Edward’s Death

‘I’m going away.’

This text reached seven people within a few minutes of half past two in the night of the last Tuesday of September, 1999.

All of them had been sent simultaneously from a cell phone number that switched off two minutes later.

With that, Edward commenced the process of witnessing his own death. And the effect of his death on the people around him. Relatives, friends, lovers, haters.

Edward wanted to find out what he really meant to people and what they inwardly thought of him. Realizing that his existence was an unavoidable hurdle in this evaluation, he had finally decided to take himself out of the equation.

Nobody had ever done this before, because nobody was sure they could hide today in a world so small for the rest of their life. But Edward had spent time working it all out.

As the phone sank into the cold midnight waters of the Pacific, Edward walked away from the moonlit beach, a bleak, unremarkable figure, receding slowly into the late night drizzle, giving himself up to the waiting darkness of the forest.

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Locating Numbers inside Bisected Interval Sequences

I think in a real analysis course in the second semester of my first year, the teacher was discussing the nested interval theorem, when one of his examples or something he was saying struck me, and I thought of this interesting problem. Well, interesting to me.

We pick any fraction, say. Now we look at the interval [0,1]. We divide it into two halves [0,0.5] and [0.5,1] and say, ‘the fraction belongs to this half.’ Say the right half. Then we divide the right half into two halves, check again, and say ‘now it’s in the left half’. We continue like this until we hit the number bang in the middle of an interval.

Now that’s not really a problem, but I thought it would be an interesting thing to look at this sequence of ‘left’ and ‘right’ for a chosen fraction. So I wrote a python program for that. Nothing very amusing came out of that. Then I thought of something else. I took evenly spaced fractions in that interval along the horizontal axis, and plotted the fraction of rights in their respective left-right sequences, on the vertical axis, using matplotlib. Here is the python source code:

#! usr/bin/python
import matplotlib.pyplot as plt
c = 0.
x=[]
y=[]
while c<=1.:
    a = 0.
    b = 1.
    dc = c - a
    d = (b-a)/2
    R=1
    L=1
    while True:
        if dc > d:
            R+=1
            a = a + d
        elif dc < d:
            L+=1
            b = b - d
        elif dc==d:
            break
        d = float(b-a)/2
        dc = c - a
    x.append(c)
    y.append(float(R)/(R+L))
    c+=1e-4
plt.xlabel('Fraction')
plt.ylabel('''Fraction of 'Right's in sequence''')
plt.plot(x, y, marker='.', markerfacecolor='blue', linestyle='None')
plt.show()

This is what I got:

graph1

Now, for example, 0.375 = 0.5 – 0.25 + 0.125. A minus sign means an L, a plus sign is an R. So 0.375 is LR. 0.625, which is the fraction the same distance from the right as 0.375 is from the left, is 0.5 + 0.25 – 0.125. So it’s RL. So as you look at fractions equidistant from 0.5 on either side of it, all the R’s and L’s in their sequence get switched. Therefore, the fraction of R’s in one should be the fraction of L’s in the other, or 1 – fraction of R’s. Thus, you expect the graph to be symmetric about the point (0.5, 0.5). (Think about this, no hurry.) What miffed me at this point, therefore, was that this graph didn’t appear to be symmetric with respect to its center point. There’s some fuzzy mess to the left and some scattered points isolated from the main band that are not symmetric at all.

Then I ran some tests with fractions whose sequences aren’t supposed to end at all. Like what? Like 0 and 1, say. If you’ve followed the algorithm, you can tell that we can never arrive at a cleaving of an interval where the separating number is either 0 or 1, because there’s nothing on one side of these numbers. So 0 should just give me LLLLL… and 1 should give me RRRRR…, never ending. However, guess what I found when I looked at the number of L’s and R’s in their sequence.

0    L: 1074, R: 0.

1    L: 0, R: 54.

So why do the sequences end? That’s fairly simple. It’s because of the limitation of storing and computing floating point numbers in a computer. Notice that with each step of the sequence we are squeezing our number tighter and tighter, into an interval that is halving its length with each iteration. Very soon, our computer (or the interpreter) arrives at a point that numbers so infinitesimally separated in that tiny interval are no longer separate numbers to it, and so it cannot differentiate between our fraction and the mid-point of the interval, and stops.

Exactly how big is this error? It is difficult to tell from looking at these numbers above. One tells you it should be 1/2 1074, the other tells you it’s 1/2 54 (which is closer to where I’d put it, owing to other checks I did and don’t want to discuss here). The final result has to do with all the calculations it is doing at every step, and so all the floating point errors that accumulate at every step. However, I think the only way the answer could still be different for a fraction and its ‘mirror-image’ is if different floating point errors are associated with addition and subtraction, because these two operations have been switched for them.

Notice, though, that the fraction of R’s for 0 is 0, and that for 1 is 1. The symmetry is preserved. So where is the final problem in the plot? Well, we’ve been lucky with these two numbers because one of the counts is 0 for both cases. I’ll give you an example of another case:

0.1    L: 28, R: 26.

0.9    L: 25, R: 27.

In this case, obviously, the symmetry will not be maintained, because the second pair is 25,27 instead of being 26,28. Thus, the graph is no longer symmetric about the center point.

Since I was stubborn about getting a symmetric graph, I decided that I’ll cut off the process before it gets to the ambiguous stage, that is stop with a wide enough interval length, and plot a graph with the truncated sequences. I finally got a symmetric one when I set the interval length at the order of 1e-13. For this, in line 20 (highlighted), instead of elif dc==d you need to write elif abs(dc-d)<=1e-13. Here is the resulting graph:

graph2

Note, however, that this error tolerance is not something fixed. It depends on the resolution (spacing) of fractions you want to do this computation for. In the images you have so far seen, the fractions were multiples of 10-4. You get a better image with one order lower, but for that the error tolerance had to be jacked up to 1e-11:

graph3

Do you see something really interesting in this graph now, in the way that it organizes itself into parallelograms within parallelograms? It’s a highly ordered fractal. I’ve marked them out for clarity:

graph

In other words, the point symmetry is repeated on increasingly smaller scales, as it should. The whole bisected nature of the nested intervals is responsible for this. More parallelograms would be revealed if we kept making our resolution finer, and the horizontal extent of these parallelograms are only exhibiting those nested intervals.

The fraction of rights, however, doesn’t reveal a lot of information. More interesting could be to see how many such bisection steps are required before we converge onto a number. For this you need to modify the source code just a bit. On line 25 (highlighted), substitute float(R)/(R+L) with R+L, and you get this:

graph5

The black dots are the data points, joined by blue lines for clarity. Again, this should have been symmetric about x=0.5 (about a line this time, not about a point), but it isn’t. Notice that the low sequence lengths for numbers such as 0.125 or 0.375 like we discussed don’t even appear. The lowest sequence length we see here is about 35. That’s because these fractions never even arrived in the incrementing loop, although they should have. This is computational error again. I can tell because I have poked around a bit. Try out this python snippet for example:

c=0.
while c<=1.:
	c+=1e-2
	if c==.12:
		print c

By the way, one data point, corresponding to the fraction 0, had to be removed from this graph, because its sequence length was very big, 1074, as we saw before.

If you zoom into the middle of this graph a bit, however, you’ll see the kind of symmetry I had been looking for:

graph6

Do you see why we should have a picture like this? Think about it, it’s not very hard. Meanwhile you can download a wallpaper I made in Photoshop out of the above graph, because I liked it so much.

graphwallpaper

That’ll be it for now. Let me know if you have any ideas or questions about all this.

My First Steps to Lucid Dreaming

I think last night I took some of my real beginning steps to lucid dreaming.

Let me briefly summarize what lucid dreaming is. It is a highly aware form of dreaming in which you have complete knowledge that you are dreaming and can willfully direct the course of events and happenings in your dream. This art can be practised.

I think I came across this phenomenon (among many other delightful things) while roaming about on StumbleUpon. I tried the exercises on the websites a long time, perhaps a year or two, ago, and I think they worked a bit. They instructed that I keep a notebook, a dream journal, handy on the bed as I go to sleep. If I wake up in the middle of a dream, I was to jot down whatever I could remember in the notebook.

I had dreamt of the death of a close relative. I woke up and wrote down in the notebook, sleepily, just the name of the relative, because I really didn’t have the strength or the will to write any more. Then I went back to sleep.

I woke in the morning to discover I’d never even tried out the plan of keeping a notebook on my bed. The whole thing had been a dream. But I remembered clearly which relative I had dreamed of as dying.

So I guess that was probably my very first step to lucid dreaming, a long time back. Then the next step was last night, or what I suspect as early this morning. (It’s weird how you still preserve a sense of time while you sleep. Maybe it’s a false, distorted sense, but nevertheless a considerably clear sense.) I’ll try to describe to you the dream I had in all the detail I can master.

I had many dreams last night, all very confused and possibly interwoven. I’ll skip the irrelevant ones and get to the one that was important.

It was a remote, wild place, possibly somewhere in the mountains. A storm was raging at night and it was raining and very muddy all around. There was either some natural or man-made calamity going on, and we were stuck and needed to get out of there. I vaguely remember that my mother and sister were there.

I think I was outside on the street in front of a wooden bench of some small tea-shop in the middle of this, when I remembered about lucid dreaming.

The very next thing I remember is that I was lying flat, face down on the ground in a comfortable position, and was feeling as if I was being sucked downwards in a strong sudden whoosh. It was a floating, weightless sensation as if I were swimming effortlessly on water, at the same time that I was sinking very fast downwards in response to some great force pulling me in. It was a very passive, light sensation, like you have when you have downed some glasses.

I was tremendously happy. I remembered an interview of a guy who’d just been able to dream lucidly and was very happy. (It was a real interview I had watched in a BBC video in my laptop. See, that’s how in lucid dreaming you are much more conscious and can access your conscious memory like you normally do. You don’t experience the reduction of consciousness or mental abilities that you usually have in a dream.) I was happy, for I knew this was happening because I had identified that I was in a dream, and as an immediate consequence was being sucked out of it. I think my flat, face down posture was actually my awareness of my own body that was sleeping on the bed, dreaming. So, my first lucid dream, I thought happily.

I started feeling an increasing pressure on top of me, though, particularly on my head, as I kept sinking like that. As if I were going down deep into water and the pressure on top of me was building. I tried to keep calm, telling myself that this is all in the dream. No matter how bad this gets, you’re actually all fine on a bed, sleeping, and no real bodily harm will be done. Keep calm, I told myself, but panic was rising.

Suddenly I was out of the sensation. But I wasn’t on my bed. I was standing in a brightly lit, expensively decorated, majestic corridor, like in some palace. It didn’t seem weird at that time that I didn’t wake up on my bed.

The corridor had no doors or windows opening laterally. It was narrow, and just went on forwards, pillar after pillar, very brightly lit, until it ended at a narrow vertical piece of wall. At the foot of this wall there was a very expensively decorated, majestic trunk.

I approached the trunk and was looking at it. I think I also crouched and touched it.

Then I think the dream ended or something. I think what I felt was that my sleep was getting lighter and I was gaining waking consciousness. You know when at the end of a dream you’re waking up and you know you’re conscious but you still try to continue the dream, adding to it consciously but it’s not as much fun any more? I think that’s what was happening. I was disappointed.

I told myself, ‘you need to get out of this. For that, you have to open your eyes.’ I had done this before when I was on the operation table for my fractured wrist and under sedative drugs. That was one hell of a trip I’ll talk about some other time.

And sure enough, as I opened my very heavy eyes with a lot of deliberation, I realized that I rose at once from all the confused layers of dreams and saw the bedsheet under me. White with interweaving patterns of green. It was a calculated, deliberate, forced action, not very pleasing, to have to wake like that. But it was very real. I could tell that I was now awake and my head was working clearly.

I was sleepy still though. Nevertheless I tried to analyze my dreams a bit. I realized that the first bit, being sucked out of that dream within a dream, was closer to lucid dreaming. I also realized that I had ‘woken’ from it into another dream which was much less lucid, in the sense that I wasn’t as conscious in it. It was very vague and I wasn’t directing the course of events or thinking too much by myself or making decisions. It had been more like watching TV. And I also realized that waking from that hadn’t been like lucid dreaming at all. The dream had just faded away and I had slowly woken up, trying to continue it using my conscious self, which is not lucid dreaming at all.

I was disappointed. I think I went back to sleep.

I woke up much later. I was on a blue, purple and red bedsheet. I checked it carefully. I couldn’t believe it. There is no white and green bedsheet in the house, nor do I remember having slept on one. This bedsheet, though, I remembered.

I think my mother was waking me up. It was midday. There was a lot of sun around. I realized that the first lucid-like dream, the second vague dream, and the third very conscious, very real, waking up and analysis, had all been dreams. Dream within a dream within a dream within a dream. Four layers. I stared at my mother’s face for a while, I think. Then I remembered to ask myself one of the lucid dreaming exercise questions: ‘Is this a dream?’ Because I was seriously not sure any more how long this would continue.

Fortunately, that’s where it stopped. It’s the same blue-purple bedsheet on which I am sitting right now as I write this. But it was one hell of an experience. I am looking forward to more adventures soon. I am pretty sure lucid dreaming works, and I will see this to the end.

Have you ever had any unusual experiences with dreams? Let me know; it will be interesting.

What is this writing?

Something funny happened today. Two of my college friends and I were lounging around in one of the classrooms of the college after hours. Because it has this sexy AC. There was this big clean inviting blackboard in front, and I just went ahead and scribbled some stuff on it in a way I had taught myself when I was in school:

Photo0037

Here it is from up close:

Photo0036m

We were taking these photos and transferring them on a laptop and doing related useless stuff when suddenly one of my junior students, A, entered the room, followed by Professor P, Head of Department of Physics and another professor, probably, who I don’t know. And I thought, shit.

They ambled into the room. Their gaze fell upon the board. Now that looks like some serious Arabic or Klingon or aboriginal script, so they kept staring at it. And I thought again, shit.

Professor P asked, ‘what is written here?’ My heart accelerated mildly. I thought at the back of my head that it’s going to take some time for them to figure this out. This can be salvaged comfortably.

A explained immediately how it was written. Shit came to my mind. He had seen me do this before some time.

Okay, we don’t know what it means, or who wrote it. We were just in the room, I thought.

P asked, ‘who wrote this?’ A pointed to me and said it was me; it couldn’t have been anyone else.

You know what came to my mind? Right.

I had to intervene before it all went out of control. I raced up towards the board from behind as the professors stood reading it. The second professor was already saying, ‘I can tell what the first word is…’

I grabbed a duster and started wiping it off, saying, ‘Sir, I’ll show you. I’ll show you right now how it’s written.’

Professor P started saying he wanted to know what was written. I kept  erasing until I had wiped the whole thing clean.

Then I took a chalk and said, ‘Look, Sir, I’ll write your name.’ And I wrote his name. He was impressed.

I came back to my seat. The professors took their seats as A started explaining some physics stuff on the board.

My friends and I stayed for a while in the room, then got the hell out of there when we couldn’t stop sniggering.