When I was growing up, television in my adoptive country of Switzerland did not run 24 hours a day, airing only test patterns for long stretches at a time. Making up for lost opportunity, visits to my American grandparents thus featured orgies of Saturday morning cartoons.
And so I would return to Switzerland with a trunkful of Looney Tunes memories, central among them Elmer Fudd’s admonition: “Be vewy vewy quiet.” Today, fellow sky walkers, like Elmer Fudd before us, we’re hunting wabbits, or more precisely constellation Lepus, the celestial hare.
A bunny-eared constellation
Lepus is perhaps the most endearing constellation visible in northern latitudes. Appropriately enough for a timid herbivore, Lepus hides most of the year, showing his twinkling bunny ears only in midwinter. To find Lepus in the sky, wait until Orion is in full stride, around 8 p.m. or so. Lepus lies below Orion’s legs, hiding as best he can from the celestial hunter and his great dog, Canis Major (see the sky chart).
You will need a moonless sky to see the comparatively dim stars of this celestial wabbit. Look for a trapezoidal body, with leaping legs swept back. Last to be visible are the tips of his ears, which twinkle into view as your eyes adjust to darkness. Once the whole Lepus emerges, you will see that like its terrestrial counterparts, this wabbit was hiding in plain sight the whole time.
Counting wabbits
Lepus is one of my favorite sights in the sky and reward enough on its own. I would be remiss, though, not to take this opportunity to bring up another favorite bit of scientific wabbit lore, the Fibonacci numbers.
In the West, the sequence of numbers that begins 1, 1, 2, 3, 5, 8, 13, 21, … was first enunciated by the mathematician known to us as Fibonacci (the sequence was also known in India). Here’s how it works. The first two numbers in the sequence are both the numeral 1. To get the next number at any point in the sequence, simply add the two prior numbers. Since the first and second numbers are both 1, the third number is 1+1=2, giving us the partial sequence 1, 1, 2, … The next number (the fourth) is the sum of the second and third, that is 1+2=3, giving us the sequence 1, 1, 2, 3, … Two additions later, we have 1, 1, 2, 3, 5, 8, … And two more additions beyond that, we have 1, 1, 2, 3, 5, 8, 13, 21… We quickly reach larger and larger numbers.
Here comes the math. Don’t panic.
In mathematical notation, we might write fn to indicate the nth number in the Fibonacci sequence, so f1 stands for the first number, f2 stands for the second and so forth. The whole sequence is compactly written as follows, which just restates what we saw above: the first two numbers are both 1, and to get the third number and beyond, simply add the prior two.
f1 = 1
f2 = 1
fn = fn-1 + fn-2
Fibonacci numbers seen as a spiral.
Fellow sky walkers, you may ask just what this has to do with wabbits. As it happens, Fibonacci discovered this sequence while trying to estimate the population growth of bunnies.
To wit: Suppose we place a pair of baby bunnies in a field, one male and one female. Assume there are no predators and no death by old age. Bunnies are the Olympic gold medalists of procreation, an accomplishment that Fibonacci modeled thus: at age one month, a pair of bunnies is ready to have offspring, whereupon they immediately mate and become pregnant. One month later (the actual gestation period of rabbits), they give birth to exactly two offspring, one male and one female, then immediately mate again. The cycle continues, with all offspring pairs joining the reproductive bandwagon after they reach one month in age.
It’s a bit of an idealization, but here’s how it all works out, one month at a time. We will count bunny pairs, writing the number of pairs on month n as bn.
b1 = 1 (the original pair)
b2 = 1 (the original pair, now pregnant)
b3 = 1 + 1 = 2 (originals plus offsprings)
That is, on month 3, three, we have the original bunny pair plus a new pair of offspring. Note that the original bunnies immediately become pregnant again, so that one month later we have
b4 = 2 + 1 = 3
That is, on the fourth month, we now have the two pairs of bunnies from the third month, (the “2”) plus a new pair produced by the original bunnies (the “1”). At this point, both the original bunnies and their first offspring are eligible to mate, since these first offspring are now one month old. They all obligingly do so, and on month five we therefore have
b5 = 3 + 2 = 5
In general, Fibonacci observed, for any given month n, the number of bunny pairs will be exactly the number of pairs alive on the prior month (month n-1) plus new offspring pairs for each pair alive on the month prior to that (month n-2) – this latter number factors in the one-month latency required for baby bunnies to reach procreation age. Writing this mathematically:
bn = bn-1 + bn-2
This is exactly the same formula as above, just rewriting the f terms as b terms! If you’ve gotten this far, fellow sky walkers, you are now in for a treat.
Spirals, spirals everywhere
The Fibonacci sequence can be recast geometrically in terms of square figures. The first and second numerals are squares of size 1, the third numeral a square of size 2, and so forth. Place each sequential square to the right of the prior square, then rotate the resulting figure 90 degrees clockwise, and continue placing and rotating. If you trace a quarter circle from corner to corner in the squares, you obtain a spiral.
This shape is ubiquitous in nature. The nautilus shell is the most common example of the Fibonacci spiral, but it also describes the rows of seeds in a sunflower, the unfolding of ferns, and many more cases that are nicely documented in the Wikipedia entries for “Fibonacci numbers” and “golden spiral.”
And to bring it all home, consider this: The spiral induced by the Fibonacci sequence is an excellent model for the arms of spiral galaxies. The universality of this simple mathematical relation is breathtaking. From lowly ferns to galactic structures, the universe keeps showing us the number sequence that we know as Fibonacci’s.
Fellow sky walkers, may your days ahead feature wabbit-rich night skies. May your stay under the stars afford you time to stop and experience the wonder of our mysteriously mathematical universe, and give you the chance, as Elmer Fudd would say, to be vewy vewy quiet.
Hubble image of spiral galaxy M101, in Ursa Major, the Pinwheel Galaxy. (Photo courtesy European Space Agency and NASA)
Marc Vilain spends his midwinter nights hunting celestial wabbits, and marvels how one year on, Fibonacci’s field has 233 bunny pairs—and 75,025 pairs the year after that.
