The Soccer Ball
Well, the world cup excitement is getting really intense with the knock-out round starting from tomorrow.
In this post, I attempt to present some interesting geometric facts about the shape of the soccer ball that I read pretty recently. I am no mathematician by any means and, therefore, the contents of this article might not be very technically precise!
The shape is that of a truncated icosahedron which is the 32-faced Archimedian solid corresponding to the facial arrangement of 20 hexagons and 12 pentagons and has 60 vertices and 90 edges[1].
Archimedian solids are convex polyhedra that are composed of two or more kind of regular polygons meeting in identical vertices. Without getting into too much technical details let me explain few simple things here.
First, a convex object is such that if you join two points lying within the object, the resulting line segment will also lie within the object. The objects could be in 2-D or 3-D. For e.g., a square is a convex polygon (polygons are 2-D objects like triangle, rectangle, rhombus, pentagon, hexagon, etc.) with 4 sides. If you join any two points that lie within the square, the resulting line segment will also lie within the square. However, if you consider the shape of an arrowhead, it is still a polygon with 4 sides but not convex anymore because if you join the two end points of the back tips of the arrowhead, the resulting line segment doesn't lie within the polygon.
A regular polygon is a 2-D shape with all its sides having the same length e.g., equilateral triangle, square, rhombus etc.
We use the above understanding to extend the definition of a convex polygon to a convex polyhedron (plural is polyhedra). A convex polyhedron is a convex 3-D object created by joining convex polygons at their vertices. Thus, a convex regular polyhedron would then mean a 3-D object created out of regular polygons.
In case of the truncated icosahedron, the regular polygons are hexagons and polygons. It seems that there are only 13 distinct Archimedian solids known [2]. These are different from Platonic solids which are convex regular polyhedrons and are composed of just one kind of regular polygon meeting in identical vertices [3]. The simplest Platonic solids happen to be the tetrahedron, consisting of 4 triangles, and the cube, consisting of 6 squares. An interesting thing is that all these shapes (which vary from trivial ones to mighty imaginative ones) have been known since the ancient times. And by ancient, we mean, there is no documented evidence of the invention of these shapes.
This figure [4] above shows one of the shapes that you would obtain if you ripped open you soccer ball and placed it on the ground. Note the uniformity in the topology of the polygons. There are 10 columns of vertically stacked regular polygons each with 2 hexagons and 1 pentagon except for column 2 & 3 (counting from left to right) which have an extra pentagon each. But if you disregard those two pentagons, the stacking arrangement of the polygons in a column is the vertical inverse of the stacking arrangements of the polygons in the adjoining columns.
Some other facts about this shape is that it was also the configuration of lenses used for focusing the explosive shock waves of the detonators in the Fat Man atomic bomb, which was detonated over Nagasaki, Japan during World War II on August 9, 1945. The truncated icosahedron is also the shape of the Buckminsterfullerene (C-60) molecule which is considered to be one of the most beautiful molecules in the community of chemists. The ratio of the diameters of the soccer ball and the Buckminsterfullerene molecule is 200 million to 1 [1].
-QT
References:
[1] http://en.wikipedia.org/wiki/Truncated_icosahedron
[2] http://en.wikipedia.org/wiki/Archimedean_solid
[3] http://en.wikipedia.org/wiki/Platonic_solid
[4] http://mathworld.wolfram.com/TruncatedIcosahedron.html
"Ig" Nobel Awards and Darwin Awards
Today morning I was at my university bookstore looking for a present for a friend and while browsing through the books in the "humor" category, I bumped into one whose title was The Ig Nobel Prizes: The Annals of Improbable Research. I had heard about these awards but never delved into the details of it. So I bought the book after reading the preface. Following that I looked up Wikipedia for the Ig Nobel prize.
Wikipedia:
"The Ig Nobel Prizes are a parody of the Nobel Prizes and are given each year in early autumn — around the time the recipients of the genuine Nobel Prizes are announced — for ten achievements that "first make people laugh, and then make them think." The name is a play on the words ignoble and Nobel. Sponsored by the scientific humor journal Annals of Improbable Research (AIR), they are presented by genuine Nobel Laureates, formerly at a ceremony in a lecture hall at MIT, but more recently at a ceremony in Harvard University's Sanders Theatre each fall. "
Examples (sources- Wikipedia and the referred book):
- The discovery that the presence of humans tends to sexually arouse ostriches.
- The statement that black holes fulfill all the technical requirements to be the location of Hell.
- To research on the "five-second rule," a tongue-in-cheek belief that food dropped on the floor won't become contaminated if it is picked up within five seconds.
- Conducting and publishing scientific study to establish that imcompetent people are completely unaware of their incompetence and others' competence.
- Fart stink absorbing underwear (this product has a US patent on it).
For more details refer to the following URL:
http://en.wikipedia.org/wiki/Ig_Nobel_Prize
I also came across another award called the Darwin awards. The awards are based on Charles' Darwin's idea of evolution - survival of the fittest. So, the organisms/species of organisms which are not fit eventually phase out or become extinct.
Wikipedia:
"A Darwin Award is a tongue-in-cheek honor given to people who improve the human gene pool by removing themselves from it in a spectacularly stupid manner. The prizes, named after pioneering evolutionary theorist Charles Darwin, are awarded over the Internet. There is no monetary prize, only (necessarily) posthumous recognition."
Some of the recipients of these awards had died (or rendered themselves incapable of reproducing) in the following ways (source- Wikipedia):
- Juggling hand grenades (Croatia, 2001).
- Jumping out of a plane to film skydivers while forgetting to wear a parachute oneself (USA, 1987).
- Trying to get enough light to look down a gun barrel using a cigarette lighter (USA, 1996).
- Cutting off one's own head with a chainsaw in a macho-contest (Poland, 1996).
- Using a lighter to illuminate a fuel tank to make sure it contains nothing flammable (Brazil, 2003).
- Heating a lava lamp on a gas stove (USA, 2004).
- Having sexual intercourse with a vacuum cleaner (USA, 2000).
For more details refer to the following URL:
http://en.wikipedia.org/wiki/Darwin_Awards
-QT
The Perfect Collaboration
WARNING: Contents of this post can ensue severe consequences if used as a topic on a date!
I was reading a book titled "Mathematical Treks - From surreal numbers to magic circles" by Ivars Peterson that my cousin (he's the true math geek) gave me earlier this year. I came across some trivial yet interesting facts on a family of prime numbers and perfect numbers.
So, what is a prime number? All of us know this one.
A prime number is a natural number what has exactly two (distinct) natural number divisors - 1 and the prime number itself.
Now, let us focus on a subset of prime numbers called Mersenne primes (named after French cleric and mathematician Marin Mersenne). Mersenne primes are the primes that can be represented in the form 2^p - 1 (read this as 2^p as "2 raised to p"), where the exponent p is itself a prime number. It seems that Mersenne primes have had a special place for mathematicians in pursuit of large primes. For e.g., (2^2 - 1) = 3 is the smallest Mersenne prime. The next is (2^3 - 1) = 7. Then you have (2^5 - 1) = 31 and so on and so forth.
So, what's so special about Mersenne primes? Since, the unsigned binary representation of a positive number of magnitude (2^n - 1) has all 1's, thus all these Mersenne primes have all 1's in their binary representation.
3 = (2^2 - 1) = 11 (sequence of 2 1's) - 1st Mersenne prime
7 = (2^3 - 1) = 111 (sequence of 3 1's) - 2nd Mersenne prime
3 1= (2^5 - 1) = 11111 (sequence of 5 1's) - 3rd Mersenne prime
...and so on and so forth.
Now lets refresh our primary school mathematics with the definition of a perfect number. A perfect number is a number whose proper divisors (i.e. the divisors of the number excluding the number itself) add up to the number itself. For e.g., the proper divisors of 6 are 1, 2, and 3 and 1+2+3 = 6. Similarly, 28 = 1+2+4+7+14.
Although, it might appear that these numbers have no association between them, Euclid proved that if (2^n-1) is prime, then (2^(n-1))(2^n - 1) is perfect.
if n = 2, (2^2 - 1) = 3 ; (2^(2-1))(2^2 - 1) = 2 x 3 = 6, which is a perfect number.
if n = 3, (2^3 - 1) = 7 ; (2^(3-1))(2^3 - 1) = 4 x 7 = 28, which is a perfect number.
if n = 5, (2^5 - 1) = 31; (2^(5-1))(2^2 - 1) = 16 x 31 = 496, which is a perfect number.
So, if you are looking for a pattern in these numbers, how about looking into their binary representations, which again follow from the magnitude formula? This is what we get:
6 = 110, 2 1's followed by 1 0's
28 = 11100, 3 1's followed by 2 0's
496 = 111110000, 5 1's followed by 4 0's
8128 = 1111111000000, 7 1's followed by 6 0's
Not bad, is it?
- Quantum Teleporter.
The Communication Barrier
This is something that I have been thinking about for quite some time now. In this post I shall pen down some of the experiences on communication problems of Indians in a foreign land, mostly United States. But I also have one interesting incident from UK as well.
I have been in the United States for just over a year now. Prior to that I was working in Cadence Design Systems, India Center. It is an American company and the work culture is very American (That is what I was told. I haven't worked in a commercial organization here). Over the period that I worked there I became aware with some of the American terms e.g.
"My two cents on this"
"I really appreciate it"
"Take it easy"
"Hi, how's it hanging?"
Moreover, American movies are very popular in India. Therefore, I never thought that i would have problems communicating with people here. Even then, it took me some time to get comfortable here. Very often I used to have the "Microsoft Internet Explorer Not Responding" look on my face.
The first experience happened even before I set foot on the United States soil. When my flight was close to Chicago airport, the pilots announced "...we'll land in twenty past the hour". I had two threads - logical reasoning ("what could this mean?") and the situational facts ("what time is it now?") - run in my brain to resolve what seemed like a profound conundrum. It took me that much effort to understand that he was indicating that he meant "twenty minutes past the hour of the day then, which was eleven" - 11:20am!
A week later, I went down to the post office to send a package to a friend of mine in Chicago. It was after I spoiled a countless number of envelopes that I finally got the position of the "To Address" and the "Return Address" correctly on the envelope. Finally, when I asked him "How much?", he said "Dollar six". I was not too surprised because I had the notion that except for electronic goods, everything was overpriced in this country. So, six bucks might have been normal for sending a packet to Chicago. It was only after I read the cash register display that I realized I was wrong. It meant $1.06, i.e. "one dollar and six cents".
OK, this one is certainly one of my favorites. This happened with a former apartment-mate of mine. It was during his first days here and he went to McDonalds and ordered a burger. The person at the counter asked him, "Here or To go?". My friend did not understand what it meant. However, instead of clarifying it, he tried to pretend as if he knew what to say and answered "whatever"!! The person at the counter gave him this completely baffled look and elaborated on the question to which he finally replied that he wanted to eat his burger at the restaurant.
I think Subway is certainly one of the places which makes any international student really nervous. They ask you so many questions whose answers are not even boolean (Boolean answer is YES/NO or TRUE/FALSE). So, new international students mess up things very often at Subway. Sample this account. This may have been over-exaggerated but as they say, "There's no smoke without fire". Therefore, I think that there's some amount of authenticity associated to this. So, this guy - first time at subway, is asking the person at the counter for the vegetables that he wants in his sub. He goes, "Lettyoos"(Lettuce), "Tomaetow"(Tomato)...(with an Indian accent)... "Al Pacino"(Jalapeno). The background to this is you don't get "jalapeno" peppers in India. And, therefore, the first few times you listen to the word you have like 5-6 different pronunciations of the same word going on in your head. And, give us a break, "Al Pacino" is very well-known in India.
Finally, this one takes the cake, I guess. This happened to a friend of mine in Sheffield University in UK. Let me set up the context by saying that most of the Indian students in graduate programs in US or Europe enroll in Engineering programs. The economy back in India is flourishing with the market developing for Engineering jobs. Of course, Sheffield is a multidisciplinary university with departments in Arts and Sciences, Engineering and even Business. Therefore, it is fair for anybody to assume that a student there is for any of these disciplines, right? Getting back to the funny incident, one evening my friend was partying out with his friends till late and when they were coming back they passed a bunch of hookers who were just looking for clients at that hour. So, one of them asked one of these guys, "You here for business, love?". And that guy nervously answered, "No, I am here for Engineering". Anybody's got anything to beat this?
If anyone of you has funny accounts of communication disasters, share them with us. I'd really appreciate it.
-Quantum Teleporter
Engineering Design of the Ladies' Purse
I hope that you agree with me about the latest structure of a ladies purse these days:
Since I am a college student and I see most of the girls, on and off campus, carrying purses like these, it is fair for me to assume that this is the current fashion in the entire nation among the college-goers, isn't it? Maybe not. Florida is a warm place. So, further up north the fashion might be different. Anyway, we shall then restrict ourselves to discussion of fashion in Florida.
The salient features in its design, barring the cosmetic looks are:
a. A small pouch made out of leather or fabric, mostly.
b. A tiny strap attached to the two ends of the pouch.
Please do not be too critical about this analysis of mine. Of course, the purses will have other designs too. But I'm just setting up a context here for what you're going to read next. And ladies, nothing personal! HONEST!
Now, why am I talking about the design of the ladies' purse? That is because the engineering seems a bit awkward. When you hang a purse like that on your shoulder the pouch will rest just below your armpit between your upper arm and rib-cage. My fellow male readers, it is fair to assume that most of you haven't experienced how it is like to hold a purse like that and walk around, isn't it? But let me help you with that! Picture this:
You buy a lot of grocery from a store and when you're in front of your apartment, you are tempted to move the whole grocery from the boot of your car into your house in one trip. What do you do? You hold some stuff between your arms and your torso and pick up the rest with your hands and just try to maximize the number of packets that you can hold maintaining the condition that you can still move. Then you somehow scramble to reach your apartment. But would you not avoid it if you could?
Ladies, I am not being a sexist here. I am asking the guys to do this just so that they can imagine how differently things work for girls than the guys. Oh, I am being a sexist, ain't I? No no no...I am just kiddin'!
Now, can you imagine running with the purse held like that? You definitely cannot keep up with the natural movement of the arms while running. When you run, you move your arms alternately back and forth with respect to your torso. And that movement is in coordination with the movement of your legs as well (I shall not get into the the physics of this). And a situation where you'd want to run is very normal in a university setting. What if you're late for a class and you want to be there on time? Well, I have observed that in USA, getting late for class is not a big deal. So, I might be asked to justify with a stronger reason. Well, how about this - you're walking towards the shuttle stop and you see the shuttle just leave the stop? And since you're approaching it from behind you cannot even beckon the driver to spot you and stop the shuttle for you! You have to run and you have to run well!
However, there is a possibility that all this is logical. A woman's torso is more curvacious than a man's. So, there's a possibility that there is enough empty space between the arm and the ribcage for girls to carry their purses like that and still be comfortable! Hmmm...what about it?
What if it is not? Should the girls stop carrying purses like that? Hell, NO!! I think that it makes the girls look very attractive! In fact, what is the logic behind high-heeled shoes? Just that it makes the women look very very attractive (and tall too)! So, do not stop carrying purses like that! Long live fashion promoters!
You see?! I just spent 45 minutes of my time to write up something that I do not even want anything done about!
- Quantum Teleporter
The shape is that of a truncated icosahedron which is the 32-faced Archimedian solid corresponding to the facial arrangement of 20 hexagons and 12 pentagons and has 60 vertices and 90 edges[1].
This figure [4] above shows one of the shapes that you would obtain if you ripped open you soccer ball and placed it on the ground. Note the uniformity in the topology of the polygons. There are 10 columns of vertically stacked regular polygons each with 2 hexagons and 1 pentagon except for column 2 & 3 (counting from left to right) which have an extra pentagon each. But if you disregard those two pentagons, the stacking arrangement of the polygons in a column is the vertical inverse of the stacking arrangements of the polygons in the adjoining columns.
