## Mindswap

”But if you don’t know anything about Cathy, I don’t see how any theory will help.”

Vadez sighed, not unpleasantly…”My friend, if you know all about Cathy, do you think you would be able to find her?”

“I’m sure I could,” Marvin said.

“Even without knowing the Theory of Searches?”

“Yes.”

“Well then, apply the same reasoning to the reverse condition. I know all there is to know about the Theory of Searches, and therefore I need to know nothing about Cathy.”

## Guards! Guards!

‘ I believe you find life such a problem because you think there are the good people and the bad people,’ said the man. ‘You’re wrong, of course. There are, always and only, the bad people, but some of them are on opposite sides.’

He waved his thin hand towards the city and walked over to the window.
‘A great rolling sea of evil,’ he said, almost proprietorially. ‘Shallower in some places, of course, but deeper, oh, so much deeper in others. But people like you put together little rafts of rules and vaguely good intentions and say, this is the opposite, this will triumph in the end. Amazing!’ He slapped Vimes good-naturedly on the back.

Down there,’ he said, ‘are people who will follow any dragon, worship any god, ignore any iniquity. All out of a kind of humdrum, everyday badness. Not the really high, creative loathsomeness of the great sinners, but a sort of mass-produced darkness of the soul. Sin, you might say, without a trace of originality. They accept evil not because they say yes, but because they don’t say no. I’m sorry if this offends you,’ he added, patting the captain’s shoulder, ‘but you fellows really need us.’
‘Yes, sir?’ said Vimes quietly.
‘Oh, yes. We’re the only ones who know how to make things work. You see, the only thing the good people are good at is overthrowing the bad people. And you’re good at that, I’ll grant you. But the trouble is that it’s the only thing you’re good at. One day it’s the ringing of the bells and the casting down of the evil tyrant, and the next it’s everyone sitting around complaining that ever since the tyrant was overthrown no-one’s been taking out the trash. Because the bad people know how to plan. It’s part of the specification, you might say. Every evil tyrant has a plan to rule the world. The good people don’t seem to have the knack.’
‘Maybe. But you’re wrong about the rest!’ said Vimes. ‘It’s just because people are afraid, and alone—’ He paused. It sounded pretty hollow, even to him.
He shrugged. ‘They’re just people,’ he said. ‘They’re just doing what people do. Sir.’
Lord Vetinari gave him a friendly smile.
‘Of course, of course,’ he said. ‘You have to believe that, I appreciate. Otherwise you’d go quite mad. Otherwise you’d think you’re standing on a feather-thin bridge over the vaults of Hell. Otherwise existence would be a dark agony and the only hope would be that there is no life after death. I quite understand.’

Vimes paused at the door.
‘Do you believe all that, sir?’ he said. ‘About the endless evil and the sheer blackness?’
‘Indeed, indeed,’ said the Patrician, turning over the page. ‘It is the only logical conclusion.’
‘But you get out of bed every morning, sir?’
‘Hmm? Yes? What is your point?’
‘I’d just like to know why, sir.’
‘Oh, do go away, Vimes. There’s a good fellow.’

## The poetry of definitions in maths

Coming up with an elegant mathematical definition is as much of an art as writing poetry. The analogy works even further, for like a well-written piece of poem, a definition in mathematics packs a lot of meaning in a few words. One has to unpack the meaning to appreciate either, and the process can be frustrating. But once the understanding is complete, it is easy to see why the definition/poem works, and it is infinitely preferable to use it rather than explain what one means.

Here are some examples.

Poetry : Definition of a Free group via the universal property – A subset X of a group F is said to be a free basis for F  if for every group G and every (set) function $\phi: X \rightarrow G$, there is a unique homomorphism $\bar{\phi}: X \rightarrow G$ satisfying $\bar{\phi} |_X = \phi$. A group F is called a free group if it has some subset that is a free basis.

Prose: Homomorphisms carry the algebraic structure of an object. If for example the Cayley group of X (i.e. F) has a loop, any (non-trivial) group homomorphism must carry it into a loop. The only way you can extend any set function whatsoever to a group homomorphism is if X has no algebraic structure to be carried over. This is the property that this definition captures.  Notice that we do not require $X$ to generate $F$. But that is exactly what will happen, because we require the homomorphism $\bar{\phi}$ to be unique.

The second example needs some set-up.

Quasi-isometric embedding. Let $(E_1,d_1)$ and $(E_2,d_2)$ be two metric spaces. Let $\lambda, \epsilon \in \mathbb{R}$ with $\lambda \geq 1$ and $\epsilon \geq 0$. A map $\phi E_1 \rightarrow E_2$ is called a $(\lambda, \epsilon)$-quasi isometric embedding if $\forall x,y \in E_1$

$\frac{1}{\lambda}d_1(x,y) - \epsilon \leq d_2(\phi(x),\phi(y)) \leq \lambda d_1(x,y) + \epsilon$.

Poetry:  A Quasi-isometric embedding is called a quasi-isometry if $\exists C \geq 0$ such that $\forall x_2 \in E_2, \exists x_1 \in E_1$ with $d_2(\phi(x_1),x_2) \leq C$.

Prose: We are looking for a notion that is not quite a surjective map, but close to being a surjective map. That is why we ask that if an element in $E_2$ is not actually in the image of $\phi$, it is not too far away from such an element.   So any element in $E_2$ is close to some element in $\phi(E_1)$.

As they say in Latin, quod erat demonstrandum.

## Crying Wolf

You might have heard this story earlier.

There is a girl who is very naughty. She goes out to play in the woods every day and every day, she cries wolf. All the villagers come running to her rescue, only to find her laughing so hard that tears ran from her eyes. One day, when a wolf really comes and she cries for help, no one comes. None of the villagers come to help her, but for a friend, who loved her dearly. Every time the naughty girl cried wolf, he came running-never doubting for a moment that the girl needed help. Yet every time the imp laughed, he laughed with her. When she played the same prank over and over again, he never stopped playing along-not in the way you humor a child, but in the way you are caught in a friend’s joke, in spite of yourself. When the wolf came, he didn’t feel he was proved right after all. He didn’t remember that there had been a joke about it. He chased away the wolf and let the girl cry on his shoulder and took her home.

“Love is very patient and kind, never jealous or envious, never boastful or proud, never haughty or selfish or rude. Love does not demand its own way. It is not irritable or touchy. It does not hold grudges and will hardly even notice when others do it wrong. It is never glad about injustice, but rejoices whenever truth wins out. If you love someone you will be loyal to him no matter what the cost. You will always believe in him, always expect the best of him, and always stand your ground in defending him.”

Maybe the girl wasn’t playing a prank. Perhaps she heard a rustling sound in the woods. May be she really thought that there was a wolf and cried out. And when you saw her laughing, may be she was laughing with relief-the tears streaming down her cheeks only confirming how scared she had been.

## Using an image as the axis text in R plot

But the one that worked for me was in one of the comments. Use the package “tikzDevice” and you can add the awesomeness of Latex to the plotting prowess of R. All you have to do is to construct a Latex directive in R, and use it in the R plot commands, wherever you want. tikzDevice will create a tex document for you. Compile it and you are ready to go.

Example
library(tikzDevice)
library(ggplot2)
axislabels=c()
for (i in seq(1,3))
{
fname=paste(“placeholder”,i,sep=””) #img is of course present in the wd.
al=paste(“\\includegraphics[scale=0.05]{“,fname,”}”,sep=””)
axislabels=c(axislabels,al)
}
dat <- data.frame(cond = rep(c(“Good”, “Bad”), each=10),xvar = 1:10 + rnorm(10,sd=1), yvar = 1:10 + rnorm(10,sd=10))
p=qplot(1:3,1:3)+scale_x_discrete(limits=c(1,2,3),labels=axislabels)
tikz(“annotated.tex”,standAlone=T)
print(p)
dev.off()

## Teaching fractals

As a a volunteer for CSIRO’s Mathematicians in Schools Program, I was asked to give a talk on fractals to girls in grade 9 at a school in Sydney.

I am not an expert on fractals, but as a doctoral candidate in mathematics, I figured I knew more than the the girls would, so I thought, why not? And I agreed.
This school was participating in the MegaMenger project. Their teacher wanted me to motivate the study of fractals by talking about some real world applications. I did talk about the usual suspects – animation, modeling natural systems etc. But I also managed to sneak in a point that mathematics educators always make – that ” what is this good for”, is not the right question to ask in a maths class. Paul Lockhart in his famous essay says it better than I could. But I like to extend his lament to the whole of education – how will this help me get a job is not quite the question that should be uppermost on young minds. The gifts of education are rather more intangible – curiosity and the ability to learn things, that’s what is going to get you a job. I know that these are not things one could put in one’s CV, but then, these are teens who have not yet outgrown their love for Beiber. And yet, theirs are the loudest voices in that dreaded chorus – “why are we doing this? When are we ever going to use it?”.

My rant aside, I think the activity went well. Here’s an appraisal of what worked and what didn’t.

What worked – I am usually sceptical of giving presentations to young students, as there’s no surer way of putting them to sleep. But this one was a talk with just a picture on each slide, with me talking around the picture. I also put in a fractal paper cutting activity. I was afraid that the students might find it boring, but everyone eagerly participated.

What didn’t – I did ask them a bunch of questions – what other examples of fractals they could think of, where did they think fractals could be applied etc. In hindsight, I think it would have been better to tie their answers into a discussion. Instead of accepting one workd answers, I should have asked them to elaborate.