The Structure of the Universe
We say that space is three-dimensional. But in the world of Mathematics, though, we are not limited by physical constraints!! We might have four, five, or six geometric dimensions, or even infinitely many!!!

Let's go back to two-space. Thos might be a plane or the surface of a smooth objects. Suppose we have little cretures on this two-dimensional surface. Their movements are limited compared to what we are used to.

Imagine one of these two-dimensional creatures, imprisoned inside a square. He cannot get out. But it is easy for us to think of a way to get him out: reach in, grab him, yank him off the two-dimensional surface that is his universe, and plop him out down outside the square! So easy for us, but impossible for him, or anyone else in his universe.

We can extend this idea to our own universe. Suppose somebody is locked in a cubicle that cannot be broken open. There is no escape unless the walls are breached. But a four-dimensional being, looking at this poor three-space person, could quite easily grab him and pull him ouot without the walls of the cubicle even being touched. We cannot envision how this could be done, but mathematically it could be, if we allow four dimensions to exist. More on this in Flatland.

There is another way a person could escape from the cubicle, and the method is easy for us to imagine. Figure it out!!

Tricky Track
Three High schools-Washington, Lincoln and Roosevelt-competed in a track meet. Each school entered one man, and one only, in each event. Susan, a student of Lincoln High, sat in the bleachers to cheer her boyfriend, the school's shot-put champion.

When Susan returned home later in the day, her father asked how her school had done.

"We won the shot-put all right," she said,"but Washington High won the track meet. They had a final score of 22. We finished with 9. So did Roosevelt High."

"How were the events scored?" her father asked.

"I don't remember exactly," Susan replied, "but there was a certain number of points for the winner of each event, a smaller number for the second place and still a smaller number for third place. The numbers were the same for all events." (By "number" Susan of course meant a positive integer.)

"How many events were there all together?"

"Gosh, I don't know, Dad. All I watched was the shot-put."

"Was there a high jump?" asked Susan's brother.

Susan nodded.

"Who won it?"

Susan didn't know.

Incredible as it may seem, this last question can be answered with only the information given. Which school won the high jump?

The Twin Paradox

This blog is as a result of my own attempt to understand a difficult-to-understand paradox, Twin Paradox which is an outcome of of Einstein's relativity (Any physicians around?!). Please tell me if you see anything insightful!(I have question-marked whatever I did not understand!)

Suppose we have two twin brothers, in spaceships somewhere in the intergalactic void where there is no reference point that can be called "stationary." The special theory of relativity asserts that the motion can be ascertained only when there is a reference frame (this one is okay for me!). Suppose these ships are initially hovering close to each other and that there is no relative motion. The twins-call them Jim and Joe-syncronize their watches so that they read the same date and time. What will happen if Joe speeds off to some distance place, at some high speed, and returns to Jim, who sits patiently waiting for him without using his engines during that time? According to the special theory of relativity, Jim will see Joe's time flow slow down while Joe is moving(???). Joe may not have to travel very far away from Jim at all; he might go around Jim in circles for a while, or zig-zag in Jim's vicinity, periodically checking the time. When Joe is through, we can conjecture that Joe's clock will be behind Jim's clock by a certain amount that depends on Joe's average speed and duration of this trip(is there a formula for this???). Joe will be younger than Jim by that amount when the journey is over. (This is okay for me, provided I understand the previous ?-marks!).

But who says Joe is moving and Jim is stationary. If Joe pulls away from Jim and goes a few light years into space and then comes back, is this different than Jim going away from Joe ? Relatively speaking, aren't these two models, one from Jim's point of view and the other Joe's point of view, iddentical? Then we should be able to say that Joe will se Jim's time frame as moving slower than his own, and should find that Jim's clock is behind his own when the trip is over. Now we have a paradox. Because both points are equally valid according to special theory of relativity, it follows that Jim is younger than Joe and Joe is younger than Jim! It is enough to make us want to trash the whole theory of relativity and start over.

We cannot trash the whole theory, however. Its conclusions have been proven to be true! An experiment was conducted not long ago in which an airplane was taken aloft having a payload that consisted of an atomic clock, among other things. The time according to the clock was compared with the time according to another, synchronized, atomic clock on the ground. When the plane returned, there was a discrepancy, though small, that exactly matched the discrepancy according the special theory of relativity.

There is only one way to resolve the paradox,find out how?. I have the answer, but the theory of relativity is still puzzling to me. If someone could explain that to me,....!

I should quote Einstein! "God may be sophisticated but he is not malicious" meaning that although the universe is elegant it is not designed with the intention of being too complicated!

Yesterday, we went out of the city and saw the meterio showers on the sky, saw the milky way. It is wonderful. Thanks to Kasthuri and two other friends. Ok I digressed a lot, I will stop.

A Band Around The World
Suppose the earth were a smooth, perfectly round sphere, with no hills or mountains. Imagine a rope around the earth's equator, strung so that it is snug and does not strech. If we were to add, say, ten feet to this rope, and then prop it up all the way around the planet so that it stood out equally far everywhere, how far above the surface would it stand? We can assume the earth is 8,000 miles diameter, and one mile is 5,280 feet. Do the simple calculation. The answer may surprise you!

A Fixed-Point Theorem One Morning, exactly at sunrise, a Buddhist monk began to climb a tall mountain. The narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit.
The monk ascended the path at varying rates of speed, stopping many times along the way to rest and to eat the dried fruit he carried with him.He reached the temple shortly before sunset. After several days of fasting and meditation he began his journey back along the same path, starting at sunrise and again walking at variable speeds with many pauses along the way. His average speed descending was, of course, greater than his average climbing speed.
Prove that there is a spot along the path that the monk will occupy on both trips at presicely the same time of day.