# Na­ture by num­bers How the Fi­bonacci se­quence of num­bers oc­curs naturally in many plants and can be used in de­sign

## The Fi­bonacci se­quence is a math­e­mat­i­cal pat­tern that seems to crop up fre­quently in na­ture, as well as in ar­chi­tec­ture and gar­den de­sign. TIM EN­TWISLE stud­ies the fig­ures

Gardening Australia - - CONTENTS -

What’s the connection between swirling spi­rals of seed in a sun­flower head, the off-kil­ter sym­me­try of pine cones, and our fond­ness for odd-num­bered plant­ings? Maybe none, as it turns out, but we can track our ob­ses­sion to find­ing one back to at least the 13th cen­tury, to ac­coun­tant and math­e­ma­ti­cian Leonardo of Pisa.

Fi­bonacci (as he was nick­named a few cen­turies later) loved a good co­nun­drum, in­clud­ing the num­ber of rab­bits you might ex­pect from a sin­gle breed­ing pair. As we know in Aus­tralia, it doesn’t take long for a few bun­nies to be­come a plague, but our friend from Pisa was in­ter­ested in the de­tail of what hap­pens next.

By Fi­bonacci’s reck­on­ing, the first cou­ple of rab­bits will beget two more pairs, fol­lowed by 3, 5, 8, 13, 21 and so on. The ‘so on’ is the sum of the two pre­vi­ous num­bers: the se­ries starts with one breed­ing pair so the next num­ber is also one (zero + one), then two (one + one), and we are off!

These are called Fi­bonacci num­bers, and they seem to pop up all the time in na­ture, ar­chi­tec­ture and art. As does the golden ra­tio, also known as the di­vine pro­por­tion or phi, which is what you get when you di­vide a Fi­bonacci num­ber by the one be­fore it (see over­page).

Look at the seeds in a sun­flower head or the scales on a pine cone and you will see they are ar­ranged in rows that spi­ral out from the mid­dle or apex. There is a clock­wise spi­ral and an an­ti­clock­wise spi­ral. If you count the num­ber of spi­rals in each di­rec­tion it is of­ten a Fi­bonacci num­ber. The num­ber of spi­rals in the op­po­site di­rec­tion is usu­ally the Fi­bonacci num­ber be­fore or af­ter the other one.

You can also find Fi­bonacci num­bers, or rough ap­prox­i­ma­tions to them, in a rosette suc­cu­lent such as an agave, aloe or echev­e­ria, or in a head of Ro­manesco broc­coli. If you count the num­ber of spi­rals, more of­ten than not it will be a num­ber from that se­ries.

The rea­son for the pat­tern is rather pro­saic. The most com­pet­i­tive and ef­fi­cient place for a new leaf is to be off­set by 137.5 or 222.5 de­grees to the last leaf, rather than di­rectly ad­ja­cent or op­po­site. This re­sults in as many leaves as pos­si­ble ac­cess­ing full sun­light. Over time, this off­set cre­ates spi­rals of leaves, and in the most ef­fi­cient pack­ing there is of­ten a Fi­bonacci num­ber of them.

In ar­chi­tec­ture, the de­sign of the Acrop­o­lis of Athens or the Parthenon in Rome are some­times said to be in­debted to the golden ra­tio. In the case of the Acrop­o­lis, the di­men­sions don’t quite match up and I gather there is no ev­i­dence the orig­i­nal de­sign took that pro­por­tion into ac­count. Whereas Re­nais­sance build­ings, such as the

Lau­ren­tian Li­brary in Florence and the Notre-Dame de Paris, do have de­sign el­e­ments based on the golden ra­tio, as have some Re­nais­sance gar­dens.

In de­sign gen­er­ally, an odd num­ber of some­thing is con­sid­ered more at­trac­tive than an even num­ber if we want a re­laxed and in­for­mal feel. In small gar­dens, this means we tend to use Fi­bonacci num­bers like three and five when plant­ing groups of trees. How­ever, two and eight are in the se­quence and a cop­pice of two or eight trees is un­likely to form a pleas­ing out­come. On the other hand, a mix of shrubs that reach 8m mixed with some that reach 5m – both Fi­bonacci num­bers – is a plan­ning tool used by some land­sca­pers.

The golden ra­tio is some­times used in gar­den­ing as well. A rec­tan­gu­lar bed with pro­por­tions ap­proach­ing 1:1.6, say 5m wide and 8m long, will look ‘about right’ to most peo­ple. Of course, if you have a long, nar­row gar­den or a curvy space you’ll have to ei­ther break it up into lit­tle golden ra­tio units or cre­ate some­thing with a dif­fer­ent aes­thetic.

The Fi­bonacci num­ber and the golden ra­tio may well pro­vide for an in­trin­si­cally beau­ti­ful ob­ject or col­lec­tion of ob­jects, but in most cases other prac­ti­cal­i­ties and other equally at­trac­tive pro­por­tions pre­vail. In na­ture, Leonardo of Pisa’s cal­cu­la­tions didn’t ac­tu­ally work in the rab­bit bur­row – he failed to al­low for more than two off­spring for a start – and they are only a rough ap­prox­i­ma­tion in flower and plant geom­e­try. Still, the math­e­mat­ics re­mains a beau­ti­ful thing.