PI day 2009

And here it is again. The celebration of that beautiful number, the mathematical uniqueness that occurs in our life repeatedly, whether you realize it or not. This is PI day: Mar 14 (3.14).

Since I’ve been doing this the past few years, I’m not going to repeat all the fun facts about PI. But I will nevertheless give you some more interesting facts and links I found recently.

1) The Pi day Sudoku puzzle.

2) Things that equal Pi. Btw, the 360 blog has some really interesting posts and you should definitely subscribe to it.

3) Oh and of course, the recent appreciation of the reverent number even by politicians. Check this out.

4) And if you want to buy some swag for your PI day party, I suggest getting these beautiful PI shaped ice maker.

5) Update: Just found out that old Al’ was born on PI day. How fitting ?! Thanks to Neatorama for the piece of trivia.

Hmm.. Another year. Another day. Time is flying I tell you …

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Some pun with math

An infinite number of mathematicians walk into a bar. The first one orders a beer. The second orders half a beer. The third, a quarter of a beer. The bartender says “You’re all idiots”, and pours two beers.

A nice unfolding of the following infinite series expansion:

Infinite series

This is a very simple result. Can’t see it ? Work it out.

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PI day again

Whatdya know ?! Its PI day again. I remember posting on this day, last year about the same event and now, here we are again ! And in case you didn’t know, today is also the birthday of ‘Dr. Einstein’ of the E=mc2 fame 😉
Here’s a tribute to this magic number, π:

History:

  • Biblical References: I Kings 7:23 II Chronicles 4:2
    In Kings, it states, “And he made a molten sea, ten cubits from one brim to the other: it was round all about, and a line of thirty cubits did compass it about.”
  • In 240 B.C, Archimedes of Syracuse, Sicily (287 – 212 BC) did the first theoretical calculation of . He used methods similar to the ones used by Euclid by inscribing a regular polygon inside a circle. He started with a hexagon and then polygons of 12, 24, 48, and finally 96 sides. He also used one of Euclid’s theorems to develop a numerical method for calculating the perimeter of the polygons. Archimedes obtained the approximation 223/71 < π < 22/7.
  • 150 A.D. Ptolemy found π to be approximately 377/120 (or 3.1416)
  • 480 A.D. In China, pi was found to be approximately equal to 355/113 or 3.1415929 …
  • 1150 Bhaskara (a Hindu) gave 3927/1250 as an accurate value of π
  • 1579 Viete used polygons having 393,216 sides to evaluate π correct to 9 places
  • 1610 Van Ceulen used 2^62 sides to compute π to 35 decimal places
  • 1949 ENIAC (first modern computer) spent 70 hours to compute π to 2,037 places
  • In September 2002, π was computed to 1,240,000,000,000 decimal places by Professor Yasumasa Kanada at the University of Tokyo. It took over 400 hours on a Hitachi Supercomputer.


Facts and interesting stuff:

  • All the digits of Pi can never be fully known.
  • William Jones, a self-taught English mathematician born in Wales, is the one who selected the Greek letter π for the ratio of a circle’s circumference to its diameter in 1706.
  • Thirty divided by ten gives a value of 3. However, it is interesting to note that the word circumference happens to be spelled with an extra letter. Since in Hebrew all letters are also numbers, if we take the ratio of the value for the word as it is written (111) to the normal spelled word (106) we get the number 1.047169811… If you multiply this number by 3 you get 3.141509434… An amazingly close approximation to π!
  • The 1983 Guinness Book of World Records lists Rajan Mahadevan from India as having recited 31,811 places of pi from memory !
  • PI poem by Lorreen Pelletier: The number of letters in each line corresponds to a digit in the number &#960, up to 35 decimal places.
  • The value of π can be computed using the fibonacci sequence ! Link.
  • How do you prove that the π exists ?? Here’s an interesting theory.
  • Can you decipher the digits of π using a Sanskrit sloka ? Here’s a demonstration. Interesting read !
  • Here’s π to 1000 digits:
    3. 14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196 44288 10975 66593 34461 28475 64823 37867 83165 27120 19091 45648 56692 34603 48610 45432 66482 13393 60726 02491 41273 72458 70066 06315 58817 48815 20920 96282 92540 91715 36436 78925 90360 01133 05305 48820 46652 13841 46951 94151 16094 33057 27036 57595 91953 09218 61173 81932 61179 31051 18548 07446 23799 62749 56735 18857 52724 89122 79381 83011 94912 98336 73362 44065 66430 86021 39494 63952 24737 19070 21798 60943 70277 05392 17176 29317 67523 84674 81846 76694 05132 00056 81271 45263 56082 77857 71342 75778 96091 73637 17872 14684 40901 22495 34301 46549 58537 10507 92279 68925 89235 42019 95611 21290 21960 86403 44181 59813 62977 47713 09960 51870 72113 49999 99837 29780 49951 05973 17328 16096 31859 50244 59455 34690 83026 42522 30825 33446 85035 26193 11881 71010 00313 78387 52886 58753 32083 81420 61717 76691 47303 59825 34904 28755 46873 11595 62863 88235 37875 93751 95778 18577 80532 17122 68066 13001 92787 66111 95909 21642 01989 …
  • Here’s a palindrome for you : “I prefer PI” !!
  • e raised to the i*pi power equals -1 (e is the base of the natural logarithm and i is the imaginary number which is the sqare root of -1)


Alternate π addendum:

Book:
The life of PI – Here PI is an Indian guy’s name who gets stranded in the sea for more than 250 days. Its a good read although it has nothing to do with the π we are dealing with here. Just thought that might be an interesting trivia !
Movie:
PI – The movie starts with the line “When I was a little kid, my mother told me not to stare into the sun, so when I was six I did…”. Now with a line like that, how could i not watch it ! I’d recommend this movie to anyone who’s a little perceptive and frankly, a bit obsessed on math or anything for that matter. I watched the movie and loved it but few of my friends hated me for recommending the movie. So, there you go. But seriously, if you get some time, and are a math fan, watch it !

Links:

Happy PI day

Yup. I didn’t make this up. Mathematicians in many places are celebrating the PI day.

Notice the time and the date of the post and see if you get it ?!

Well if you don’t, not a problem. Here’s a brief explanation …

The number PI to first 6 digits is given as


which in terms of time (in a very crude way) would be March 14, 1.59.20 PM. So in memory of this beautiful number, i dedicate this post to PI.

And here is another trivia about PI which most people would not know.


Alright then. Off to do some more reading now …

Logic and math riddles

Today on Slashdot, a message board challenge to assemble a catalog of favority math and logic riddles. Well not all of them are that great but they have some really interesting puzzles among the lot.
There is a king and there are his n prisoners. The king has a dungeon in his castle that is shaped like a circle, and has n cell doors around the perimeter, each leading to a separate, utterly sound proof room. When within the cells, the prisoners have absolutely no means of communicating with each other.

The king sits in his central room and the n prisoners are all locked in their sound proof cells. In the king’s central chamber is a table with a single chalice sitting atop it. Now, the king opens up a door to one of the prisoners’ rooms and lets him into the room, but always only one prisoner at a time! So he lets in just one of the prisoners, any one he chooses, and then asks him a question, “Since I first locked you and the other prisoners into your rooms, have all of you been in this room yet?” The prisoner only has two possible answers. “Yes,” or, “I’m not sure.” If any prisoner answers “yes” but is wrong, they all will be beheaded. If a prisoner answers “yes,” however, and is correct, all prisoners are granted full pardons and freed. After being asked that question and answering, the prisoner is then given an opportunity to turn the chalice upside down or right side up. If when he enters the room it is right side up, he can choose to leave it right side up or to turn it upside down, it’s his choice. The same thing goes for if it is upside down when he enters the room. He can either choose to turn it upright or to leave it upside down. After the prisoner manipulates the chalice (or not, by his choice), he is sent back to his own cell and securely locked in.

The king will call the prisoners in any order he pleases, and he can call and recall each prisoner as many times as he wants, as many times in a row as he wants. The only rule the king has to obey is that eventually he has to call every prisoner in an arbitrary number of times. So maybe he will call the first prisoner in a million times before ever calling in the second prisoner twice, we just don’t know. But eventually we may be certain that each prisoner will be called in ten times, or twenty times, or any number you choose.

Here’s one last monkey wrench to toss in the gears, though. The king is allowed to manipulate the cup himself, k times, out of the view of any of the prisoners. That means the king may turn an upright cup upside down or vice versa up to k times, as he chooses, without the prisoners knowing about it. This does not mean the king must manipulate the cup any number of times at all, only that he may.

via Slashdot and Boing Boing

Also, found this great resource of riddles over at UCB’s site from the slashdot post. Definitely worth checking out if you have an hour or two to spare on some good grey cell petrifying puzzles.

Update : While we are at solving puzzles, here is one more awesome question that i had worked on, a long time back. Dig this.

You have a port that you are reading numbers from. You know that there is one number that is generated in more than half of the cases. You keep reading numbers arbitrarily long until you are given a command to stop. When you stop you have to return the number that has occurred in more than half of the cases.

(Hint: you don’t have enough memory to store all the numbers)

Here’s the actual link.

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4D Cube

A very interesting, perspective invoking picture of a 4D cube. Well ironically, its name is still a Cube in 4D.

Here’s a 4-D visualization of the cube in a raytraced Povray version of the picture.

A 4D visualization of a cube[/caption]” />

Check out this site for more interesting trivia, facts and pictures related to math !

Sines And Cosines are history !

I am not drunk and babbling gibberish. This is news. One fundamental theory which aims to make the use of trigonometry easier and more accurate. Proposed by Dr Norman Wildberger, a professor at University of New South Wales, this theory replaces angles to which we are so much used to by now, with a concept called as ‘spread’.

Here’s an excerpt from an article about this theory.

Established by the ancient Greeks and Romans, trigonometry is used in surveying, navigation, engineering, construction and the sciences to calculate the relationships between the sides and vertices of triangles.

“Generations of students have struggled with classical trigonometry because the framework is wrong,” says Wildberger, whose book is titled Divine Proportions: Rational Trigonometry to Universal Geometry (Wild Egg books).

Dr Wildberger has replaced traditional ideas of angles and distance with new concepts called “spread” and “quadrance”.

These new concepts mean that trigonometric problems can be done with algebra,” says Wildberger, an associate professor of mathematics at UNSW.

He has also written a book called ‘The Divine Proportions : Rational Trigonometry to Universal Geometry‘ by N J Wildberger. There is a chapter available for preview.

On first look, the concepts are straightforward in a logical sense. But i do not see how it simplifies and eliminates the calculations that are presently being done with sines and cosines. Well that’s just me and i could be wrong ! Maybe this is a revolutionary theory that is going to change how we look at things in the future.