Anne M. Burns

Department of Mathematics

Long Island University

C.W. Post Campus

Brookville, NY 11548

This paper illustrates a number of ways that recursion and replacement rules can be used to create aesthetically pleasing computer generated pictures. Starting with imitation of forms found in nature, we move to more abstract designs, first designs derived from the nature imitations, and finally a purely abstract example.

Many of the forms and shapes found in nature exhibit some form of self-similarity; the larger form appears to contain smaller copies of itself at different scales. Examples abound in the plant world; we see it also in mountains, clouds, the branching structure of rivers and blood vessels, patterns on animal skins, etc. Nature imposes restrictions on growth rules, but that doesn’t mean that the artist needs to. First imitating the forms and shapes in nature, the artist finds herself changing a shape, a scale or a color to produce a more abstract but visually appealing picture. We will start with algorithms that produce imitations of forms found in nature; next we combine them into what a colleague of mine termed "Mathscapes", and finally we will abstract the forms into visually appealing designs.

In mathematics and computer science a recursive function is a function that calls itself; by calling itself more than once a function can produce multiple copies of itself. This makes it an excellent technique for creating figures which are defined by "replacement" rules. Consider the following examples of replacement rules: In each of Figures 1 and 2, start with the leftmost figure. Then replace it by the second figure. This gives you the replacement rule. For example, in Figure 1 start with a line segment. At step 2 replace the line segment with 5 line segments as pictured, each 1/3 the length of the original. At step n replace each segment in step n-1 with a reduced copy of the step n-1 figure.

Figure 1: Example of a replacement rule

In Figure 3 the replacement rule is a little different; it is an example of a random displacement rule. At each step a random number is multiplied by a scale factor and then added to or subtracted from the average of the height at the left endpoint and the height at the right endpoint; the result is assigned to the height of the midpoint. Figure 4 shows the result of carrying out the rule until the left and right endpoints of each interval are the same. To produce more realistic looking natural forms we make much use of random numbers; in the example in Figure 1, instead of each line segment being 1/3 the length the length of the segments in the previous step we could make the line length a function of a Gaussian random variable with mean 1/3.

Figure 2: A geometric replacement rule

Figure 3: A random replacement rule

Figure 4: Result of replacement rule in Figure 3

Figure 5: Some common inflorescences

Figure 6 shows the compounding of some of the inflorescences. These pictures were all done with simple recursion.

Figure 7 shows some imaginary inflorescences obtained by using random numbers to vary segment lengths and angles and taking artistic liberties with the above.

Figure 7: Imaginary inflorescences

Another model for plants can be found in the article by deReffye et al [6]. In this model at each stage of growth a "bud" can do one of three things: (1) branch, (2) flower and ultimately die or (3) sleep (do nothing until the next stage). A probability is assigned to each of the three outcomes, such that the probabilities add up to 1. We start with a single bud; at each stage, for each bud that is still alive, we generate a random number between 0 and 1; the number determines the next state of that bud. For added realism we can make the probabilities functions of time, so that at later stages the probability of a bud dying is higher. Figure 8 shows three imaginary "trees" using a branching number of 2, that is, when a node branches it produces two new branches. The exact same program drew the three trees; using probabilities and random numbers we can make it look as if the three trees come from the same family, but are not identical.

Figure 8: Trees modeled using deReffye’s method

Another of my favorite ways to model botanical growth is a method called L-systems, or string rewriting. It was discovered by Aristid Lindenmayer, a Dutch biologist who had the remarkable idea of using concepts from formal language theory to describe plant growth, and developed by Premislaw Prusinciewicz [9]. In this model each geometric part of the plant is assigned a character. Here is a simple example: let I represent an internode, L a leaf and F a flower. We use brackets and parentheses to indicate branches. Square brackets enclose a branch to the left, parentheses enclose a branch to the right. For example, the string of characters I[I[L](L)IF]I(II(L)IF)IF might represent the imaginary plant in Figure 9a. At each stage we use a set of rewriting rules (productions) to successively replace each character by a string of characters. Figure 9 illustrates several stages in the development of an imaginary plant using this method.

Another method of modeling plant growth and other branching phenomena uses a stochastic matrix of probabilities; this method is reminiscent of a Markov chain. Space considerations preclude a description here, but the interested reader can find more in [5].

2.2. Mountains and clouds. The midpoint displacement rule illustrated in Figures 3 and 4 can be extended to a two dimensional model in which we obtain a height field over a two-dimensional grid. We can use this to model both mountains and clouds. To model clouds, we assign a ramp of colors to the heights, while to model mountains we use some trigonometry and calculus to project the three dimensional height field onto a two dimensional surface.

First we outline the recursive midpoint algorithm for generating the height field. We start with a 2ⁿ+1 by 2ⁿ+1 grid of points in the plane, where n could be 9, for example, for a 513 by 513 grid. We assign a number to each of the 4 corner points. At step 1 we assign a value to the point labeled 1 in Figure 10a; the value is the average of the numbers at the 4 corner points plus a random number. At step 2 we assign values to the points labeled 2 in Figure10b, by taking the average of the three closest previously assigned points plus a random amount. Figures 10c and 10d show the order in which the next several points are assigned. At each stage the random amount is scaled down.

Once the height field is generated we can render it as three dimensional mountains using some calculus and trigonometry, illustrated in Figure 11a, or we can assign a continuous ramp of colors to the heights and render it as clouds, illustrated in Figure 11b. By changing the scale factor that we use in scaling down the random amount that we add at each stage, we can produce sharp fractal-like mountains and clouds or soft rounded mountains and cumulus clouds.