Fern leaves and cauliflower curds are not fractals

Abstract

The popular demonstration of drawing a mature fern leaf as expressed by Barnsley's fractal method is mathematically and visually very attractive but anatomically and developmentally misleading, and thus has limited, if any, biological significance. The same is true for the fractal demonstration of the external features of cauliflower curds. Actual fern leaves and cauliflower curds have a very small number of anatomically variable and non-iterating bifurcations, which superficially look self-similar, but do not allow for scaling down of their structure as real fractals do. Moreover, fern leaves and cauliflower curds develop from the inside out through a process totally different from fractal drawing procedures. The above cases demonstrate a general problem of using mathematical tools to investigate or illustrate biological phenomena in an irrelevant manner. A realistic set of mathematical equations to describe fern leaf or cauliflower curd development is needed.

Introduction

Barnsley's fern leaf¹ has become a popular demonstration of the proposed fractal nature of plant form^2-8 and many sites on the internet. The fractal description of a mature fern leaf form is achieved by running a simple chaos algorithm many times and the accumulated resulted random points from each run finally form the shape of a typical fern leaf.⁹ Similarly, the magnificent Romanesque cauliflower variety was proposed to be a self-similar “heaven's gift to fractalists.”¹⁰ Later, Strogatz¹¹ described the surface of a Romanesque cauliflower curd as “exquisitely symmetrical, sporting dozens of knobby florets, each a miniature version of the entire structure, and each built from even smaller copies of the whole,” positing it as a self-similar structure.

Here I point out the structural-developmental difficulties in positing fern leaves and cauliflower curds as self-similar structures.

Discussion

The common computerized descriptions of fern leaves and cauliflower curds are mathematically and visually very attractive but developmentally misleading, and thus have very limited if any biological value. Superficially, both fern leaves and cauliflower curds appear to be self-similar systems, but this is simply not the case. In actual fern leaves and cauliflower curds it is impossible to increase the resolution and find infinite or even many repeats of similarly shaped smaller units, as can be done with real fractals. Mature fern leaves develop through a process totally different from the formation of fern leaf shapes by fractal procedures. In actual fern leaves, groups of marginal meristems form coils (organs known as croziers)¹² that gradually uncoil while forming new tissues from their dividing meristems to establish the fully developed leaf. The fern leaf thus develops from the inside out and not by randomly dispersed dots that gradually fill the leaf area, as is done with chaos computer programs. Moreover, a closer look at the vascular tissues of fern leaves shows significant differences in vascular structure when compared along the branching orders, beginning in the central vein through the pinnae and finally the pinnules¹³ and therefore, structurally, fern leaves are not self-similar organs.

Similarly, a careful look into a cauliflower curd structure shows that it is composed of up to seven branching orders, of which only the outermost carries flower buds.¹⁴ The fact that the flowers compose only the outermost layer makes the whole system dissimilar. Like fern leaves or any other plant branching system at the organ level, the cauliflower curd develops from the inside out through a process totally different from fractal drawing. Branching in plants is a process and not just an end product¹⁵ and the branched fern leaves or cauliflower curds formed by Barnsley's method or related mathematical procedures are just a beautiful but irrelevant final condition.

The elegance of using computerized fractal-forming programs to describe general plant morphology has nothing to do with the actual anatomical structure and the patterns of developmental processes that result in the formation of such plant organs. It is a mistake to use the final developmental shape of the fern leaf and cauliflower curd for the mathematical analysis, ignoring actual organ anatomy and patterns of development, as done by fractalists and other mathematicians.

Dissecting developmental processes, such as the formation of fern leaves and cauliflower curds, into stages that are mathematically manageable and developmentally and structurally correct would make the mathematical procedures more relevant for biologists. A set of mathematical procedures that reliably describe the development of fern leaves, cauliflower curds or any other plant organ will not be just an elegant visual demonstration, but probably an important lesson in developmental biology. I look forward to seeing it.