# Cycad Cone Mathematics

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## Contents |

### Introduction:

The large and compact cones or generative bodies of many
species of *Cycas* are ideal materials to observe arcs
and spiral patterns on them. These patterns are formed from
the way the individual bracts (sporophylls or carpellary
leaves) are arranged. In some slender cones, one may notice 3
spirals veering to the left and 5 spirals moving oppositely,
or vice versa. In larger ones, 5 and 8 spirals can be noticed,
and some still larger cones may display 8 and 13 spirals:
each set moving opposite to the other. Giant cones may have
as many as 21 spirals as the one seen in Fig. 1. These
numbers (3, 5, 8, 13 & 21) form part of a numerical sequence
known as Fibonacci Numbers which came to be known by the work
of a 13th century Italian mathematician, Leonardo da Pisa.
The sequence commences with these terms: 1, 1,2,3,5,8,
13,21,34, and so on. Except the first, any term in the
sequence is obtained by adding the two preceding numbers. The
sequence is thus non-ending.

### The Fibonacci Numbers:

Leonardo da Pisa (1175-1250) wrote a book **LiberAbaci** in 1202
which remained obscure for centuries. In the 19th century, a
great French Number Theorist, Edourd Lucas (1842-1891 ) who
discovered it, was fascinated by a problem described in the
first chapter of the book known as the famous 'breeding of
rabbits problem'. When a female-male pair of adult rabbits is
put in an enclosure to breed, and if the rabbits are assumed
to produce a female-male pair of young ones every month, and
when it is also assumed that the young ones start reproducing
at the same frequency from the end of the second month of
their birth, the number of pairs of rabbits will increase in
the following manner: 1,2, 3, 5, 8, 13, 21, 34, 55, reaching
233 pairs at the end of one year. Any term in the sequence
(excepting the first) is the sum of the two preceding terms.
Lucas was very fascinated by the breeding of rabbits problem,
as he could visualise some of their properties. He named the
sequence as Fibonacci Numbers, or the son of Bonaccio numbers
since Bonaccio was Leonardo's father(and filius or filio
means son). Such a nickname was chosen for Leonardo's numbers
because Lucas felt that he was not worthy of mentioning the
real name of such a great mathematician who has given the gem
of numbers. From the middle of the current century, the
Fibonacci concept has taken great momentum thanks to the
efforts of two great Californian mathematicians, Brother
Alfred Brousseau and Prof. Verner E. Hoggatt. In 1962,
a society, Fibonacci Association, was founded which started
publishing a journal, "The Fibonacci Quarterly" from the
following year. This periodical celebrated its silver jubilee
year in 1987.

### Some Properties of Fibonacci Numbers:

The elementary-looking numerical sequence (Fibonacci
Numbers) has profound mathematical properties as evidenced
from the innumerable articles that appear in The Fibonacci
Quarterly contributed mostly by the over two thousand members
of Fibonacci Association. However, my interest is in the
applicaiton of some of Fibonacci properties to plants and
animals. For example, the number of foliar spirals of any of
the 2,70 0 species of palms included under the family
Arecaceae, is always a Fibonacci number. *Areca catechu*
has a single spiral, *Arenga pinnata* displays two
spirals,
* Borassus flabellifer*
depicts on the stem three clear spirals,
*Cocos nucifera*
has five distinct spirals, and the leaf bases of
*Elaeis guineensis* are arranged in eight spirals. Giant
stems of
* Phoenix canariensis*
have numerous, prominent leaf scars from which thirteen
spirals can be made out. It is also surprising that there is
no palm showing 4, 6, 7, 9, 10, 11 or 12 spirals on its crown
or stem. The flower heads of daisy, pansy, marygold,
chrysanthemum, dahlias and sunflower, all belonging to the
family Compositae, show spirals or arcs in the arrangement of
individual flowers on the head. Their numbers always match
with the Fibonacci numbers. So also the peduncles of most
anthurium species, pineapple, pinecones and
*Cycas* cones display spirals matching with Fibonacci
numbers.

The reason for such an occurrence is because of a
relationship between *F. numbers* and Golden Ratio (or
Divine Proportion). The most remarkable property of Fibonacci
series is that the ratio between two consecutive numbers is
alternately greater or less than the Golden Ratio (0.618). As
the series continues, the differences become less and less
and the ratio approaches the Golden Ratio of 0.618033 9 ...
(or its reciprocal 1.618033 9 ... which is an irrational
number). Thus, when we work out the ratio between consecutive
F. numbers, the values ultimately get reduced and
reach very close to 0.61 8 as seen below.
Hereafter, the value will continue to remain much close to
the Golden Ratio. In other words, Golden Ratio is the value
of Phi (0) whose expansion is (_/5-1)/2 = 0.618033 9 ... Such a
Golden Ratio also appears when we consider the manner the
various flowers in a sunflower head, pineapple or in the
arrangement of bracts in cycas cones are arranged. If the
points of origin of any two consecutive flowers are
considered, the narrower angle between the two flowers or
bracts will be approximately 137.5° and the wider angle
(to complete a full revolution) will be 222.5°. The ratio
between 137.5 and 222.5 makes the familiar Golden Ratio of
0.618. That is the reason why the cycas cone shows spiral
patterns, and the number of spirals matches with a
F. number.

### The *Cycas* Cone and the Pineapple:

In Fig. 1 is shown the photograph of a pollen cone of
*Cycas circinalis* collected from Waltair, India. From
the compactly arranged bracts, spiral patterns become apparent.
One of them is the 8 spirals running counter-clockwise. That
is, by following this and the adioining 7 other spirals, the
entire surface of the cone can be covered. Similarly, the 13
spirals which move opposite to the 8 spirals and at a steeper
angle will cover the entire surface of the cone by moving
clockwise. Again the 21 spirals which move much steeply, veer
opposite to the 13 spirals. Thus, the rhythm of changing the
direction of alternate spirals running to the left and right
followed again by left, right etc. suggests one of the
properties of Fibonacci Numbers. That is, the shifting of the
Golden Ration between consecutive F. numbers to the
plus and minus sides of 0.61 8 (or 1.618).

In pineapples too, one can trace out 3, 5, 8, 13 or even 21
spirals according to their size, consecutive sets of spirals
running opposite to each other. It is also true that spirals
representing smaller numbers veer less steeply, but those
that represent higher and higher F. numbers move
steeper and steeper. This implies that no two oppositely
moving sets of spirals will have the same steepness. But look
at the giant pineapple dome (Fig. 2) constructed at the
Sunshine Plantation at Nambour, Queensland. The marked
spirals running opposite to each other move upward with the
same steepness which is unrealistic. When I pointed out the
difference between the inaccurate pattern on the giant dome
and that on real pineapple (Fig. 3) to some workers at the
Sunshine Plantation, they suggested me to write to their
manager, Mr. Tony Jakeman as he was not on duty that day.
When I wrote to Jakeman offering my assistance to design a
realistic and scientifically accurate dome for the proposed
new giant pineapple, he sent me on 11 February 1987 the
following reply. "The article on the Pineapple in the
**Fibonacci Quarterly** was fascinating, and I much appreciate
your offer of assistance in redesigning the shape for the new
one. Regretably the contract for the construction of the new
skin for our "Big Pineapple" was let last November and the
costs involved in changing the design now would be
prohibitive. However, if at any time in the future we should
require another Pineapple to be designed I would most
certainly contact you to discuss the matter. " Mr. Jakeman
could as well have a deeper look at the pine cone to design
an efficient Fibonacci dome.

### Contributed by:

T.A. Davis, JBS Haldane Research Centre, Nagercoil,
Tamilnadu, India (from **Palms & Cycads** No 23. Oct-Dec 1989).