Phyllotaxis – patterns of leaf growth

The coolest thing I have read all year explains patterns of leaf growth. I had not heard of Fibonacci numbers, or the Fibonacci sequence, or at least I don’t remember reading about it. The subject came up in an interesting book my brother sent me about mental math after watching our older son do some figuring at the dinner table during a recent family reunion. Sam has been manipulating numbers in his head for as long as I can remember, largely because we were always outside or on a trip without other distracting devices to keep him busy. We talked, and computed, to pass the time. Mailboxes demonstrated number sequence and place value, dividing up snacks allowed us to compute fractions and serving sizes. So discovering the math underlying the leaf patterns was just eye-popping for me this morning – it’s another way to talk about the symmetry found in nature, and the principles which explain that symmetry mathematically.

The view of the sky from our tent in the woods.

The Fibonacci sequence is the sum of the preceding two numbers. Thus, 0, 1, 1, 2, 3, 5, 8, 13 and so on. If you Google the Fibonacci sequence and click on “images” you’ll see the perfect spiral and other patterns. What really blew me away, though, is that leaf patterns integrate this sequence! There are very good explanations of this, such as this lesson on botany. I will just summarize here, but I encourage you to dig deeper.

Fibonacci sequence and leaf growth patterns
Take a look at the bushes and trees in your yard. Count the number of leaves it takes to complete one rotation around the stem, so that the last leaf is stacked vertically right over the first leaf. Recall how many times you had to circle the stem to do so. This is hard to do because stems twist and turn, obscuring the count. But generally speaking, leaf patterns adhere to the Fibonacci sequence – the number of leaves to complete a rotation is the denominator and the number of times we circle the stem is the numerator.

For example, if two leaves arise from opposite sides of the stem, creating a vertical stack of leaves, each one completes 1/2 rotation, or 180 degrees separation. The most common pattern occurs in cherry, poplar and oak – we count 5 leaves to reach one standing over the first (a full rotation). However, we have to circle the stem twice to do so. This means each leaf occupies twice the amount of space in the rotation. The denominator is 5 (# leaves), and the numerator is 2 (1/5 + 1/5, # times to circle the stem). The pattern continues – beech and hazel are 1/3, oak and apricot are 2/5, sunflowers, poplar, and pear are 3/8, and in willow and almond the angle is 5/13.[From Coxeter H.S.M., Introduction to Geometry, 1961] Da Vinci was the first to suggest this principle evolved to maximize exposure to sunlight and dew. Modern botanists tend to agree, however you will find some debate continues.

What angle do we have to each other – how much of the circle do we each represent? [Answer: 1/3 angle of rotation]

I am thrilled to learn about this because I had been wondering just how long I could keep bringing math into the outdoors – could it keep up with my kids’ growing understanding of mathematical principles? Or would I have to resort to technology and other more “gimmicky” endeavors to keep them engaged? Yes, I enjoy geocaching and explaining how satellites work, but it is somehow less gratifying than teaching strictly from the natural world around us, completely unplugged. I may need to start bringing a small white board out on the trail, though 🙂

Fractions in the woods! WOW!

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