The Fibonacci Series in Plants

Source: Nelson, Dawn. “The Fibonacci Series in Plants.” Sussex Botanical Recording Society Newsletter, no. 58 (May 2004). http://sussexflora.org.uk/wp-content/uploads/2016/03/Newsletter_May_2004.pdf.

 

(Members who attended Rod’s ‘Local Change’ meeting near West Stoke in the summer were surprised when Dawn suddenly picked a plant from the verge & used it to demonstrate the Fibonacci series. We were fascinated and so, for those who were not at the meeting, Dawn has given us a written version)

 

Many formations in nature, although constituted and caused by different phenomena, are not only similar to look at, but have identical mathematical descriptions.

The Fibonacci series is a number series which has fascinated mathematicians, artists and mystics since its first introduction by Leonardo Fibonacci, who brought it to Europe in 1228, having studied mathematics with the Arabs. It begins with any number, say 1. This and the previous number 0 are added together so that their sum forms the third number in the series 1. Then the process is repeated, thus 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 etc. What is fascinating is its relationship with the ratio of the Golden Mean 1:1.618. If one divides a number of the Fibonacci series by the preceding number (this can be any combination) one gets nearly 1.618 but never exactly – larger numbers get closer.

The Fibonacci spiral is generated by reducing a golden mean rectangle to a square on its smaller side, leaving each time another golden mean rectangle, of which the larger side is now equal to the previous short side. (see fig.1)

Spiral patterns appear in all forms of nature, but are especially noticeable in plants and flowers. Think of the centre of a hybrid tea rose or a dahlia. Look at the formation of the florets or seeds of a sunflower. In fact, if you look carefully at these you find that there are two sets of spirals, one left-handed and one right-handed. Each set is made up of a predetermined number of spirals, overlapping each other to form the most intricate pattern. Most Asteraceae have 21 and 34 sets. Pine-cone scales have 5 one way and 8 the other. Pineapple bumps have one set of 8 and one of 13. The ratios of these spirals correspond to two adjacent Fibonacci numbers 5:8, 8:13, 21:34, and the same is true of many other plants with a spiralling leaf growth pattern.

I have been around the garden to see what examples of these spirals I could find. First was Euphorbia myrsinites – this wonderful succulent produces great spiralling snakes of scaly glaucous leaves. The patterns are most obvious in rosette-forming plants – all the small Sedums and Saxifrages form spiralling rosettes of leaves, as do the many varieties of House Leek Sempervivum. The way the leaves of Teasel Dipsacus fullonum curl up as they dry off in winter reflects the way fern fronds emerge tightly curled in the spring. Think also of the way Prickly Sow-thistle Sonchus asper leaves form a spiral where they clasp the stem. This spiral pattern is also apparent in fruits, such as some Medicago and Erodium species.

Flowers exhibit this form too. Look at the way Changing Forget-me-not Myosotis discolor opens from its tightly curled spiral of buds. And have you ever watched the petals of Evening-primrose Oenothera biennis jerking open in the evening from their spirally rolled buds. Another interesting fact is that the petals on daisies and roses are usually a Fibonacci number (5: Rosa, Malus; 8: Dryas octopetala, Geum rivale). The more numerous-petalled flowers with 34 or 55 are not always exact but, if you count several of them, the average will always be a Fibonacci number.

Almost all plants with alternate leaves describe a spiral. The amount of turning from one leaf to the next is a fraction of a complete rotation of the stem. It is nature’s way of allowing as much light as possible to reach each leaf, but the resulting rotation is always a Fibonacci fraction: 1/2, 2/3, 3/5, 5/8 etc. In the example shown, (see figs.2&3) there are five complete turns with eight spaces between leaves 1 – 9, so the ratio of the spiral is 5:8.

Flowers in racemes or lateral clusters often spiral in this fashion too, as can be seen in Genista flowers (Stace). Allegedly 90% of flower structures visibly relate to the Fibonacci series.

 

Illustrations from Ardalan & Bakhtiar (1973). The Sense of Unity. The University of Chicago Press.