Modeling of Phyllotaxis

Phyllotaxis is the spatial arrangement of leaves in blooms and was introduced in Sect. 3.4. Also, we previously discussed different procedures for the pro­duction of phyllotaxial patterns. In the following, a solution with L-systems is illustrated.

Golden Section ^ We will therefore return to the arrangement of the seeds in a sunflower, which was likewise addressed in Sect. 3.4. With a parametric L-system the typical angle of rotation is modeled according to the Golden Angle of 137.5° in com­bination with the square root increase of the radius (see also [66]). Thereby D is the L-system for the description of a small circle, and w = A(n):

A(n) ::= +(137.5)[/(^П) – D]A(n + 1).

Accordingly, a complete head of a sunflower can be generated, as shown in Fig. 5.9, with the aid of L-systems for the creation of the form of the seeds S, the inflorescence R, and the petals M, N, O, and P:

A(n) ::= +(137.5)[f^/n)C]A(n +1) C(n) : n < 440 ::= – S

The system differentiates the six forms to be produced according to “the control variable” n (in this case also an index). If the control variable is smaller than 440, seeds are produced, then inflorescences, and finally different forms of petals.

Figure 5.9

Realistic plant models, generated using parameterized L-Systems (Courtesy of D. Fowler, P. Prusinkiewicz)

 

Modeling of Phyllotaxis

Contrary to the sample models in Sect. 4.13, the L-system again is a descriptive model, which does not relate to natural processes, but only approximates the outer appearance of the corresponding object.