Christmas, Hanukkah, and Fibonacci

January 4, 2012

Happy New Year everyone!  As usual I’m way overdue adding to this blog, and intend to write a proper post shortly on one of the subjects that have recently piqued my interest.  However, in the meantime, I can’t resist quickly pointing out something which came to my notice today, courtesy of my daily email from arXiv; a quirky little article that neatly ties together 3 of the things I have written about recently: calendars, religion, and number sequences (in particular, Fibonacci numbers).

The paper in question is a very short one, attributed to one Shalosh B. Ekhad.  A quick search reveals that this “person” is in fact a computer belonging to a mathematician called Doron Zeilberger, who is quite well known in mathematical circles for his love of computers, and for not being entirely serious all of the time.  However, I will humour him by writing as if it was indeed the entity named Shalosh B. Ekhad who wrote this article.

So, Ekhad became interested in how often the Jewish holiday of Hanukkah coincides with Christmas, and began to run computer searches (which I suppose equates to just thinking about it, if you are actually a computer).    For the non-Jews reading this (I had to look it up myself) Hanukkah is an 8-day holiday which begins on the 25th day of the month of Kislev in the Hebrew Calendar.   The Hebrew calendar is an example of a lunisolar system, in that it takes into account the relative motions of both the sun and the moon. Each year consists of 12 lunar months of either 29 or 30 days, apart from leap years which have an extra month.   Leap years occur 7 times every 19 years; if we think of the first year of a 19-year cycle as being year 1, then the leap years are years 3, 6, 8, 11, 14, 17, and 19.

As you might expect, this means that dates of the Hebrew calendar vary quite wildly in relation to those of our strictly solar Gregorian calendar.  In particular it means that the pattern of years in which Christmas falls within the Hanukkah period is highly unpredictable.  What Ekhad found was that it will happen in 27% of the years of this (3rd) millenium, with this figure falling for subsequent milleniums, until the 9th millenium, when it will stop happening altogether until at least the year 20000 AD.  So far, so mildly diverting.  Much more interesting is the following observation: during the period when the gaps between  between the years 1801 and 7390, that is, in the time-span in which the gaps between years in which Christmas falls within Hanukkah are relatively small, the number of years making up these gaps are always Fibonacci numbers!  (I talked about Fibonacci numbers in this post).  In particular they are always either 2, 3, 5 or 8.  Ekhad then goes on to point out that exactly the same phenomenon occurs for years in which Christmas falls within Sukkot, another Jewish holiday lasting 7 days.

This seems quite incredible…perhaps slightly less so for those who are used to Fibonacci numbers cropping up in the most unexpected places, but it cries out for explanation all the same.  So what is going on here?  Well, the regular occurrence of gaps of 2 and 3 years between these special Christmas-in-Hanukkah years surely has something to do with the fact that the number of non-leap years between leap years in the Hebrew calendar is always either 2 or 3.  The sequence of gaps between leap years (3,2,3,3,3,2,3) would also go some way to explaining the occurrence of 5 and 8 as well, as both of these numbers can be made from sums of consecutive numbers from this sequence.  But then so can 6,9,10,13….

So there’s more to it than that.  Any insights?


Mathematicians and religion

September 26, 2011

Where was I?  Well, last week* we established, among other oddities, that diclofenac is bad for the religion of Zoroastrianism.  That post didn’t really have anything to do with mathematics (although I did at least attempt to tenuously link it chaos theory), so I will make up for it by at least mentioning some mathematicians this week, if not actual mathematics.  However, I will stick with the topic of religion for the time being.

This is partly inspired by a book I’ve just read: Galileo’s Daughter, by Dava Sobel.  It doesn’t really match up to Longitude, but is a good read nonetheless.  It is really about the life and work of Galileo Galilei, although Sobel gives us the hard science and history in a more easily digestible form, by interweaving commentary on his relationship with his daughter.  She seems to have been a quite extraordinary woman: sent to a convent at age thirteen due to her illegitimacy (and hence lack of marriage prospects), she spent her whole life in extreme poverty within those walls,  but still managed to be a doctor, playwright, composer, musician and prolific correspondent in the little time she had which wasn’t dedicated to prayer, labour and general suffering.

Anyway, one of the things which struck me most about Galileo’s life was his relationship with the all-powerful Catholic church at this time.  He was a very devout Catholic: publicly, of course (claiming Catholicism is, after all, preferable to torture and painful death), but more surprisingly, given the utter ignorance and persecution he suffered at the hands of the Inquisition, he remained privately devoted to the church.  He even said, near the end of his life:

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How conkers created Israel, cats cause wars, and painkillers are bad for religion

November 22, 2010

I’ve recently been reading a book called Chaos, by James Gleick.  It is a nice, easy-to-read overview of chaos theory in all its forms.  Chaos theory is not really a proper mathematical field, more of an ideology, which has applications in all walks of life.   The phrase seems to be bandied about less these days, perhaps because the ideas have become so accepted that it is no longer considered a theory, but just “how things are”.  It takes the form of turbulence, entropy and unpredictability; it has great influence on the weather, the traffic, the stock markets…indeed it is hard to imagine how science worked before the notion of chaos.  As one physicist in the book puts it:

“Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurement process; and chaos eliminates the Laplacian fantasy of deterministic predicability”.

Poor physicists! Always having their work eliminated by something or other.  Luckily this doesn’t happen in mathematics.  Chaos in mathematics is studied in the form of dynamical systems,  in which small perturbations in initial conditions can have a dramatic long-term effect.  This sensitivity is known in popular culture as the “butterfly effect”, from a paper by Edward Lorenz – a pioneer of chaos theory – titled: Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?

Lorenz was  a meteorologist, and first noticed chaotic effects whilst running weather simulations.  Weather is notoriously chaotic (see, for example, long-term forecasts by the Met Office for evidence of this), and one day, whilst trying to restart a simulation where he had left off, he fed in data which had been output from the middle of a previous session.  He noticed that the outcome was wildly different from his previous results, a consequence of the computer having rounded his output to what he had thought was an insignificantly fewer number of decimal points.    Gleick goes into weather patterns in some depth, as well as delving into such interesting topics as the fractal – and by implication, infinite – nature of coastlines (the closer you get the more little “bays” there are), and the chaotic behaviour a human heart displays while fibrillating (basically what a defibrillator does is to reset a chaotic system with a massive jolt of electricity).*

But I am not actually going to talk about chaos theory today.  Well, not quite.  Instead I am going to share a few odd and interesting freakonomics-style chains of events I’ve learnt about recently.  They all involve seemingly insignificant things – conkers, diclofenac and a cat parasite, to be precise  – which have (arguably) had a huge impact on world events.   In that sense you could possibly claim that this was some kind of chaos in practice.  But that would be quite a tenuous way to try and link it with what I’ve written so far, so I won’t.

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