"The poet only asks to get his head into the heavens. It is the logician who seeks to get the heavens into his head. And it is his head that splits." G.K. Chesterton

Wednesday, March 23, 2011

Youth's Embarrassment - aiding the maintenance of an image of fitness.

First I need to justify my usage of the term "youth" -  having reached the "age of reason", I have the perspective of the years to make valid (i.e. experiencially substantiated) comparisons. That is all. And now to the topic at hand.
I have always felt this to be true, as when observing teen dynamics or some of my young university collegues who are still in their early twenties. In the latter case, usually when a group engages in a discussion of even something remote and theoretical, what seems to dominate is the character of the delivered statements, not their content alone. Nicely wrapped and eloquently delivered packages, with suprisingly average content. This is not an evaluative statement, merely an observation; it appears that we tend to learn how to send envelopes, before we learn what they should contain.

Last week, as I was leaving a train station, and upon approaching the stairs leading down to the underpass I noticed a group of four 18-20 year old women. One of them slipped on the wet stairs (as it was raining earlier), and as she landed on her backside - which looked quite painful - I was amazed how she started laughing unnaturally and nervously, obviously giving her girlfirends a sign that nothing had happened. Fiercely blushed cheeks however, betrayed her embarassement. Her social status as a "non cluts" was more important (and of immediate value) to her than her physical wellbeing, apparently. You don't see this kind of behavior equally prevalent in adults or the elderly. I can imagine fear in their eyes in such situations, if anything.

Why this apparently irrelevant embarassement in cases of small, physical danger? Embarassement is certainly an unpleasant sensation. It is a hormonally induced state telling us "you look less attractive (i.e. less fit)". Since our social bonds are fundamentally being shaped and often established in our youth, and also due to the fact that this is the peak of our sexual activity, it is imperative that we project an image of a fit, healthy, socially sophisticated and mentally capable individual. Embarassement tells us when we fail to maintain that image.

Sunday, March 13, 2011

Wake Up, Kitty!

The story below was written for the New Lotus magazine (at Buddhistdoor International). An attempt at Buddhist fiction: Puss In Boots meets Alice in Wonderland meets Saṃsāra. Needless to say, a condition to fully appreciate the metaphors used therein requires some familiarity with Buddhist philosophy.


Felis loved his lazy afternoon naps by the front door, despite the mysterious recurring dream they offered each time. Those naps were the culminating point to the morning routine spent chasing geckos around the house and climbing the living room curtains each time he had the mischievous impulse to do so, and Alice happened not to be looking. Alice was the most caring and loving owner a cat could ask for, but she certainly was careful not to spoil her pet and let him run wild and do as he pleased. This in part was the condition Alice's parents insisted on: "Darling, remember that having a pet means plenty of responsibility. So it would be good for everyone if you made sure the little kitty behaves."

Nausea had always been the cue upon which Felis would "wake up" within his notorious lucid dream. In fact it was quite unbearable nausea, induced by dizziness. It would then soon become apparent that Alice is holding him close, pressed to her chest, and she herself is strapped to a seat in a gigantic merry-go-round situated at the Luna Park which has been seasonally erected in front of their suburban house. The long arms of this mechanical contraption were always decorated with tacky multicolored lights which blinked to the rhythm of a popular nursery rhyme blasting out of the large speakers attached to the ride’s main column. "She must have smuggled me in, under her sweater" - Felis would realize each time, since pets are not allowed on the grounds of the Luna Park. "It's okay Felis, don't be so nervous - this is supposed to be fun!" - Alice would in vain attempt to reassure her little whiskered friend as he clutched anxiously to her arm with claws tearing the sweater's fabric and scratching her skin.




Aside from the extremely unpleasant predicament of being helplessly spun around on this enormous carousel - it appeared very large to this little kitty - the noise of other passengers’ cries consisting of grotesque combinations of hysterical laughter and desperate screams of horror, was quite unsettling. To top this off, this disturbing mixture of cries and jeer was occasionally accented with sounds of painful convulsions of regurgitation - unsurprising consequence of the ride-induced "sea sickness". "Mheeeoowgrr" - was Felis' response to the surrounding human cries as he buried his fluffy head within the folds of Alice’s jumper. "I'm sorry darling - I wasn't thinking when I took you on this ride" - Alice would reply, realizing that her beloved pet is actually terrified, and not just being obnoxious.

Things worsened as it started to become obvious to Felis that the ride wasn’t going to end at all - it just continued on, and on. People's ghostly cries rang like delirious mantras uttered by a group of helpless lunatics - "Luna Park!” flashed through Felis' little, witty head, but the sight of his mistress pale face brought him back at once to sobriety again. The sickly appearance of Alice's fixed and wide-eyed stare, framed by an unnatural smile overshadowed by pain, alarmed Felis. This unmoved, mask-like expression of his dear owner failed to obscure the imminent danger and urgency of the situation, as Felis' ears, pressed against Alice’s chest, were suddenly flooded with the desperate hammering of her heart. As the frequency of her heartbeats intensified, it seemed that her chest would burst open any second. It was at that moment that Alice would start singing the lyrics of the ride’s nursery rhyme theme. She did that whilst maintaining her unmoved and comatose gaze and unnatural grin:

“Ring around the rosie,
Pockets full of posies,
A-choo, a-choo,
We all fall down”

-"This is a nightmare!" realized Felis, as he looked nervously around the Luna Park, which is no easy feat when done from a spinning carousel, since the whole world rushes past like a blurred and phantasmagorical vision. There was no one to be seen! The entire park was completely deserted except for the people trapped on this spinning, monstrous contraption."Can't anyone see what's going on?! This is unbearable!" were the last thoughts Felis had before freeing himself from Alice's embrace and leaping off the carousel seat. Despite being flung out like a projectile out of a slingshot – his little furry body somersaulting through the air, he managed to land on all fours unscathed.

Only then, once on the ground, shortly after his dizziness subsided, the nausea finally vanished for the first time since entering his lucid dream. Felis became aware of the surrounding stillness. It was quite overwhelming, and offered a clear perspective of the entire Luna Park. This serene repose was suddenly terminated by a sharp and foul smell of vomit painfully piercing Felis’ sensitive nostrils – it reminded him why he was here, on the ground. Compelled to stop the mechanism, he was relieved to finally locate the main switch protruding out of the carousel’s column. Knowing what remains to be done – Felis at this point of the dream would flick the switch and wake up before seeing the carousel slow down to a halt.

Waking up this time, he was pleasantly greeted by the delicate touch of Sun’s mid afternoon rays. Opening one of his eyes lazily he noticed a gecko on the door frame stealthily following a giddily fluttering moth. Felis leaped toward the door frame, but the agile lizard swiftly evaded this sudden feline pounce and wriggled outside onto the grass. After his surreal, yet reinvigorating nap Felis felt ready for a few more playful laps around the house in chase of the amusingly frightened geckos. Having ventured deep into the backyard bushes in pursuit of the swift reptile, Felis was too far to hear Alice’s soft voice singing, as she was busy cutting some flowers blooming in the front yard:

“Ring around the rosie,
Pockets full of posies,
A-choo, a-choo,
We all fall down”.

Saturday, March 5, 2011

Waking up each day: "A Rebirth"

A thought I have had since the time I had been working nights in a large studio. Our brain has a way of reconstructing reality each day analogous to a rebirth, and so a waking day (interval) could be seen this way as a "life". In the morning certain planning and anticipatory procedures take place. The entire day is punctured by certain routines which could be considered as equivalent to "rites of passage".

Having worked nights, and seeing some fellow workmen/women returning to work in the morning, I became aware of how our perceptions of "any given moment" differed. This become obvious not only from the general behavior patterns, but also from interactions and conversations with them. The morning arrivees would generally tend to interact with each other, as would the bunch of those who stayed overnight. This observation verifies an extension of the analogy, since youth generally "sticks" together, as do the more mature folk.


Anyway, the thought I had today was this. If we permit the analogy of each morning being a kind of "re-birth", and consider "the by now non controversial thesis" that life experiences (childhood in particular) determine to a degree how we live out the rest our lives, then we could make an equally uncontroversial hypothesis that our experiences from the earlier part of the day (the few hours after waking in particular) will to a degree influence how the rest of the day plays out for us. For example, I'm sure that most of you had an experience in the past where some annoying or frustrating experience ruined the rest of your day.


Also, and this is the interesting bit, I have observed that older people, including myself as I age, tend to gradually fall into a more rigorous and fixed morning routine. This routine set also tends to grow over time. Now if we consider the above hypothesis, then it should be expected that such limitation of early morning experiences inadvertently limits the way the rest of the day will be "lived out". The self imposed narrowing of the morning experiences narrows down our outlook and hence experiences and set of behaviors for the rest of the day.


That's not to say that this is a bad thing. It seems that this way we self determine how each day plays out - I guess in some sense this is an emphasis of our uniqueness - we fall into our unique and self determined patterns of behavior and being in general (this includes how we "take in" the World i.e. our word constructs). It may also be interpreted as a way that our body settles to a more efficient way of functioning i.e. it follows patterns that have proven to work in the past.


On the other hand, in an epistemic sense this observation transcends those behavioral patterns (it becomes aware of them), and it may be useful to anyone who feels their life has entered an uncomfortable infinite loop, and they would wish to free themselves from it. What would be the best strategy to break out? - Diversify your "daily childhood"!

Tuesday, March 1, 2011

Interesting Properties of Fractals


KOCH SNOWFLAKE
This object has the property of having a finite area enclosed by an infinite perimeter. Don't believe it? Then see the proof below.




The recursive construction of this fractal is achieved by extending the middle third of each straight line by an additional equilateral triangle. The first three iterations (plus the starting – zeroth iteration) are shown below.

And each additional triangle tn is 1/9th of the area of the one it appears on  tn-1 , as shown in the diagram below for t0 and t1 .





The entire area A of the entire Koch’s Snowflake can be calculated as follows, where A0 is the area of the initial triangle:



So the area is finite, as it is obvious by inspection. But how about the perimeter of this object?
Let Ln be the perimeter at the n’th iteration where L0 is the perimeter of the initial triangle. Let l be the length of the side of the initial triangle.


So an infinite perimeter encloses a finite area.


THE CANTOR SET
This object has the property of countably infinite cardinality and measure zero.
Below are seven iterations of the Cantor Set, n+1, below n.

The Cantor Set shares similar odd properties, since its cardinality is countable infinity, but it’s measure is zero, as shown below. The construction of the set, proceeds recursively, starting with the unit interval [1,0]. At each iteration the middle 1/3rd open interval is subtracted from any previous intervals. So the first three iterations are the following sets:  ℵ ∪∈



Cantor has shown, in his famous diagonal argument, that the cardinality of infinite binary sequences ∆ is uncountable. Below we present a bijection between ∆ and the Cantor set, defined as:

 
Thus demonstrating that the Cantor set cardinality is uncountable and hence by the continuum hypothesis of cardinality of the continuum i.e. 1


Define the functions on the closed intervals:
L([a,b]) = [a, a + 1/3(b-a)],
R([a,b]) = [a + 2/3(b-a), b]

And with each binary sequence  d  we associate a sequence of closed intervals:
Fd0 , Fd1 , Fd2 ,

Defined recursevly as follows
Fd0 = Cn = [0,1]
Fdn+1 = L(Fdn), if dn = 0
Fdn+1 = R(Fdn), if dn = 1

Example: The mapping of the binary string 00101… onto the sequence Fd0, Fd1,... emphesized in red:

 
Now we define the function   
f : C

 
Now,  f is clearly surjective by construction since for each d we find a unique intersection of the sequence Fd0, Fd1,... 

To complete the proof  we need to show that f is injective. Consider have 2 binary strings red and blue. Suppose n is the least number such that rednbluen. and suppose w.l.o.g. redn= 0. To complete the proof it needs to be shown that f(red) ≠  f(blue).
The illustration below we see how red and blue are mapped onto C, and f(red) ∩ f(blue) is denoted with purple (violet?) color.


We have Fbluen= Fredn from the choice of n,

f(red) ∈ Fredn+1 = L(Fredn),  f(blue) ∈ Fbluen+1R(Fredn), but L(Fredn) ∩ R(Fredn)  =
Hence f(red) ≠  f(blue). Which proves that,
C   =c  1

So what is the length (measure) of C? The measure of the Cantor Set. One way to answer that is to subtract from the measure of the unit interval:


Hence the cardinality of the Cantor set is equal to the continuum, and it has measure zero.