SCIENCE CAPSULES

If you go on increase the number of  sides of a polygon. the circle will be the limit.      1/2^1 + 1/2^2 + 1/2^3 + 1/2^4+........  =1                            or      Sum of all the terms of 1/2^n  = 1      When n goes from 1 to infinity, the sum of the above series approaches 'one'; but never touches 'one' or crosses 'one', however bigger the 'n' may be.  Hence we can say, the limit of the sum of the series is 1. (1+1/1)^1+ (1+1/2)^2 +(1+1/3)^3 + .....     =2.71... = e The limit of the above series is the famous Euler constant e.     The concept of limit is the corner stone of calculus.  Let us see an example. Let y= x^2 Let dx be the smallest change in x , let dy be the corresponding small change in y. I want to find dy/dx Let x be 3, let the small change be 0.1 (dx) The new x is 3.1 correspondi...

       Today, we live in a world of digital images.  Ads, Media, movie, legal documents, CCTV or just picnic photos.  Everywhere, there are images.  We know, this images can be easily doctored and engineered by any body with the computer.  Further, when the photos are taken at different perspectives, the lengths, the angles and their relationships are distorted.  Then, how to find the truthiness of the image?  There is a mathematical way.  Let us go into detail.      Take a torch.  Switch it on.  Allow the cone of light to fall on the wall.  If you hold the torch straight, you see a circular patch of light.  If you tilt the torch slightly, you see an ellipse of light.      The circle is there in the light cone always. The circle is distorted into an ellipse, when the light patch hits the wall.  I want to prove, the ellipse is only the projection or an image of ...

    x = V - E + F Euler characteristic = Vertices - Edges + Faces. Draw triangle on a sheet of paper.  It has 3 vertices, 3 edges and 1 face. Hence the Euler char. = 3 -3 + 1 = 1 For a square Euler char. = 4 - 4 + 1 = 1 For a circle, imagine two points or vertices on the circle. So, Euler char. 2 - 2 + 1 = 1. The circle, square and triangle all have same Euler characteristic 'one'.  They are geometrically different, but topologically same.  That means, all the shapes can be converted into one another, if they are made of rubber sheet. " Topology is concerned with the properties of space that are preserved under continuous deformations; such as stretching, crumpling and bending, but not tearing or gluing" (Wikipedia).      Now we get the idea of topology.  Let us see one more example.     Take a 'tetrahedron' solid (prism).  It has 4 vertices, 6 edges, 4 faces.     Euler char. = 4 - 6 +4 = 2 ...

         We know well about the circle, the distance between any point on the circle and the center is constant and it is called 'radius'.  Let us now consider an ellipse.  There are two centers (focus) in an ellipse.  The total distance between any point on the ellipse and two foci is constant.      Kepler discovered that all planets revolve around the sun in the elliptical orbits.  The comets appear to us at some point of time.  They reappear after a gap of long time(for example 76 years).  People were not able to predict the motions of comets for a long time.  Then it is found that the comet travel in long elliptical path with sun at one of the foci.  Hence the long time gap for every appearance.      There are two  famous whispering gallaries.  1. United states statuary capitol hall. 2. London's st. paul's cathedral.  They are designed in the shape of e...

        First, we must know about relative velocity.  Say, you are travelling in a express train at the speed of 100km/hour.  In the next track another train is moving at the speed of 120km/hour.  Now subtract your speed (100) from 120 km/hour.  You get 20km/hour.  That means, another train is running at the speed of just 20km/hour with respect to you.  Hence it will apear to go slowly for you.      Say, rain is falling vertically at the speed of 4 kmph.  Draw a 4 cm arrow pointing downwards.  It is the vector representing rain's speed.      You are fast walking at the speed of 6 kmph towards east.  Draw 6 cm arrow from the head of rain vector representing your speed.  To find the relative velocity, we have to subtract.  Hence reverse your vector (-6cm) towards west side.  If we now add both the vectors, we will get the relative velocity.      To ...

        Water, a peculiar liquid.  It is the only liquid that can exist in all the three forms in room temperature.  That is, ice, steam, and water.            It is a life giving substance since it is one of the ingredient needed to make energy giving glucose by the life forms.  It also regulate the body temperature.  Water helps to transport nutrients, clear waste products in our body.  It goes on and on.  Yet, the science books misses to term it as a nutrient.      We have, spherical drops of water, slashing sheet of rain, interesting shapes of ice crystals, ice mountain called ice bergs, flood of water which sweeps away everything, giant waves called Tsunami, cloud forms pregnant with rain drops, a placid lake, the deep ocean, water dust raising due to water fall, high energy steam-so many avatars of water.     Men has made very high pressure jet of water which...

        A cricket game in progress.  The batting team has scored 60 runs so far in 10 overs.  Dividing 60 by 10 we get 6.  6 is the average number of runs per over taken so far.  . That is , the team seems to have scored 6 runs in each over  (Practically, they might have scored 12 runs in a over and zero runs in another over).  'Average runs' is also called run rate.  If the rain bring the game to a halt, run rate is used to decide the winner.  Remember, as the game processes, the run rate changes.      Toss a coin 5 times  .  4 times you may get the head and 1 time tail may appear.  So the mean is 80% for head and 20% for tail.  Toss the coin 10 times.  6 times  may be head.  4 times may be tail.  Now the average is 60% and 40%.      Throw the coin for 1000 times.  (A computer simulation helps).  You may get the head 502 times and tail 498...