The Mathematics and Computer Graphics of Spirals in Plants

This is a study of the pattern known as spiral phyllotaxis-literally, "leaf-arrangement", but applied to the arrangement of seeds, florets, petals, scales, twigs and so on-and which is very widespread in plants. A number of stages in the refinement of model-making are recorded. In particular, the model illustrated in the literature on the subject indicating uniform growth is taken a step further in generality to indicate non-uniform growth. The intention has been to arrive at a wholly visual statement. Andyet the mathematical approach is integral to this end. The study of flower forms becomes a study of the relation between the circle and the golden ratio. A computerised drawing system was used to draw forms that are algebraically defined. This solved a technical problem, but also gave me a glimpse of the enormous scope opened up by this electronic tool. To draw, for example, a simple equi-angular spiral (as found in snail shells) is difficult and time-consuming by traditional means, but elementary by computer. This is a study of what I call the "green spiral", the spiral phyllotaxis-literally, "leaf arrangement", but applied to the arrangement of seeds, florets, petals, scales, twigs, and so on-and which is very widespread in plants. I began by painting an idealised daisy based on the pattern shown in Fig. 1. It is found in Islamic art and in Escher's artwork. The following are its principles of construction: On a circle of any radius twelve smaller and identical circles may be drawn such that they touch. They will bear an exact ratio [1] in size with their host circle. And then an exactly similar necklace of circles can be drawn to touch the first-inside the first or outside. Each circle nests between two circles of a neighbouring necklace and bears an exact ratio [2] in size with circles of a neighbouring necklace. This ratio of increased size remains uniform and the pattern may be extended forever outwards, or forever inwards. It is a pattern of growth. This pattern of uniform growth is that of a geometric series e.g. 1, 2, 4, 8, 16, ..., or 5, 15, 45, 135, .... In general si = ksi_l for some fixed k, giving si = s0k. The patterns found in plants, unlike the static regular lattices of crystalline structures, must exhibit members of a sequence at different stages of growth. The pattern in Fig. 1 offers a closepacking of twelve identical sequences growing uniformly. Spiral and helical phyllotaxis is an archetypal arrangement studied in the patterns of sequential branching around plant stems. Pine cones, daisies, sunflowers, pineapples and cacti are some of the clearest examples. Twigs, leaves, fruits or florets may constitute the elements of the sequence. One possible gestalt of Fig. 1 is the pattern of opposing spirals, twelve in each direction. Analogous spirals are seen in daisies (Fig. 2). However, the spirals seen in plants differ in number clockwise to anti-clockwise [3]! Indeed the numbers thus exhibited are consecutive pairs taken from the celebrated Fibonacci series [4]: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, .... The nth term of the series is the sum of the two previous terms,f =fn +f,-2; fo = 0,f = 1. The cone of a redwood presents to the eye a pattern of 3 X 5 "helico-spirals". The daisy and sunflower offer 21 x 34 and 34 X 55. Therefore, returning to the drawing board: about a single pole 21 X 34 spirals were drawn (programme 1) (Fig. 3). Each spiral has the form r=ab?, a logarithmic spiral. Over equal angular increments the spiral grows by a fixed proportion. It is also called the exponential spiral or the proportional spiral or the *Artist/mathematician, 42 Hemingford Road, London N.1, England. (Received 25 June 1981) Fig. 1. Touching Circles. A pattern of symmetry and uniform growth. equi-angular spiral. The last name may be explained like this: At any point on the spiral, the angle between the tangent (the momentary direction of the spiral) and the direction to the "centre" of the spiral remains the same. It is the spiral found in snail shells. The uniform rate of growth is determined by the constant b. Making b small tightens the spiral, degenerating to a circle when b = 1. The spiral approaches a straight ray when b becomes large. In order for a differing number of spirals in each direction to give rise to an array of approximately regular and tessellating rhombi, each set is given a different and appropriate value for b. This pattern, however, gives rise only to uniform growth. The seeds go on growing by the same ratio without limit. Clearly no sunflower does this. The seeds must slow and cease enlargement, and yet maintain packing. Are the Fibonacci spirals lost as this happens? Intriguingly, no. What seems to happen [5] is that the ratios switch over from 8 X 13 to 13 X 21, or from 21 X 34 to 34 x 55!