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 .