Tag: xkcd

Fun With Permutations and Asimov’s Three Laws of Robotics

I’m not a big fan of science fiction, but I know enough to know that Isaac Asimov was one of the great science fiction novelists of the 20th century. The following was written by him in the October 1980 issue of The Magazine of Fantasy and Science Fiction and was reprinted in his book Counting the Eons, which was published in 1983. (I’m now holding the battered and torn pages of my copy of this book; I devoured Asimov’s musings on mathematics and science when I was young.)

Robotics has become a sufficiently well development technology to warrant articles and books on its history and I have watched this in amazement, and in some disbelief, because I invented it.

No, not the technology, the word.

In October 1941, I wrote a robot story entitled “Runaround,” first published in the March 1942 issue of Astounding Science Fiction, in which I recited, for the first time, my Three Laws of Robotics. Here they are:

  1. A robot must not injure a human being or, through inaction, allow a human being to come to harm.
  2. A robot must obey the orders give it by human beings except where those orders would conflict with the First Law.
  3. A robot must protect its own existence, except where such protection would conflict with the First or Second Laws.

Clearly, the order in which the Three Laws of Robotics matters. Shuffling the order leads to 3! = 6 possible permutations, and xkcd recently had some fun about what the consequences would be of those permutations.

Source: http://www.xkcd.com/1613/

Was There a Pi Day on 3/14/1592?

March 14, 2015 has been labeled the Pi Day of the Century because of the way this day is abbreviated, at least in America: 3/14/15.

I was recently asked an interesting question: did any of our ancestors observe Pi Day about 400 years ago on 3/14/1592? The answer is, I highly doubt it.

My first thought was that \pi may not have been known to that many decimal places in 1592. However, a quick check on Wikipedia (see also here), as well as the book “\pi Unleashed,” verifies that my initial thought was wrong. In China, 7 places of accuracy were obtained by the 5th century. By the 14th century, \pi was known to 13 decimal places in India. In the 15th century, \pi was calculated to 16 decimal places in Persia.

It’s highly doubtful that the mathematicians in these ancient cultures actually talked to each other, given the state of global communications at the time. Furthermore, I don’t think any of these cultures used either the Julian calendar or the Gregorian calendar (which is in near universal use today) in 1592. (An historical sidebar: the Gregorian calendar was first introduced in 1582, but different countries adopted it in different years or even centuries. America and England, for example, did not make the switch until the 18th century.) So in China, India, and Persia, there would have been nothing particularly special about the day that Europeans called March 14, 1592.

However, in Europe (specifically, France), Francois Viete derived an infinite product for \pi and obtained the first 10 digits of \pi. According to Wikipedia, Viete obtained the first 9 digits in 1579, and so Pi Day hypothetically could have been observed in 1592. (Although \pi Unleashed says this happened in 1593, or one year too late).

There’s a second problem: the way that dates are numerically abbreviated. For example, in England, this Saturday is abbreviated as 14/3/15, which doesn’t lend itself to Pi Day. (Unfortunately, since April has only 30 days, there’s no 31/4/15 for England to mark Pi Day.) See also xkcd’s take on this. So numerologically minded people of the 16th century may not have considered anything special about March 14, 1592.

The biggest obstacle, however, may be the historical fact that the ratio of a circle’s circumference and diameter wasn’t called \pi until the 18th century. Therefore, both serious and recreational mathematicians would not have called any day Pi Day in 1592.

Circumference

Source: http://www.xkcd.com/1184/

Further comments, from Nicholas Vanserg, “Mathmanship,” The American Scientist, Vol. 46, No. 3 (1958):

In an article published a few years ago, the writer intimated with befitting subtlety that since most concepts of science are relatively simple (once you understand them), any ambitious scientist must, in self-preservation, prevent his colleagues from discovering that his ideas are simple too…

The object of… Mathmanship is to place unsuspected obstacles in the way of the pursuer until he is obliged, by a series of delays and frustrations, to give up the chase and concede his mental inferiority to the author…

[U]se a superscript as a key to a real footnote. The knowledge seeker reads that S is -36.7^{14} calories and thinks, “Gee what a whale of a lot of calories,” until he reads to the bottom of the page, finds footnote 14 and says, “oh.”