W 4th St Station, NYC
I had a lovely day with dear friends Marlowe, Yasmina, and their little ones, Elia and Aviva. Marlowe encouraged Laurie and me to go to Babbo for dinner. I had the mushroom sformato, squab, and panna cotta. It was delightfully delicious. The spirits of New York and Europe were alive and well in us as we left the restaurant. We were heading to Times Square to have a brief encounter with bright lights, big city, tourists, and to leave a little math. We made our way to the W 4th Street subway station.
Heading down to the F train, I entered a cavernous hall with a floor of 2 x 2 foot concrete squares. I couldn’t resist. With panna cotta in my belly and the F train in my future, Fibonacci was looking to be conjured.
Once upon a time there was a man named Leonardo of Pisa who was born in the late 12th century in what is now Italy. This Leonardo, though Italian, was not a painter.1 He was a traveler and a mathematician. Most people know him as Fibonacci. You may have heard of the Fibonacci numbers and the connections to the golden ratio, golden rectangle, and golden spiral.
If you haven’t, you will soon. Just keep reading. There is a lot to say about Fibonacci, his namesake sequence of numbers, the goldens, and his gifts to “western” mathematics.2
To get those nice little Fibonacci numbers, start by adding 0 and 1 to get the next number, 1, add 1 and 1 to get the next number, 2, add 1 and 2 to get the next number, 3, add 2 and 3 to get 5, add 3 and 5 to get 8, and so on.
0, 1, 1, 2, 3, 5, 8, 13, …
As the Fibonacci numbers grow, the ratio of one of them to its smaller next-door neighbor heads towards a number known as the golden ratio, Φ ≈ 1.618.
1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 ≈ 1.666, 8/5 = 1.6, 13/8 = 1.625, … , 144/89 ≈ 1.618, …
Just as the ratios approach the golden ratio, the golden rectangle begins to take shape when squares with sides of Fibonacci number length are put together just so. Start with two 1×1 squares, lay a 2×2 square next to the pair, put a 3×3 square next to those, and keep going with subsequent squares. The more Fibonacci squares you include, the closer the outermost rectangle gets to the golden rectangle.
Now, what is really cool is to embed little quarter circles inside each of the Fibonacci squares. Guess what. As you include more and more arcs, the large spiral gets closer and closer to achieving golden status. (Forever failing to do so, but forever willing to approach.)
I love the look and feel of the Fibonacci spiral.
Fibonacci left zero out of his book, and I left it out of my spiral in the W 4th Street subway station in New York City. Poor zero.
The photographer for this post is Laurie Zimmerman Mann. Check out her site: www.lzmstudio.com.
1 It is said another Italian, also named Leonardo, the one from Vinci, employed the Golden Ratio and the Golden Rectangle in an attempt to capture objective aesthetic beauty in his Vitruvian Man.
2 Fibonacci brought to “western” mathematics when his book Liber abaci was published in the early 1200s. Fibonacci returned from his travels in the east with remarkably powerful tools for mathematics including the Hindu-Arabic use of place value and the Arabic numerals themselves. Being able to write “94” where the “9” is interpreted as “9 tens” and the “4” is “4 ones” is a huge step forward from writing “94” like the Romans had done, XCIV. Where the “X” is in front of the “C” to note that 10 should be removed from one hundred to take us to 90 before we slap on 1 less than five (known to you and me as four.) “94” is way better, right? If you disagree, than check out 1998 in Roman numerals: MCMXVCVII, not to mention what algebra equations used to look like. (Check out MacTutor Math Archives for more.)