### Geeky Fun at Christmas

*Science News / The Mathematical Lives Of Plants*

The seeds of a sunflower, the spines of a cactus, and the bracts of a pine cone all grow in whirling spiral patterns. Remarkable for their complexity and beauty, they also show consistent mathematical patterns that scientists have been striving to understand.

A surprising number of plants have spiral patterns in which each leaf, seed, or other structure follows the next at a particular angle called the golden angle. The golden angle is about 137.5. Two radii of a circle C form the golden angle if they divide the circle into two areas A and B so that A/B = B/C.

The golden angle is closely related to the golden ratio, which the ancient Greeks studied extensively and some have believed to have divine, aesthetic or mystical properties.

Plants with spiral patterns related to the golden angle also display another curious mathematical property. The seeds of a flower head form interlocking spirals in both clockwise and counterclockwise directions. The number of clockwise spirals differs from the number of counterclockwise spirals, and these two numbers are called the plant's parastichy numbers (pronounced pi-RAS-tik-ee or PEHR-us-tik-ee).

These numbers have a remarkable consistency. They are almost always two consecutive Fibonacci numbers, which are another one of nature's mathematical favorites. The Fibonacci numbers form the sequence 1, 1, 2, 3, 5, 8, 13, 21 . . . , in which each number is the sum of the previous two.

*The Fibonacci numbers tend to crop up wherever the golden ratio appears, because the ratio between two consecutive Fibonacci numbers happens to be close to the golden ratio. The larger the two Fibonacci numbers, the closer their ratio to the golden ratio. But this relationship doesn't fully explain why parastichy numbers end up being consecutive Fibonacci numbers.
Scientists have puzzled over this pattern of plant growth for hundreds of years....*

As you struggle to engage in the tedium of Christmas behold the sprout stalk or the festive pine cones and observe and wonder, let others debate Vince, Dave and Nick, let your mind be on higher things.

## Comments

Fascinating.

Closely related to growing and packing on spherical surfaces (or at least, curved surfaces), and the resulting need for maximum efficiency in the use of space. I suspect.

Isn't evolution wonderful?

Incidentally, it wasn't just the Greeks who were intrigued by golden means, golden angles, golden ratios, etc. The geometry and layouts of great medieval buildings (such as Chartres Cathedral) are also full of such numbers.

Have a good Christmas, y'all!

Posted by: Andrew Duffin | December 23, 2010 3:02 PM

Love it - you're sitting there, contemplating spirals.

Posted by: jameshigham | December 23, 2010 3:39 PM

The Magical Mystical Maze of Mathematics holds wonders beyond compare for the mind willing to learn.

Posted by: Michael | December 23, 2010 6:23 PM

That, in the limit, the ratio of two consecutive Fibonacci numbers is the Golden Ratio is a consequence of Binet's formula, which is a closed-form solution to finding the nth Fibonacci number. Binet's formula falls out as a necessary consequence of diagonalising the Fibonacci generating matrix {{1,1},{1,0}} by eigenvector decomposition. One of the neatest little proofs I know, and one coincidentally on my whiteboard right now from where I was proving it last week to my cleaning lady's daughter, who is studying maths and computer science.

Here's another fun one: add squares of Fibonacci numbers. What do you get?

1 + 1 = 2 = 1 x 2

1 + 1 + 4 = 6 = 2 x 3

1 + 1 + 4 + 9 = 15 = 3 x 5

1 + 1 + 4 + 9 + 25 = 40 = 5 x 8

etc.. There is a very simple geometric proof of this.

Fibonaccis are fun.

Posted by: David Gillies | December 23, 2010 7:46 PM

To go off on a slight tangent (ahem), have you seen this video of multiplication Japanese style?

Posted by: nbc | December 23, 2010 11:17 PM