Twig and Leaf Phyllotaxy

by Carol Levine • Fall 2000 (Volume 28, nos. 2 & 3)

Leonardo da Vinci once wrote "The leaf always turns its upper side towards the sky so that it may be better to receive the dew over its whole surface; and leaves are arranged on the plants in such a way that one covers another as little as possible. This alternation provides open spaces through which the sun and air may penetrate. The arrangement is such that drops from the first leaf fall on the fourth leaf in some cases and on the sixth in others." This observation call attention to the fascinating, naturally occurring mathematical phenomena: leaf phyllotaxy and the Fibonacci numbers.

Leaf phyllotaxy explains how leaves are arranged on the twig, why they are opposite, whorled or alternate: 

Opposite leaves have the leaves opposite in respect to each other, and usually the adjacent tiers cross at right angles to the original opposite leaves. This is described as opposite-decussate, or pairs alternately crossing at right angles. Examples are maples, ash, horse-chestnut and most viburnums.

Whorled means three or more leaves at one node. A good example is the catalpa.

Alternate leaves are distributed along the twig in a spiral. If we draw a line from the point of attachment of one leaf to the next, this line will wind around the twig as it rises. The same species will always bear the same number of leaves for each turn around the twig. An equal portion of the circumference of the stem will always separate the successive leaves from each other. Thus is created an equiangular spiral.

The alternate arrangement of leaves in each species is also explained in terms of ranking. 

Two-ranked describes those species that have leaves that alternate 180 degrees on the two sides of the twig. This means that the third leaf is found directly over the first leaf and the fourth leaf is on the other side of the twig, directly above the second leaf. (See illus. right.) If a line is drawn spirally from the first leaf through the second leaf to the third leaf, the line will have circled the twig once. Two-ranked is described in the fraction 1/2. The numerator is the number of revolutions (1) around the twig and the denominator (2) means that two leaves were encountered in this one spiral of 360 degrees (not counting the first leaf). Examples are the grasses, sycamore, birch, elm and linden.

Three-ranked is shown as 1/3. (See illus. left.) Examples are the sedges, false hellebore and sometimes beech. 

Five-ranked is shown as 2/5. This arrangement is the most common among the hardwoods as oaks, cherry trees, tulip trees, walnuts, hickories, sweet gum, and others. Here we have two circles around the twig encountering five leaves. (See illus. right.)

Eight-ranked is 3/8. Examples are holly, bayberry and sweet fern.

Thirteen-ranked is 5/13. Examples are willows and almond.

These ratios (1/2, 1/3, 2/5, 3/8, 5/13, 8/21...) are the angular spiral divergence of one leaf from another. Upon analyzing these figures, one realizes that each number of the numerator and each number of the denominator is the sum of the two preceding numbers. In other words, 2+3=5, 3+5=8, 8+13=21 and so on. A man called Fibonacci for whom they have since been named first analyzed this series of numbers in the 13th century. Although no one has explained their significance, mathematicians have chased these numbers through art, architecture, botany, astronomy and music. It was discovered that each number bears a special relationship to the numbers surrounding it. If you divide a Fibonacci number by the next highest number it will be 0.618034 times as large as the number that follows. The Greeks called this ratio the "golden mean." This golden proportion of 0.618034 to 1 is the mathematical basis for the shape of the Parthenon, playing cards, sunflowers, pine cones, snail shells, breaking waves, the comet's tail and the spiral galaxies of outer space.

Johann Kepler, the astronomer, said, "Geometry has two great treasures; one is the theorem of Pythagoras; the other the division of a line into extreme and mean ratio (the golden mean). The first we may compare to a measure of gold, the second we may name a precious jewel."

In the 17th century Jakob Bernoulli suspected that the golden mean was connected to the spirals in nature. He found that any section an equiangular spiral is 0.618034 as large as the remainder of the spiral. He named it the logarithmic spiral and noted that any line drawn from the center of the spiral will intersect it at precisely the same angle as any other line. For this reason it is known as an equiangular spiral. Bernoulli ordered the spiral engraved on his tombstone.

Botany is a gold mine of spirals and Fibonacci numbers. The arrangement of seeds in the sunflower (or in other Asteraceae for that matter), the scaly plates in the pineapple and the bracts in pinecones are other examples of Fibonacci numbers.

None of these examples is perfect in the field. But it seems to be a fascinating natural tendency that appears far too often to be discounted as mere chance.

We can conclude that phyllotaxy is another means by which to identify woody plants in the field, especially in winter. The leaves may be gone, but the leaf scars and buds remain. Knowing the phyllotaxy of woody plants adds to the fun of field trips in the winter.




  • Cook, Theodore A., The Curves of Life, Dover Pub.
  • Hoffer, William, "A Magic Ratio Recurs Throughout Art and Nature," Smithsonian Magazine, (Dec. 1975, p. 111).
  • Oster, Gerald, "The Spiral Way," Natural History Magazine, (Sept. 1974).
  • Stevens, Peter S., Patterns in Nature," Little, Brown & Co.
  • Thompson, D'Arcy, On Growth and Form, Cambridge Univ. Press.