Monthly Archives: October 2003

Olympus Mons

 

      Which is the highest mountain? The mount Everest? Ha! It is only

8.8 Kms high. I know one that is 27 kms high!!! Its name is Olympus Mons. It is located on Mars. And it is the highest mountain in the solar system.

      Friends, in today’s VERITAS we will explore the highest and most exotic peak in the solar system. I promise that it would be an out of the world experience.

      Olympus Mons is a shield volcano. And it is huge! It has a diameter of 624 Kms. That is more than twice the distance from Delhi to Chandigarh. It is located in the Tharsis region of Mars. At the summit of Olympus Mons there is a crater( or calrera) much like the volcanoes found on earth. But the crater are huge! It is almost 80 kms across.

The base of the mountain is rimmed with a 6km high wall of scrap!

      A shield volcano a volcano with the shape of a flattened dome. It is built with flows of liquid lava.

        ANother interesting thing about this mountain. The width is 625 kms but the hight is only 27 kms. So it is not steep. The maximum slope is about 6 degrees.

       What makes this mountain and other mountains on Mars more special is the fact that Mars is such a small planet compared to earth( only 1/9 the mass of earth) and it can boast of mountains that would put the ones on earth to shame.

      How did such a small planet get such huge volcanoes? One reason is that at some time in history the eruption rates of volcanoes on Mars was much higher than that on earth. But all this happened millions of years ago. Some Geologists claim that Mars surface has been quiet for at least a million years. Other scientists put this figure to 200 million years!

      That is one reason. But there is a bigger reason: When a volcano erupts on earth it reaches a certain height but the gravity of earth pulls down the magma and it reaches a balance beween the force of eruption and the pull of gravity. But Mars is ighter than earth, so everything weighs less. A volcano can reach a greater height without collapsing under its own weight. So the lower mass of Mars is responsible for its huge volcanoes.

         Another reason for the huge valcanoes on Mars is that the surface of the planet does not move that much. On earth the plates constantly move against each other and the eruption takes place a little further from the initial point every time. On Mars the surface is less mobile and so it erupts at the same place every time. These eruptions contribute to the initial height and the volcano grows higher. SO on earth volcanic chains are formed and on mars the same volcano grows higher and higher.

      Olympus Mons is named after Mount Olympus, the home of Greek Gods.

It was discovered in 1879 by Giovanni Schiaparelli. The region around Olympus ( Tharsis) has several other huge volcanoes ranging from 8 to 10 Kms in height.

       IN this VERITAS we have begun a quest to discover the tallest mountans in the Solar system, Future VERITAS may take us to the hot Maxwell mountains on Venus or the mystereous Boosaule Montes on Jupiter’s moon Io or even the icy mountains on Uranus’ moon Miranda. Please dont go away, we will be right back.

      See the attached pictures: The first one is a top view of Olympus Mons. See how huge it is! The scrap that surrounds the valcano is clearly visible. The second picture is a 3d image of the mountain from the side. See the wall of scrap on which the whole mountain rises. Also see the crater on the top.

        These heights remind me of Alexander Popes lines:

 

        SO pleased at first the towering Alps we try

        Mount over the vales, and seem to tread the sky

        The eternal snows appear already passed

        And the first clouds and mountains seem the last

        But, those attained, we tremble to survey

        The growing labours of the lengtened way

        The increased prospect tires our wandering eyes

        Hills peep over hills, and Alps on Alps arise!!

 

Olymp1Olymp3d

Kanwar

 

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  Go, wondrous creature! mount where Science guides:                 

  Go, measure earth, weigh air, and state the tides:                 

  Instruct the planets in what orbs to run,                          

  Correct old time and regulate the Sun;                             

|=======================================

Reductio ad absurdum

 

                  Friends, in today’s VERITAS we will try to understand a technique of mathematical proof. This is a very powerful technique and helps prove some very complicated mathematical theorems. The technique is called “Reductio ad absurdum” which means reduction to an absurdity. This technique is also called proof by contradiction.

        This technique is very old. It has been there since the time of Euclid(330 – 275 BC). Euclid used this very effectively in the proof of several theorems. This is how it works: suppose we want to prove a theorem which can be stated by a statement S. We assume that S is not true i.e we assume that our theorem is false. Then we show that the consequences of such an assumption is something absurd which can never be true. So this means that our assumption that S is false is not valid. Thus S is a true statement.

 Theorem proved.

  The mathematician Hardy( Ramanujan’s friend) wrote in his book, A Mathematician’s apology: “Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game”

        Notice that our technique makes use of the law of the “excluded middle”. According to this law a statement can either be true or false.

And if a statement is not true then it MUST be false. Not every school of mathematical thought accepts this technique as a valid method of mathematical proof. The school of intuitionism does not take the law of the excluded middle to be universally valid and thus does not agree that Reductio ad  absurdum should be a valid method of mathematical proof.

        Lets look at an example of a proof by Reductio ad absurdum. This is Euler’s proof that the number of prime numbers are infinite i.e there is no largest prime number. But before we start our proof we should remember that every non prime  natural number can be written as a product of primes in essentially one way. This is called the fundamental theorem of arithmetic. We will use this theorem in our proof that the number of prime numbers is infinite:

proof: Lets assume that our theorem is false i.e there are a finite number of prime numbers. That means that there is a largest prime. Now lets multiple all the primes together and add one to the product. This new number leaves a remainder of 1 when divided by ANY prime. Therefore it cannot be  divided by any prime ie it is not divisible by ANY prime.

So it HAS to be another prime. But that is absurd because we had already taken all the primes and multiplied them . And now we have found another prime! SO this contradicts the statement with which we started. So our statenment that  the number of primes is finite is wrong. So we have proved that the number of primes is infinite!!

        See what a powerful technique this is! But it is also a very dangerous one. Euclid made a mistake in his proof of the fifth postulate(the parallel postulate). The problem is caused when a statement looks false but is not really PROVEN false.

      A question to you: do you think that God listens and acts upon prayers?? Should be apply Reductio ad absurdum here? Ivan Turgenev once said ” Whatever a man prays for, he prays for a miracle. Every prayer reduces itself to this: `Great God, grant that twice two be not four'” !!!!!

How very absurd ! ūüôā

Kanwar

 

|======================================================|

  Go, wondrous creature! mount where Science guides:                 

  Go, measure earth, weigh air, and state the tides:                 

  Instruct the planets in what orbs to run,                          

  Correct old time and regulate the Sun;                             

|======================================================|

 

Reductio Ad Absurdum

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Friends, in today’s VERITAS we will try to understand a technique of mathematical proof. This is a very powerful technique and helps prove some very complicated mathematical theorems. The technique is called “Reductio ad absurdum” which means reduction to an absurdity. This technique is also called proof by contradiction.

       This technique is very old. It has been there since the time of Euclid(330 Р275 BC). Euclid used this very effectively in the proof of several theorems. This is how it works: suppose we want to prove a theorem which can be stated by a statement S. We assume that S is not true i.e we
assume that our theorem is false. Then we show that the consequences of such an assumption is something absurd which can never be true. So this means that our assumption that S is false is not valid. Thus S is a true statement.
 Theorem proved.

¬†The mathematician Hardy( Ramanujan’s friend) wrote in his book, A Mathematician’s apology: “Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or
even a piece, but a mathematician offers the game”

¬† ¬† ¬† ¬†Notice that our technique makes use of the law of the “excluded middle”. According to this law a statement can either be true or false. And if a statement is not true then it MUST be false. Not every school of mathematical thought accepts this technique as a valid method of mathematical proof. The school of intuitionism does not take the law of the excluded middle to be universally valid and thus does not agree that Reductio ad ¬†absurdum should be a valid method of mathematical proof.

¬† ¬† ¬† ¬†Lets look at an example of a proof by Reductio ad absurdum. This is Euler’s proof that the number of prime numbers are infinite i.e there is no largest prime number. But before we start our proof we should remember that every non prime ¬†natural number can be written as a product of primes in essentially one way. This is called the fundamental theorem of arithmetic. We will use this theorem in our proof that the number of prime numbers is infinite:

proof: Lets assume that our theorem is false i.e there are a finite number of prime numbers. That means that there is a largest prime. Now lets multiple all the primes together and add one to the product. This new number leaves a remainder of 1 when divided by ANY prime. Therefore it cannot be  divided by any prime ie it is not divisible by ANY prime. So it HAS to be another prime. But that is absurd because we had already taken all the primes and multiplied them . And now we have found another prime! SO this contradicts the statement with which we started. So our statenment that  the number of primes is finite is wrong. So we have proved that the number of primes is infinite!!

       See what a powerful technique this is! But it is also a very dangerous one. Euclid made a mistake in his proof of the fifth postulate(the parallel postulate). The problem is caused when a statement looks false but is not really PROVEN false.

¬† ¬† ¬† ¬†A question to you: do you think that God listens and acts upon prayers?? Should be apply Reductio ad absurdum here? Ivan Turgenev once said ” Whatever a man prays for, he prays for a miracle. Every prayer reduces itself to this: `Great God, grant that twice two be not four'”
!!!!!
How very absurd ! ūüôā

Kanwar

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 Go, wondrous creature! mount where Science guides:
 Go, measure earth, weigh air, and state the tides:
 Instruct the planets in what orbs to run,
 Correct old time and regulate the Sun;
|======================================================|

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