Monday Morning Math: the Winter Solstice

December 12 was going to be the final Monday Morning Math of the semester, but finals had started two days earlier and with one thing and another (well, really just one thing – the aforementioned final exams), it didn’t happen.  

It felt a bit odd to take a break without announcement, however, so here is one final MMM for 2022.

(Arriving on a Monday at least, even though it’s not quite morning anymore.) And the timing is perhaps good for a math-adjacent topic: the Winter Solstice, which happened on Wednesday, December 21.  This is the shortest day of the year in the northern hemisphere, with just under 9 hours of daylight (technically 8 hours, 59 minutes, and 10 seconds) here in Rochester.  But there are two things about the solstice that I find interesting mathematically.

The first is that if you google “When is the Winter Solstice?” you get not just a day but a time: 4:47pm here.  This feels a little weird to me if I think about it being a day, but it has to do with northern hemisphere being tilted as far away as possible from the rays of the sun, as in this tweet from NASA below:

Or, if you want to envision the Earth with its axis vertical, it’s moving along a plane that is not horizontal, as explained on NASA’s blog: 

This is the image that makes the idea of a moment for the solstice make sense to me: it’s at the very peak of the ellipse, and that happens at one particular moment rather than a full day.

The second math adjacent thing is about subtraction.  You might think that the shortest day must have the latest sunrise and the earliest sunset, but actually neither is true: the lastest sunrise doesn’t happen until January 3, 2023, at 7:42am, which is about 3 minutes later than it rose on Dec 21.  So that’s kind of a bummer, in terms of how dark the mornings are.  But that’s compensated by the fact that the earliest sunset happened several weeks ago, back on December 9.  The sun set at 4:35pm that day, about 3 minutes earlier than it set on the solstice.  The sunrises and sunsets don’t quite change symmetrically, and that’s why the shortest day is about halfway in between. 

Happy Solstice and Happy Holidays to everyone! We’ll start again in about a month – see y’all in 2023!

Monday Morning Math: Whose sequence?

Today’s post is a follow-up to last week’s about Leonardo Pisano, also known as Fibonacci.  And it is specifically about the sequence that bears his name: 0, 1, 1, 2, 3, 5, 8, … where each number is the sum of the two before it.  This appears as the answer to a puzzles that he proposed in his book Liber Abaci:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

The sequence was given the name “the Fibonacci Sequence” by Edouard Lucas (1842-1891), a French mathematician for whom a similar sequence: 1, 3, 4, 7, 11, … is named.  Blog commenter S shared that leading up to that time “it was rarely known and when used was called by many names including “the sequence of Lamé” (as Gabriel Lamé had used the sequence for analysis of the Euclidean algorithm)” (Thanks S!)  

I recently got the book Finding Fibonacci by Keith Devlin, and while I haven’t read the whole book yet (something for the break!), he does talk briefly about the history of these numbers, which poetically were used in prosody (the study of rhythm and sound/stress in poetry):

They first appeared, it seems, in the Chandahshastra (The Art of Prosody) written by the Sanskrit grammarian Pingala sometime between 450 and 200 BCE.  Prosody was important in ancient Indian ritual.  In the sixth century, the Indian mathematician Virahanka showed how the sequence arises in the analysis of meters with long and short syllables.  Subsequently, the Jain philosopher Hemachandra (ca. 1150) composed a text about them.

S, too, linked to some photos of the sequence appearing in these sources.  I am wondering if the long and short analysis is similar to some problems we’ve given in our classes, which essentially boil down to the number of ways that you can add to 1, 2, 3, 4, 5, 6, etc. using just the numbers 1 and/or 2, where order matters: for 1 there is only 1, for 2 it can be 2 or 1+1, for 3 it can be 2+1, 1+2, or 1+1+1, and for 4 it can be 2+2, 2+1+1, 1+2+1, 1+1+2, or 1+1+1+1.

The idea that the sequence comes up in poetry makes me happy.  The idea of continuing to call it the Fibonacci sequence doesn’t, however, since despite what may have been Lucas’s good intentions, it ignores these earlier contributions.  I’ve seen reference to calling it the Fibonacci-Hemachandra numbers, but even that would bypass Pingala and Virahanka, as well as some other mathematicians, as noted in the abstract of “The so-called fibonacci numbers in ancient and medieval India” by Parmanand Singh

What are generally referred to as the Fibonacci numbers and the method for their formation were given by Virahṅka (between a.d. 600 and 800), Gopla (prior to a.d. 1135) and Hemacandra (c. a.d. 1150), all prior to L. Fibonacci (c. a.d. 1202). Nryana Paita (a.d. 1356) established a relation between his smasika-paṅkti, which contains Fibonacci numbers as a particular case, and “the multinomial coefficients.”

I will admit to still feeling like I don’t have a handle on the history of the sequence – just enough to be uncertain how to call it.  I’ll close with a photo of a bust of Acharya Hemachandra at Hemchandracharya North Gujarat University (CC license)