Saturday, August 25, 2012

GKC, the Beer Can Formula, the 13, and related matters

Yes, though it must sound hilarious to some, I really am a Catholic, a computer scientist, a Chestertonian - and a writer. All interwoven, intertwined, cross-linked and so forth, as befits a Medieval Man, who tries to look at the world as a unified Good... But this is not going to be an autobiographical post - I am (alas) comparatively BORING, as there are plenty of other INTERESTING things to ponder.

Like Beer Cans... and the formula which governs towers built with them.

And so, in my attempt to resolve an extremely detailed interrelation regarding the Bad Guys from the Syllo Republic in my Saga, I was forced to consider the topic of beer cans, and the very famous equation which tells you how many beer cans will be in a complete triangle given the number across the bottom - which will also be the number of rows.

The equation is the famous "Sum of the integers from one to a given number N", which can easily be derived as:

SumOfIntegers(N) = N * (N+1)/2

There is a famous funny story told of the young Gauss relating to this, but I will tell it another time. There is also the observation - made by one of the Church Fathers - that the number 153 (the fishes caught near the end of St. John's gospel) is the sum of the integers from one to seventeen:

153 = 1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17


17*(17+1)/2 = 17*18/2 = 17*9 = 153

Well, that's very nice. Nice equation. Easy to derive, easy to prove correct - by induction, which is how we prove so many things in computing.

But what does THAT have to do with those bad guys from the Syllo Republic?

Well... the one branch of the Bad Guys is called "The Thirteen" - and I began to wonder whether they had thirteen groups, each of thirteen... I won't explain their inner organization - it's quite secret, and even speaking as the author I have not yet been enlightened about it. However, I wanted to know whether A Certain Number (thirteen squared, or 169) could be arranged as a triangle, in the style of beer cans, and if so, what that triangle would be.

Hence, I had to work out the INVERSE of that famous Beer Can Formula... That is, I wanted the equation for InverseOfSum(Z) = N such that SumOfIntegers(N) = Z.

I did it - it was a lot of fun to do. I won't give the steps, but I will state the result:

InverseOfSum(Z) = (sqrt(8*Z+1)–1)/2


InverseOfSum(169) = 18.39157 (approximately), so we have

SumOfIntegers(17) = 153 <= 169 <= SumOfIntegers(18) = 171

Thus the triangle formed from 169 beer cans will have a base of 17 and contain 153 cans (a biblical number, you know?) with 16 left over. Alternatively, we could consider there being a group of 171 (a triangle of 18) which gives 13 groups of 13, and then have two supernumeraries... all sorts of possible uses for them... but herewe note that this is a literary plot issue, and not a mathematical matter.

Very nice. The mathematician bows politely and heads for the nearest supply of cold beers, intent on implementing his theory... (hee hee)

But what does all that have to do with Chesterton?

Well, it would be very interesting to discuss the matter of Story, or of antagonists in Boys' Books, or the relation (so very Newman, and so very Medieval) between the so-called Liberal Arts like writing and the so-called Hard Sciences like math and computer science... but I prefer to give you a curious excerpt and let you ponder it. I was well familiar with GKC's famous reference to the "Binomial Theorem" in relation to "talking about God" - but I found there was another place where he mentions it, somewhat less well known - so I have chosen that. Ihope you will think about it:
The brain is simply an object we perceive when we happen, every now and then, to split open the skull of some social acquaintance with a chopper. It is as much an object in the landscape, so to speak, as a blue lamp-post or a green tree. By inference or analogy we argue that there is something of the sort in our own skulls also. But the brain and the blue lamp-post are still merely two of the ten thousand things we see and experience as objects. The mind is not one of those things. The mind is an absolute; the mind is the thing that sees them. All those objects in the landscape can only exist, as we know them, in a field of consciousness called the mind; and it is tenable even that they do not exist, or do ot exist as we know them. But the mind exists; and we have not, in that sense, the same certainty that anything else exists. Now there is not, and cannot be, any bridge of imagination between the brain and the mind. We cannot form any conceivable notion of how the grey cells we find in Mr. Smith's skull (when we split it open) can possibly be a field of consciousness in which there are trees and lamp-posts. As an idea, that identification is much harder to entertain than the mystery of the Trinity or the Dual Nature of Christ. Minute changes of grey matter cannot be the Binomial Theorem or the memory of last Wednesday. The best that can be said for it is that it is a mystery; and the only thing to be said for the materialist is that he is a mystic. [GKC ILN July 7 1928 CW34:553]
In case you didn't know, the above "Beer Can" equation forms one of the terms in the Binomial Theorem - which is one of the reasons why I picked this excerpt.

Next time maybe I will tell you more about the Saga... and the other ways computer science has played a major part in its writing. Yes, QUITE Medieval - and quite suitable for Getting Things Done.

And if you are still wondering what any of it had to do with Catholicism, just note the proximity of words like "Trinity" or "Dual Nature of Christ" fall with respect to "Binomial Theorem"... it may be only poetry to note that we ARE talking about a TRIANGLE... be it made of beer cans, or wicked Syllanese...

For in talking about stories, and enemies, and equations - yes, and beer cans - "Whether you are talking about pigs or the Binomial Theorem, you are still talking about Him." [GKC, DN quoted in Maycock]

* * *

Oh gosh, I forgot to point something out, lest you think I am a triskaidekaphobe. (Hold on, it'll be a shock.)

The GOOD GUYS in my Saga also have thirteen in their group.

But then, it's always funny to think about this number 13 in light of the "Acts of the Apostles"... since (1) after they selected Matthias (at Pope Peter's orders) to replace the Traitor, and (2) after Saul converted, there were Thirteen apostles.

(Yes, I know there's one or two others considered as such, like Barnabas, and even Mary Magdalen, the apostola apostolorum... but ALL of us are SENT... and our mission can even be in writing, or proving theorems - or stacking beer cans.)


At 26 August, 2012 11:46, Anonymous some guy on the street said...

It's one of those funny things that N and (N+1) simply cannot share ANY prime factors; so the only way for half their product to be a square is for one to be a square and the other twice a square; and this means solving the equation M² = 2N²±1; I think Descartes stole someone's solution of this puzzle, using the curious fact that (√2 + 1)(√2 - 1) = 1. Now, 1=2×1-1, one has (1,1,-) as a solution, and the purloined method then suggests (1+1+1,1+1,+) as the next solution: 9=2×4+1, one has (3,2,+) as a solution indeed, and then (3+2+2,3+2,-)=(7,5,-)... BUT! The tower for, e.g., the (7,5,-) solution is built in 49 rows, and contains 49 * 50/2 = 1225 = 35² cans. Ah, well.


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