# A Mathematical New Year

## Katie Steckles

Amidst all the usual New Year festivities, this year you may have felt slightly more mathematical than usual, but not been able to quite put your finger on why. That’s because this year, we celebrated a Fibonacci New Year, marked on the date 1/1/23: a once-in-a-lifetime date coincidence linked to a particularly interesting and famous sequence of numbers.

**Fibon-who?**

Leonardo Fibonacci, also known as Leonardo of Pisa, was an Italian mathematician born circa 1170 CE, who – like many historical mathematicians – is primarily remembered for one book he wrote. In Fibonacci’s case, the book was called “Liber Abaci” – literally, “The Book of the Abacus” – although it was actually presented as an alternative to the then-common use of abaci for calculation. Published in 1202, this was the first European work covering Indian and Arabian mathematics, and introduced the idea of Hindu-Arabic numerals – the standard digits 0-9 with a decimal system we use today – to Europe for the first time.

Fibonacci (Image Credits: PD-OLD, Wikimedia Commons)

As well as the use of these strange new numerals, Fibonacci’s book contained some important ideas from number theory, a selection of calculations for commerce like currency conversions and interest calculations, as well as some work on approximations of irrational numbers. It also included a selection of mathematical problems, and one of them has been instrumental in securing Fibonacci permanent fame and glory:

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?

While this might seem like a simple numerical puzzle, this is based on a piece of mathematics known in India as early as 400-200 BCE and links to a particularly elegant sequence of numbers, which has become known as the Fibonacci sequence. (If you’d like more time to think about the puzzle, do it now before I explain the answer).

The single pair of rabbits will obviously beget (details omitted) another pair in the first month, and in the second month while there are technically two pairs of rabbits, only one pair is of breeding age, so the second month only one new pair of rabbits is begotten. The following month, we’ll get two new pairs of rabbits (one from our original pair, and one from their first set of progeny) and then the month after that there’ll be three rabbit pairs in breeding mode.

Laying all of this out clearly, and considering only the number of new pairs produced each month (not counting the original pair of rabbits), we get the following sequence:

1, 1, 2, 3, 5, 8, 13, 21, 35, …

If we examine the terms of this sequence, we can see that each term is, intriguingly, the sum of the two terms before it (assuming an imaginary zero term at the start so that the second term can be generated).

With the rabbits problem, it can be a little fiddly to see how the sequence comes out – you need to add the requirement that rabbits can’t reproduce the month after they’re born – but once you’re happy with that, it makes sense that the number of new rabbits generated depends on the number of breeding pairs, which would be related to the number of rabbits who were alive two months earlier, and the ones who were alive and reproduced the previous month.

The Indian mathematicians who originally came up with the idea weren’t even thinking about this in terms of rabbits – they derived the idea from the Sanskrit poetic tradition, in which poetry consists of long (L) syllables of length 2 beats, and short (S) syllables of length 1 beat. Scholars were deeply interested in listing all the possible patterns of syllables – for example, in a short poem four beats long, I could have SSSS, SSL, SLS, LSS or LL. This is a total of five possible patterns, and it turns out that for a poem of N beats long, the number of possible rhythms is the N+1-th number in the Fibonacci sequence.

**Fibon-everywhere**

Seeing the same set of numbers pop out from two completely different approaches shouldn’t surprise you; I’ve written enough columns here about how mathematical ideas can connect across approaches and applications and coincide unexpectedly when you abstract an idea down to its mathematical roots. These numbers also have plenty of other interesting properties, of which I’ll try to summarise a few:

- The problem of arranging syllables of length 1 & 2 beats in a poem of length N beats, known as a composition problem, has plenty of other applications: I’ve also seen it used to count the number of ways to climb a staircase by 1s and 2s; it turns out it’s also the same as the number of different ways to tile a 2 by N rectangle with 2 by 1 dominoes.

- The number of ways to write a string of binary digits of length N without having more than one 1 in a consecutive run is the N+2th Fibonacci number. Such numbers are called Fibbinary numbers, and have plenty of interesting properties. There are various other problems in this area, concerning strings of binary digits with certain constraints, many of which turn out to be connected to the Fibonacci numbers.

- The ratio of successive pairs of terms in the sequence (1, ½, ⅔, ⅗, …) get closer and closer to the Golden ratio – a value of \(\frac{\sqrt{5}+1}{2}\) considered to be a beautiful ratio, and discussed in this post I wrote for my Golden Ratio wedding anniversary.

- Every positive whole number can be written as a sum of Fibonacci numbers, using each number at most once. This means the Fibonacci numbers are an example of a complete sequence – the other main example being the powers of two, used in binary to make each integer using 0 or 1 of each. (It’s also true of the prime numbers, if you add in 1!)

- And finally, you might also recall the Fibonacci numbers crop up in Pascal’s triangle – mathematical structures not named after the person who actually first discovered them gotta stick together, I guess? I discussed this in fact 5 of my ‘Twelve Facts of Christmas’ back in 2018.

I hope I’ve sufficiently convinced you that this New Year – the Fibo-New Year to some – was one of the most auspicious occasions of recent times, assuming you write the date in the format 1/1/23 (which, luckily, most people actually do, at least for that one day). Hopefully, you found a mathematically appropriate way to celebrate, like a Sanskrit poetry reading, or just running and jumping up the stairs!

The post A Mathematical New Year originally appeared on the HLFF SciLogs blog.