Ah! Geometry

Today is World Poetry Day. 

Poems are a fun way to look at the lighter side of Mathematics. Still remember reading aloud some silly math poems with my son when he was younger. Here is one for all of you.

                     What's The Point?

by Noah Weisz, Rockville, Maryland

Geometry begins with a measly little dot, Smaller than even the faintest ink blot, Tinier even than the brain of a bee, Even littler than a drop in the sea, More mini than even the thinnest nose hair , So minutely minute that it's not even there. But if there are two of these curiosities, They make a line segment with length, if you please.

Add one more, and there's a triangle in store, Then a square is there as soon as there are four. Dot after dot builds a circular place; But three dimensions would take up more space. So many little dots step outside the ring, And the final shape is an interesting thing. For when all is connected, what does appear? A big ball of points: a genuine sphere!

Can you think of any interesting mathematical poems ? Do mention it below in the Comment Box.

FoxTrot

Haven't we all gone through these Math nightmares! 

Have a good day !

Pi Day

Today, March 14 is celebrated as Pi Day. Do you want to make a guess as to why? No its got nothing to do with the blockbuster "Life of Pi". If you know your Circles well and aced your geometry while growing up you should know a constant called Pi (π) whose value is approximately 3.14. Hence, today is Pi Day!

What is Pi ?

Pi (π) is the mathematical constant that has been known for almost 4000 years. Its value is the ratio of any circle’s circumference to its diameter  or the ratio of a circle’s area to the square of its radius. The value of pi is approximately equal to 3.14159265, but it is an irrational number and its decimal representation never ends or repeats.

                                  Mosaic outside the mathematics building at the Technische Universitat Berlin                                       

Some cool facts on Pi :

• The official celebration on Pi Day begins at 1:59 p.m., to make an appropriate 3.14159 when combined with the date.

•  Albert Einstein was born on Pi Day (3/14/1879) in Ulm Wurttemberg, Germany

• William Jones introduced the symbol “π” in the 1706, and it was later popularized by Leonhard Euler in 1737.

• Pi was first rigorously calculated by one of the greatest mathematicians of the ancient world, Archimedes of Syracuse (287-212 B.C.). He was so engrossed in his work that he did not notice that Roman soldiers had taken the Greek city of Syracuse. When a Roman soldier approached him, he yelled in Greek “Do not touch my circles!” The Roman soldier simply cut off his head and went on his business!

• In the Greek alphabet, π (piwas) is the sixteenth letter. In the English alphabet, p is also the sixteenth letter. 

• We can never truly measure the circumference or the area of a circle because we can never truly know the value of π. Pi is an irrational number, meaning its digits go on forever in a seemingly random sequence.

• Since there are 360 degrees in a circle and pi is intimately connected with the circle, some mathematicians were delighted to discover that the number 360 is at the 359th digit position of Pi.

• In 2005, Lu Chao of China memorized 67,890 places of Pi and is considered the current Pi champion.

• A mysterious 2008 crop circle in Britain shows a coded image representing the first 10 digits of Pi.

• Potentially every possible number you can think of is hidden somewhere in Pi– your date of birth, phone number, or even your bank details.

• Want to find your birthdate in Pi? Try this out www.facade.com/legacy/amiinpi  (My birthdate is at this location in Pi : 30035)!

• Pi has inspired a particularly tricky form of creative writing called Pilish. These are poems – or "piems" – where the number of letters of successive words is determined by pi.

Here's something for you to smile about !  “What do you get if you divide the circumference of a jack-o'-lantern by its diameter? Pumpkin π.”

Have a great Pi Day everyone!

The Konigsberg Bridges

It is strange how Mathematics can pursue one even when out on a leisurely stroll ! This is exactly what happened to the people of  a laid back city called Konigsberg a couple of centuries back.

Konigsberg is a city formerly in Germany but now in Russia and is called Kaliningrad. River Pregel flows through the city and divides it into two islands and two banks There were seven bridges connecting the different part of the city with each other. The people of Konigsberg always wondered whether it was possible to start from one point and cross all the bridges only once and come back to the same point.

The Konigsberg Bridges

What were they thinking?? Couldn't they just enjoy the stroll around the river peacefully and not make life so complicated? Apparently not and thanks to them a new field in Mathematics was subsequently developed called Topology.

It so happened that a Swiss mathematician called Leonhard Euler was the first to solve this problem. In the process, he introduced the branch of Mathematics called Topology. He used an area in Topology called Networks. A network is a group of points, which are called vertices, and a collection of lines, called edges, connecting these points. 

The Konigsberg Network 

He must have got both tired and fed up of walking the bridges to solve the problem that he decided to draw the map of the bridges on a paper and solve the problem sitting at home! So this is what it looked like. A,B,C And D are the land mass (vertex) and a,b,c,d,e,f and g are the bridges (edges). 

He made the remarkable discovery             ( The Graph Theory) that whether a network is traversable depends on the number of odd vertices. In the Königsberg network, there are an odd number of edges (a,b,f) at point A, so A is called an odd vertex. If the number of edges meeting at a point is even, the point is called an even vertex.  

Traversable Networks

1. A network with exactly two odd vertices is traversable. 

2. A network with no odd vertices is traversable.  
3. A network with more than two odd vertices is not traversable.

Since the Königsberg network has four odd vertices ( all vertices A,B,C AND D have odd edges meeting them) it is not traversable. As this Graph, is not Eularian, therefore, it is not possible to take a walk over the bridges of Königsberg and cross each bridge only once. 

A Network has a Euler Path if every edge in the network is traversed only once without lifting the pencil from the paper.

  

Now do you remember those tricks that our friends challenged us in childhood?? If we could draw a certain figure without lifting our pencil from the paper. I particularly remember this one but there were more impossible ones as well . If only we knew then that it was pure Mathematics playing up!

*If you can remember any more figures do post them here. There will be more exciting posts on the Euler Paths .

The Palindrome Numbers

Palindromes are words or numbers that read the same forwards and backwards eg. EYE, RACECAR, 1221.

The Sator Square

A Roman graffiti artist, in the first century AD, made a square grid of some mysterious words in the city of Pompeii. This is what it looked like

                                            
                                            SATOR
                                            AREPO
                                            TENET
                                            OPERA
                                            ROTAS

The words can be read horizontally or vertically and if all the words are put together they form a sentence which reads the same forwards and backwards :

                   SATOR AREPO TENET OPERA ROTAS

This when translated reads " The sower Arepo holds the wheels with effort". There are many more interpretations to this. 

But enough of this historical trivia ! The point is that Palindromes are so much fun. Although,  most of us have been aware of the palindromes in words, very few know that some of the numbers can be palindromic as well. There are many ways in which  numbers can be toyed around with to get palindromes. Quickly grab your calculator and punch the following :

                                              3 x 7 x 11 x 13 x 37

So what did you get? Isn't it amazing! You can make patterns in numbers appear from nowhere. Are you able to figure out anything more intriguing about this palindromic answer you got ? Confused? Let me come to your rescue.

The answer to the above operation is 111,111.

Interestingly, the number 11 makes some striking patterns.

1  x 1 =                                               1

11  x 11 =                                         1 2 1

111  x 111 =                                   1 2 3 2 1

1111  x 1111 =                             1 2 3 4 3 2 1

11111  x 11111 =                       1 2 3 4 5 4 3 2 1

This will keep on continuing till you reach number 111111111. Do you want to have more fun with this palindromic pattern? If you square the above number, you will get 12,345,678,987,654,321 which is of course a Palindrome. Only problem being, you will have to do the above calculation manually as your regular faithful calculator will not display numbers beyond 12 digits !

As mentioned earlier, there are many palindromic number patterns out there, which would be constantly discussed here.  Do you know of any interesting number palindromes you would like to share here? 

Bubble Magic

Photo courtesy Neetu Aggarwal

Now there would hardly be a being on this planet who hasn't played with Bubbles. Ah those refreshing childhood memory of dipping the wands in a soap solution and blowing air through them and Whoosh!!  Hundreds of transparent balls flying everywhere. There is surely something magical about Bubbles that even adults rejoice in this activity. I would personally look forward to this activity with my toddler. During one of those times, i was drawn into the science and math of Bubbles. Yes, even these innocent frolicky transparent balls are Mathemagical. You would say "Whats in a Bubble?"
Apparently a lot ! Lets just start with how various people look at Bubbles:

Child                    :  Lovely transparent balls bouncing around 
Parent                  :  Some silly bouncing balls that keep the kids entertained
Artist                    :  Lovely transparent balls trapping the colors of the rainbow
Scientist               :  A stretched thin film of soapy water filled with air.

Mathematician      : Spheres, almost always Spheres!



Simple, isnt it !? Now if you thought that Bubbles were as simple as that, then think again! 

First, some facts. Bubbles consist of a thin film of soapy water filled with air. When you blow a bubble the film expands outward to its maximum capacity and then blows away. 

Did you know that whatever the shape of the wand the bubbles are always spherical in shape. You take a triangular or a rectangular wand but the bubbles that escape out are always Spheres. The Sphere is the shape that minimizes the surface area of the structure, which makes it the shape that requires the least energy to achieve i.e. it encloses the most volume with the least surface area.

Picture courtesy Wikipedia

But wait till you see these Bubbles come together. The moment they form a a cluster (as in foam) they act differently. Lets see the 2-D drawing of layer of bubbles. In a cluster they always stick to each other tightly and hence form hexagonal shape. Regular hexagons fit together perfectly, leaving no gaps thus minimizing surface area. Three neighboring bubbles will together form a Triple Junction i.e. three 120 degree angles. 


  "Soap films always meet in threes along an edge called a Plateau Border, and they do so at an angle of 120 degrees" -Plateau's Law on soap films formulated by the Belgian scientist Joseph Plateau in the Nineteenth century.

         

         

       
Triple Junction 

2-D illustration of layer of bubbles

The detailed mathematics of bubbles is more complex and beyond the scope of this blog.

But why this immense interest in the mathematical behavior of bubbles? Foams are of immense importance in firefighting, mineral processing, radioactive dust recovery, oil recovery, crop spraying, food technology and also the brewing industry( where the formation of foam is not beneficial). 

 

Did you know

   

Many of the processed food we eat are actually foams: bread, chocolate, ice cream, cake, coke, soda and beer !   

          

Trivia

One of the Olympic venues, at the Beijing 2008 Olympics, was the National Aquatic Centre, named the Water Cube. It seems to be sliced from a giant foam of bubbles, an effect that is enhanced when it glows blue against the night sky.

The Water Cube at night. Image © Chris Bosse.       

                                 
So are you ever going to look at bubbles the same way? No way !

Mathematically yours,

Let me confess today publicly that I HATED MATH for the better part of my student life. It bullied me  persistently and unrelentingly through out my exemplary education. It was clear to me very early in my academic life, that this relationship would not last long. Later  in my student life, we parted ways. I was greatly relieved and started enjoying my association with Ms. Life Sciences. I had vowed that I would maintain my ignorance to Mr. Math for good.

I should have known better! Did I mention that Mr. Math is a creepy character who almost always manages to find its way around you? It has a knack of sneaking up on you when you least expect it. As an adult, i started having visions of Math everywhere around me in everything i saw; flowers, buildings, paintings etc. I could feel myself getting drawn towards it against my will. The more i read about its presence around me the more fascinated i became.

I started gradually sharing my learning with children in various talks and workshops. I love to see their enthralled faces once they experience the marvels of Math. It has also been a life changing event personally for me to overcome my fear and hatred for a certain Mr. Math!

Through this blog, i would like to demystify this subject and try to eliminate any negative emotions like fear and hatred associated with it. I would like to further maintain that i am no Math expert !! I just like to look at math from a lighter point of view.

So come join me, adult and children, into this fascinating and fun journey into the World of Math.

Mathematically yours,