# Tetration

I stumbled over this problem, thinking it to be just another tricky math problem:

$\displaystyle \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}}$ = ?

Trying out by regular method of solving this problem, i.e removing one iteration in this infinite exponentiation, we get

$\displaystyle (\sqrt{2})^x = x,$

Now the above equation has two solutions which we see intuitively. x=2 as well as x=4 seem to satisfy the equation. This too doesn’t take me by surprise because in such situations one of them will be right and the other wrong because the LHS in the problem is a well defined real number and has to be unique. But, I recollected,

Any number greater than 1, raised to a power greater than 1 would always grow exponentially. Hence, an infinite exponentiation(tetration) of such number would yield infinity as the result.

I felt it to be quite intuitive! But here, I ended up with 2 solutions and both appear to be converging, while I expected it to be diverging. After a bit of googling I hit the right terms of Tetration and Infinite Power Tower Problem. I found out to my dismay, that it so happens,

Infinite tower power $x^{x^{\cdot^{\cdot^{\cdot}}}}$ is continuous as a function, whose domain is the interval $[e^{-e}, \sqrt[e]{e}]$ and whose range is $[e^{-1},e]$.

I’m taken aback by the surprises math has in store to offer me. I thought I knew basic math and believed in some things as a thumb-rule which were so intuitive that they could not be denied. Its these mysteries, that drive us towards understanding things. Lesson learnt is,

However intuitive a thing might be, if you cannot prove it, you better not believe in it!

Jeremy Kun has a nice post, False proof 2=4 on validity of the solution and a link on detailed proof of the same.

# LED flash for mobile phones

How many times have you felt bad about the spooky images taken by your mobile phone camera? Well, nobody cared about the poor souls who were stuck up with a low-end mobile phone, which supports a camera but not flash. Most mobile phones today come with the flash too at economical price, but our focus was to complete a mini project which was somewhat unique, which many had not tried earlier (probably for a reason that its dumb!). Whatever, it would still be a good learning experience to design a product altogether from scratch.

Our requirement was to simulate the inbuilt flash in cameras by an external one. Basically, we need to turn on the LED flash exactly at the time, sensor is active and keep the LED on-time as small as possible(to save power). Our idea was to use the Camera ‘click’ sound that comes if the sound is enabled. the sound generated can be tapped by audio port of mobile phone and can be given as a trigger signal to 555 timer circuit, configured as monostable multivibrator. But unfortunately, the ‘click’ sound in Nokia phones is produced after the picture is captured. So, we went for a simpler way of tapping the keypad tone. After some amount of trial and error by adjusting RC for preferred duration of LED on, we could get it working. We used  two 3W LEDs in series to get light bright enough.

Then, the Industrial Design part of bringing up a proper mechanism to cling on to the phone, its adjust-ability were designed. With a few acrylic sheets, glue and some work in workshop, we came up with a nice box (bigger than the phone itself!). It fit well on the phone.

Its a pleasurable experience to design and build things however small or insignificant they are. Hope to build more things not just out of compulsion, but just for the sheer pleasure of it.

# ಬಾಳ ಬವಣೆ

Poems take a hard time to write. Putting the thoughts across the reader in minimal set of words is yet another art. Attempting to write one helps us to get to see our thoughts personified and gives us a sense of companionship to our thought process. As J.Krishnamurti says,

You must understand the whole of life, not just one little part of it.
That is why you must read, that is why you must look at the skies,
that is why you must sing and dance, and write poems and suffer and understand,
for all that is life.

Here is a poem written by my sister..

ಮೊದಲ ಅಳುವಿನೊಂದಿಗೆ ಶುರುವಾಯಿತು ಕಥೆ
ಕೊನೆಯುಸಿರಲ್ಲಿ ಬರೀ ವ್ಯಥೆ..

ಅಮ್ಮನ ಬೆಚ್ಚನೆ ಮಡಿಲಿನಲ್ಲಿದ್ದಾಗ, ಹಸಿವು ದಾಹದ ಅರಿವಿರಲಿಲ್ಲ
ಕಣ್ಣೆದುರಿಗಿನ ವಸ್ತುವೆಲ್ಲವೂ ವಿಸ್ಮಯ, ಪ್ರತಿಯೊಂದನ್ನೂ ಕೆದಕಿ ನೊಡುವ ಹಂಬಲ
ಚಿಗುರೊಡೆದ ಮನಸ್ಸಿನ ಸಸಿಗೆ, ಎಲ್ಲವನ್ನೂ ಅನುಭವಿಸುವ ತವಕ
ಬಾಳ ಈ ತಿರುವಿನಲ್ಲಿ ಹೊರಟೆ ನೀರಿನ ಅಲೆಯೊಡನೆ..

ಈಗ ನಡೆದಂತೆಲ್ಲ ಬೆಳೆಯುತ್ತಿದೆ ದಾರಿ,
ಎಷ್ಟು ಓಡಿದರೂ ಮುಗಿಯದ ಪಯಣ
ಸರಿ ತಪ್ಪುಗಳ ಅರಿವಿಲ್ಲದೆ,
ಎಡವುತ್ತ ಸಾಗುತ್ತಲೇ ಇರುವೆ ಈ ಜನಜಂಗುಳಿಯಲ್ಲಿ..

ಜೀವನ ಜೇನಿನ ಗೂಡು ಎಂದು ನಂಬಿದ್ದೆ
ಇದು ಸುಳ್ಳೆ? ಎಂಬ ಆತಂಕ ಮೂಡಿದೆ
ಎಲ್ಲರೂ ನನ್ನವರು ಆದರೂ ಯಾರೂ ನನ್ನವರಲ್ಲ
ಎಂಬ ವಿಪರ್ಯಾಮತಿ ಬೆಳೆದಂತೆ, ಮಾಯವಾಯಿತು ಕುತೂಹಲ..

ಚಿಂತೆ ದುಃಖದ ಸುಳಿಯಲ್ಲಿ, ಎಲ್ಲರ ಪ್ರತಿದಿನ ಸಾಗುತಿದೆ
ನಗುವಿನ ಮುಖವಾಡದ ಹಿಂದೆ, ಹನಿಯೊಂದು ಕಣ್ಣ ಅಂಚಿನಲ್ಲಿದೆ
ಶಾಲೆಗಳು ಕಲಿಸದ ಸತ್ಯ, ಜೀವನ ಅರ್ಥೈಸಿದೆ
ಎಷ್ಟು ನಕ್ಕೆವೊ ಅಷ್ಟು ಕಣ್ಣೀರು ಸುರಿಸಲೇಬೇಕೆಂಬ ಕರಾರು ಬಾಳು ಮೊದಲೇ ಬರೆದಿದೆ..

ಈ ಸಂತೆಯಲ್ಲಿ, ತಮ್ಮನ್ನು ತಾವು ಅರಿತವರೇ ಧನ್ಯರು
ಜೀವನದ ತಕ್ಕಡಿಯಲ್ಲಿ, ಸುಖ ದುಃಖವನ್ನು ಸಮನಾಗಿ ತೂಗಿದವರೇ ಮಾನ್ಯರು..

ಶಾಂಭವಿ.ಕುಲಕರ್ಣಿ