Half Dome

View of Half Dome from Glacier Point. At some point, I will process the rest of the images from Yosemite.

Reciprocity

There is no reciprocity. Men love women, women love children, children love hamsters.

- Alice Thomas Ellis

Gaps in History

Junot Diaz who won last year's Pulitzer Prize for The Brief Wondrous Life of Oscar Wao, a not-so-brief but wondrous book, says that he is fascinated with gaps in history. His book was is set in Newark and the Dominican Republic during the time of the infamous Trujillo. A dictator so brutal and merciless that there are almost no true reports of his republic. He once had a graduate student murdered for writing a thesis on the true nature of his regime. Junot Diaz's point is that if you look at somebody's or a country's history, the gaps in the reporting are the most interesting.

The current presumptive Democratic candidate oozes so much charisma that people who never voted before are now lining up to help him get elected. Of course, armed with an incredible story like Obama's, impossible is nothing. The image that is portrayed is that of a newcomer, a fresh face, someone who represents a new kind of politics. In the same breath one also cites that as a sign of his inexperience and naivety about Washington and the world.

Obama and Trujillo has poles apart, but there is a certain gap in Obama's self-reporting. The recent controversy over the satirical cover of last week's issue of the New Yorker obscured, in ironical fashion, facts that are relatively unknown about the junior senator from Illinois.

Ryan Lizza's story traces the making of Obama in his years as a Chicago politician. To forge an identity as a a nobody in in a city "which doesn't take kindly to political carpetbaggers" would be commendable enough, but Obama's sights were always higher than the tallest buildings in the Windy City. He always knew that he was slated for bigger and better things. People have been calling him President Obama for over a decade.

Of course, politics is always a Faustian bargain. On his way up, Obama has eschewed many of his old principles and let his old friends down. As President, things are not going to be any different even if Obama projects a different sort of image. Lizza writes:

Perhaps the greatest misconception about Barack Obama is that he is some sort of anti-establishment revolutionary. Rather, every stage of his political career has been marked by an eagerness to accommodate himself to existing institutions rather than tear them down or replace them. When he was a community organizer, he channelled his work through Chicago’s churches, because they were the main bases of power on the South Side. He was an agnostic when he started, and the work led him to become a practicing Christian. At Harvard, he won the presidency of the Law Review by appealing to the conservatives on the selection panel. In Springfield, rather than challenge the Old Guard Democratic leaders, Obama built a mutually beneficial relationship with them. “You have the power to make a United States senator,” he told Emil Jones in 2003. In his downtime, he played poker with lobbyists and Republican lawmakers. In Washington, he has been a cautious senator and, when he arrived, made a point of not defining himself as an opponent of the Iraq war.

Like many politicians, Obama is paradoxical. He is by nature an incrementalist, yet he has laid out an ambitious first-term agenda (energy independence, universal health care, withdrawal from Iraq). He campaigns on reforming a broken political process, yet he has always played politics by the rules as they exist, not as he would like them to exist. He runs as an outsider, but he has succeeded by mastering the inside game. He is ideologically a man of the left, but at times he has been genuinely deferential to core philosophical insights of the right.

Only the naive would call Barack Obama naive. If he does win in November, like all victors he can write his own history.

Speed dating issues

I would like to think that my dating days aren't over, but just in a perpetually suspended state. However, I was at a speed-matching event yesterday which attempted to have a bunch of people meet everybody else one-on-one in a space of an hour.



First, there were only fifteen ppl and it was apparent that at every round someone would have to sit out. Then another guy showed up and we were sixteen. Since that was an even number no one would have to sit out. The person in charge did the obvious thing - he made two concentric rings of chairs splitting ppl two groups - 'inners' and 'outers'. The 'outers' rotated around the inner ring. After the first rotation, everybody had met half of the people in the group, except the ppl in their own ring. Then he did the next obvious thing, made another two concentric circles of the rings themselves and repeated the process. After the second rotation, everybody had met 3/4th of the ppl. For the next round, these rings needed to be split further into similar rings. As you can see, this process stops when each of the two concentric rings have exactly one person.

Of course, since we had the fourth power of 2, i.e. 16 ppl, this scheme works beautifully. Since, I was part architect of the idea, I wondered if this arrangement would work for other even numbers, leaving the rather odd case of odd numbers aside for now. Very quickly, you can see that this scheme breaks down for 6 people.

Round I            Round II
A D             A-B       D-E
B E             C (sits out)       F (sits out)
C F

After the first rotation, you have Ring I meet everyone in Ring II. Proceeding as before, you end up with 3 ppl (an odd number) in each new sub-ring. Now every subsequent iteration will have one person in each group simply sitting one session out. As you can see from the example, we can form a pair with C and F in Round II, but they have already met each other in Round I. So, every subsequent period, one person will be sitting out and wasting his time. This would require 6 time periods.

Thus, the concentric circles thing is inefficient for all non 2ⁿ even numbers. I tried to come up with a pairing scheme for n=6. Since ⁶C₂ = 15 total pairs, and we have 3 pairs at each stage, we should need at a minimum five rounds. With the schedule below no one sits out and we achieve efficiency.

Table for 6 ppl

Station	1	2	3	4	5
S1	AD	AE	AB	AF	AC
S2	BE	BF	CE	BC	BD
S3	CF	CD	DF	DE	EF

The movement of people is non-intuitive and the schedule is complicated. You would need to hand people a schedule map: A would not move at all. B would move to stations: S2-S2-S1-S2-S2. I was certain that there was a solution to this problem, floating in graph theory or combinatoric literature. And indeed there is! But, this is no simple can of worms as I was to discover.

Famously, in 1850 Reverend Thomas Kirkman sent a query to the readers of a popular math magazine, Lady's and Gentleman's Diary:

Fifteen young ladies in a school walk out 3 abreast for seven days in succession: it is required to arrange them daily, so that no two will walk twice abreast.

1 of 7 possible solutions

The more general case of problem is called the Social Golfer Problem: Determine the maximum number of days 'w' that 'n' golfers can play in groups of 'r' each without meeting each other. This is still an UNSOLVED mathematical problem!!! If you are interested in reading more see: Social Golfer Problem

My original question of dividing 'n' ppl in pairs has been solved at least up to n=200. Round-robin scheduling is a wonderful site for those scheduling matches, or speed-dating style events. There is no simple movement order that can be prescribed. You have to pretty much follow the schedule blindly. Some schedules are unique, in other cases there are over 1000 solution, usually when 2ⁿ numbers are involved.

Yeah! Even speed-dating has issues!

Ignoble Research

While my so-called 'real' publications are limping along, one of my side-projects got published in this month's Annals of Improbable Research. A journal that is self-styled as " the journal of record for inflated research and personalities" . These are the good folks who dish out the Ig Noble Prizes each year

For the past three years, at the Society for Neuroscience conference my labmates and I present a 'joke' poster in the vein of The Onion. The Cingulate Cortex Does Everything started off as a satire on the field of fMRI research in neuroscience. There are tons of papers in journals like Nature and Science that implicate the cingulate cortex in all kinds of behavior. Brain fMRI scans detect oxygenation levels in the blood and determine if blood flow to particular part of the brain increases or decreases with respect to a behavioral event. It seemed rather interesting to us that the use of fMRI correlates so strongly with cingulate cortex sightings. We suspect that since the cingulate cortex is above the saggital sinus, a major drainage vessel for the brain, it seems to light up in fMRI studies as an artifact.


On scanning recent literature on the subject, we saw an explosion in cingulate cortex research and reached our startling conclusion - "Cingularity", i.e. if current trends continue the cingulate cortex will not only take over neuroscience research, but everything!!

Another interesting fact is that the word cingulate is derived from the Latin word cingulum meaning belt. Specifically, a belt protecting your family jewels.

Quiz blog

Dinesh K. aka Dinky has set things rolling with the U of M Quiz blog. And for all you shameless people out there - Googling is allowed!

Dawkins's Mirror Question

A really good book is one that makes you want to put it down for a while so that you can think.

This is taken from Richard Dawkins's latest book The Oxford Book of Modern Science Writing. See the review in Science (Mail me if you need access).

For years as a college tutor at Oxford, I would try the intelligence and reasoning powers of entrance candidates by asking them at interview to muse aloud on the conundrum of why mirror images appear left-right reversed but not upside down. It is a provocative puzzle, which is hard to situate among academic disciplines. Is it a question in psychology, in physics, in philosophy, in geometry, or just commonsense? I wasn't necessarily expecting my candidates to "know the right answer." I wanted to hear them think aloud, wanted to see if the question piqued their interest and their curiosity. If it did, they would probably be fun to teach.

The book is maybe good bedtime reading, but not the paragraph above. I read the paragraph and I couldn't sleep till had some sort of satisfactory answer. It's rather intriguing and wanted to discuss this below. My friends in Mathematics have a better idea of this, but I am taking a stab at it here.


STOP HERE IF YOU WANT TO THINK ABOUT THE PROBLEM ...



It is about particular symmetry? That lead me away from thinking about mirrors to drawing mirror images and considering that we reflect images about a particular axis. This is a practical issue when it comes to making stamps, or printing for example.

(Pardon my sloppy drawing. It's much harder to draw accurate mirror images than is popularly believed.)
Upon drawing this simple figure, I realised that mirroring (reflection) is a not really a 2D transform, but is a 3D transform since there is no strictly planar or 2-D operation(rotate, shift) can be performed that convert A to B. Of course if you reverse the X-axis then you automatically reflect. It's like looking at the 2-D figure from the bottom instead of the top.

Thinking about it in another way, if you had simply an outline figure (shown on top), you wouldn't really see any lateral inversion. If viewed from one perspective, there is no lateral inversion. That's because the figure being hollow does not have a correct viewing side.

So, according to me it boils down to a frame of reference. By imposing your frame of reference onto the mirrored image you get the perception of lateral inversion. In that parallel universe of mirror images, the X axis is reversed and hence correctly speaking left is actually right! The absolute values of all the points along X have not changed, only the signs have.

Coming back to the other question of top and bottom inversion, the simple four figure shows that top bottom is achieved by rotation around the Z-axis, or by two repeated mirrorings on the Y and X axis. So, you can ask why rotation only changes top and bottom, but not left and right. They are simply two different transformations.


More generally speaking, there is really left or right, or top and bottom. It's all relative all perceptions are due to the frame of reference. So, if you perform a transformation then you have to choose the appropriate frame of reference or grid to view it correctly. In a more extreme case, even in case of a distortion, the square will be a square in the distorted frame, but appears warped only because you impose your original frame on the 'new' square.

So, I think the problem is one of pure geometry and not of psychology or even physics. Perhaps, humans are predisposed to think of particular frames of references which cause this effect.