We’re in Venice’s great museum, Gallerie dell’Accademia, standing in front of what might be the most famous drawing ever made – Vitruvian Man. What is it that drives the world’s keen interest in Vitruvian Man? It may have something to do with a special number that Leonardo da Vinci and Luca Pacioli called the Divine Ratio.
When we stopped last time, math teacher Luca Pacioli had told his student, Leonardo da Vinci, about a famous problem involving rabbits. Well, to you and I it was about rabbits. To Luca and Leonardo, it was about an infinite series and ultimately, that special number.
Back to the Garden
Our version of the problem, (see below), asked how many pairs of rabbits there’d be in the garden at the end of July. The answer is 13. The month-by-month sequence of numbers looks like this…
1 1 2 3 5 8 13
Does this sequence of numbers look familiar to you? In the movie, “The Da Vinci Code”, these numbers were scrawled into the floor of the Louvre near where its famous curator had been murdered. When Tom Hanks and Audrey Tautou figured out what the numbers meant, they were astonished.
Luca Pacioli didn’t make this problem up. In fact, 300 years before Luca and Leonardo, a slightly more complicated version had been offered in a math competition in Pisa. That’s right – a math competition. (Hey – there was no TV back then). A youngster by the name of Fibonacci solved it.
Fibonacci Numbers
The problem Fibonacci solved was to figure out how many pairs of rabbits there would be in “n” months, where “n” can be any number. If n = 7, like in our problem, the corresponding number is “13.” The answer for any “n” is a sequence of numbers called the Fibonacci Series. It looks like this:
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610…
There are some surprising things about these numbers, (called “Fibonacci Numbers”). For instance, if we add any two adjacent numbers together, the result is the next number in the series. Hmmm.
If we divide any number in the series by the one that precedes it, the result is always in the ballpark of 1.6. Five divided by 3 is 1.67. The further out we go, the result approaches 1.618…. The 3 dots tell us the numbers to the right of the decimal point go on forever.
But guess what? Fibonacci wasn’t the first guy come up with the ratio, 1.618…. Nope. The first person to find it was Euclid. Bet you haven’t heard that name in awhile.
Let’s skip the part where Euclid comes up with this ratio. On the other hand, maybe you’re curious. Maybe you’d like to get in touch with your inner nerd? If so, you can read about it here: Euclid and the Golden Ratio.
The Divine Ratio
Luca and Leonardo liked this special ratio so much, Luca christened it “La Divina Proportione” – The Divine Ratio. Today it’s usually called the Golden Ratio. Luca wrote a whole book about it entitled “De Divina Proportione.” Leonardo did the illustrations for it.
How cool would it be to have Leonardo da Vinci do the illustrations for your math book? There are only two original copies of De Divina Proportione remaining. One of them is in Milan.
There’s something magical, or perhaps mystical, about Fibonacci numbers and this ratio, 1.618… It shows up in all kinds of places.
Nature is full of Fibonacci numbers and the Divine Ratio, 1.618…. The number of petals on flowers, more often than not, turns out to be a Fibonacci number — 3, 5, 8, 13… Oleanders have 5 petals. Field daisies have 34. The sequence of branches on a tree or the veins in a leaf.
In classical architecture, structures based on this ratio were thought to be especially pleasing to the eye. And that would take us back to Vitruvius, eh?
There are plenty of other examples, but you know how people are… there are more bogus examples than real ones. The ratio, 1.618… , has been linked to the the great pyramids, the stock market, and you guessed it – Vitruvuan Man.
Leonardo da Vinci – Man of Mystery
Leonardo da Vinci was a man of mystery, known for embedding secrets and codes into his work. And there’s certainly something mysterious about the Divine Ratio. Leonardo studied it. There has to be a link!
Nope. No link. There is something special about Vituvian Man, but no one has ever been able to legitimately tie him to the Divine Ratio. In fact, Luca Pacioli states explicitly in his book that the proportions of Vitruvian Man aren’t based on 1.618…
Leonardo simply used the detailed description of human proportions from Vitruvius’ book on architecture and made a drawing from them. No matter. If you Google “Golden Ratio,” don’t be surprised if you see Vitruvian Man pop up.
It’s OK that Leonardo’s drawing isn’t based on the Golden Ratio. Vitruvian Man is still pretty cool.
To be continued…
Here’s the original problem:
“In January, a lady put a pair (one male and one female) of baby rabbits into an enclosed garden. It takes two months for rabbits to be, ahem, productive, and after their second month they produce another pair of rabbits. Assume that each pair of rabbits in the garden bears a new pair of rabbits every month, starting after their second month, (just like the first ones). How many pairs of rabbits were there in the garden at the end of July?”
