums of the most basic repeating functions, the ones that are most simple are the sine and cosine functions. These are used to model waves of any sort which are of course ubiquitous in nature. It turns out you can build up more odd shapes like triangular and square waves out of these sinusoidal ones. Being able to do this is important in many aspects of Science and Engineering, and eventually this theory leads to all sorts of cool stuff like the way data is compressed in photos and more.

The graphs of Sine and Cosine

This graph is labelled in degrees, so you can imagine if we change it to radians as we should then the 360 would become 2 pi. This is where again, proponents of tau will argue that replacing the 360 with a simple tau makes things easier. And it does, if all you want to do is to label that diagram. But the foundation of Fourier theory is building functions up in combinations of these:

 1, \sin t, \cos t, \sin 2t, \cos 2t, \sin 3t, \cos 3t, \ldots

The formulae you need to be able to deal with to do this are (among others):

 a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(t) \cos nt dt \quad ; \quad b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(t) \sin nt dt

which I grant, strike fear into the hearts of many. But they don’t look nicer with tau (and are a little more awkward):

 a_n = \frac{2}{\tau} \int_{-\frac{\tau}{2}}^{\frac{\tau}{2}} f(t) \cos nt dt \quad ; \quad b_n = \frac{2}{\tau} \int_{-\frac{\tau}{2}}}^{\frac{\tau}{2}} f(t) \sin nt dt

Remember, the people arguing for tau are claiming it simplifies formulae, not making them look worse.

Euler’s Identity

Finally, I cannot leave this without talking about Euler’s identity considered by most mathematicians (including myself) to be one of the most beautiful results in Mathematics.

e^{i \pi} + 1 = 0

This result can be written in a few ways, but this way is very commonly used. This is because in this form you can see how this identity connects the five most important numbers of Mathematics: 0, 1, pi, i and e. With tau, it just doesn’t have the same beauty:

e^{i \frac{\tau}{2}} + 1 = 0

so I will stick with pi. Thanks all the same.

The Academic Descent to Me

How interesting, today I learned that Derek Burgess, the PhD supervisor of my PhD supervisor (Brian McMaster) was himself supervised by Frank Smithies. With a little help from the Mathematical Genealogy Project this has helped me work out my academic “parentage”.

  1. Colin Turner, Queen’s University of Belfast, 1997
  2. Brian McMaster, Queen’s University of Belfast, 1972
  3. Derek Burgess, University of Cambridge, 1951
  4. Frank Smithies, University of Cambridge, 1937
  5. G. H. Hardy, University of Cambridge
  6. Edmund Whittaker, University of Cambridge, 1895
  7. Andrew Forsyth, University of Cambridge, 1881
  8. Arthur Cayley, University of Oxford / University College Dublin / Universiteit Leiden, 1864,1865,1875
  9. William Hopkins, University of Cambridge, 1830 (Note his many famous students)
  10. Adam Sedgwick, University of Cambridge, 1811
  11. Thomas Jones, University of Cambridge, 1782
  12. Thomas Postlethwaite, University of Cambridge, 1756
  13. Stephen Whisson, University of Cambridge, 1742
  14. Walter Taylor, University of Cambridge, 1723
  15. Robert Smith, University of Cambridge, 1715
  16. Roger Cotes, University of Cambridge, 1706
  17. Isaac Newton, University of Cambridge, 1668
  18. Isaac Barrow, University of Cambridge, 1652
  19. Vincenzio Viviani
  20. Galileo Galilei, University of Pisa

I think I’ll stop there. I found this absolutely fascinating, many notable figures, and Hopkins supervised many famous figures. It’s a shame I could never hope to live up to such a line!

Black Hole Radius, or How I came to love the LHC

I allowed myself to become rather distracted by my second years last week as the class was finishing. They were talking about an episode of Horizon that discussed General Relativity and theories of Quantum Gravity. What followed was a free ranging discussion on the nature of infinity, mentioned briefly in the program. But we also talked about the nature of a black hole and its size.

It’s surprisingly easy to calculate this with reasonably elementary maths and physics. I first did this when I was about 17 (how very sad) using classical physics equations, and was astounded to discover that even so, the answer was correct (I checked it in the Encyclopedia Britannica in the library at the time).

Here is Newton’s universal law of gravitation, between two bodies. It describes the force F between two bodies that are r metres apart. Let’s take the one with mass M to be the black hole. G is a small (though mysterious) constant.

F = \frac{GMm}{r^2}

You can work out the energy needed to escape the black hole using the old stand by equation that work done is the force times distance traveled against that force, but that only works with a constant force, this force will change as we move, so we need to use the big daddy of multiplication, integration.

 WD = \int_a^b F dr

Specifically, we will work out the energy needed to escape from the event horizon, the surface at which the escape velocity is the speed of light, which is c (299,792,458 m/s). So the energy will be given by moving my little mass m from the radius of the event horizon, let’s call is R to infinity, to show we have broken away.

 WD = \int_{R}^{\infty} \frac{GMm}{r^2} dr = \left [ -\frac{GMm}{r} \right ]_{R}^{\infty} = \frac{GMm}{R}

Now, this should just balance the kinetic energy possessed by my little mass m traveling at the speed of light.

 \frac{1}{2} mc^2 = \frac{GMm}{R}

and if we rearrange for R we get that

 R = \frac{2GM}{c^2}

In other words, the radius of the event horizon, the bit we think of as the “hole” is dependent entirely upon the mass of the object. Please note this is based on a very simple model of a non rotating black hole. Nevertheless we can do some nice calculations from this.

The Sun, would have to be compressed from its diameter of about 700 million kilometres into a radius of just under 3 kilometres. The Earth’s mass would need to be compressed so much to form a black hole you would need to squeeze its radius of over 6 thousand kilometres into a radius of around 9 millimetres. That’s how dense we’re talking here.

We can also consider the radius as described by the contained energy of the black hole, since we know that

E=mc^2

and so, replacing our M in our above equation we get

 R = \frac{2GE}{c^4}

Wow. Remember c is a big number, taking it to the power of four is a lot. So why do this? There’s been a lot of speculation about the possibility the Large Hadron Collider (LHC) could create a black hole. This has caused a fair degree of panic, and at least one suicide. It’s a physicist’s dream that a black hole might be created. I just looked up the “high” energies used by the LHC, and high is a relative term. It plans to bash protons together with 7 TeV (Tera electron volts) of energy each, or lead nucleii with 574 TeV each, let’s take the latter. Just how much energy is that in a collision? Well, doubling and converting to good old Joules gives 184 micro Joules. That’s really not a lot, 184 millionth’s of a Joule. A 100W light bulb uses 100 Joules each and every second. How big would the radius of such a black hole that might form be, from that energy? Check the maths, because so far I haven’t but I get.

R = 3.040 \times 10^{-48} metres

which is 0.000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,003 metres wide (I hope, I didn’t double check the zeros either), which is probably not the planet swallower of people’s imagination (but this is fun). But the problem is people think this tiddler will grow very rapidly, but that’s because they don’t know about Hawking Radiation. This is an interesting quantum effect that means black holes aren’t really black, they do emit a little radiation. Large holes would gather surrounding matter faster than their low radiation rate, but small holes have the opposite situation, they radiate more rapidly. The maths for all that is pretty complex, and you need to make lots of assumptions, but the time taken for our little black hole to “evaporate” is (hurriedly calculated by me)

t = 7.21 \times 10^{-79} s

a tiny, tiny fraction of a second. Even allowing for the ambient temperature and some fall in of matter, this little baby is not in equilibrium, it’s not getting mass fast enough to accumulate more. It’s safe*.

 

* All disclaimers apply. No liability is assumed for foolish unvalidated experiments done by you or other members of your species. Do not attempt to create black holes in your garage. Any subsequent destruction of your civilization, planet or solar system is at your own risk, and any “EPIC FAIL” signs placed by aliens on the remains is not due to me or my calculation. No calculations have been done on the matter of strange matter either. If you break the planet / system / galaxy or universe you own all the parts.