 |
 |
 |
 |
Let's begin by looking at an interesting plant from a geometric point of view:
Photo 1. A Fern. The three photographs, D, E and F, are three views of the same fern. Photo E shows a close-up view of photo D, and F is a close-up of E. If we look at the overall appearance of the fern in D, we can see that it is made up of one central stem from which green leaves with pointed tips emerge. If we zoom in on one of the leaves (E), we see that the pattern of central stem and band after band of green leaves is repeated. Finally, if we zoom in on one of the leaves (F), we can observe the same pattern, and so on ad infinitum. The property, whereby the geometric form of the fern is independent of the scale on which it is seen, is called self-similarity. The geometric shapes that it forms are called fractals. Other examples of fractals appear in nature:
Photo 2. The fractal geometry of nature. Photo A shows a bolt of lightning, B a tree and C, the human circulatory system, taken from an early drawing. The concept of fractals has been introduced so that we may understand the geometrical form of the poem presented here below, called insula smaragdina (hereafter called insula). As can be seen on the homepage, the insula is a visual poem, one that "expresses its spiritual content through words laid out according to metrical laws". In this particular case, the laws will actually be "mathematical" rather than "metrical" since the words of the work are written in accordance with fractal geometry. It is therefore a visual poem in that the words are laid out according to this geometry. All of these images have been scanned from "Mundo Cientifico" ("World of Science"), no. 201, May 1999, p. 58
The words of the poem are laid out in such a way that they form a fractal called Koch Island (hence the name insula, which in Latin means island). The Koch Island is part of a simple union of fractals called "Lindenmayer Systems":
Figure 1. Koch Island (3rd iteration) Koch's island and other fractals can be explored through the following software (click here for installation instructions): Fractal visualization software (soon in english) It is important to note the self-similar structure of the Koch Island. To examine it more closely we can look at it in different scales to see if the resulting structures bear any similarities. First of all, let's sketch the Koch Island to the fourth iteration. Let's imagine that we have a camera and that we photograph one part of it at maximum, at medium, and at minimum zoom:
Then we can carry out a succession of superimpositions between these figures to demonstrate their similarities:
When we superimpose figure 3 onto 2 (by changing the size of 3), figure 3 almost follows the perfectly-drawn rectangles of figure 2, but does not do so perfectly. By superimposing figure 4 onto 2 (again adapting the size of figure 4), we note a similar result to the previous one: figure 4 also almost follows the perfectly-drawn rectangles of 2, and does so a little better than 3, but it still does not follow it perfectly. When we also superimpose figure 4 onto 3 (and change the size of 4), we note that they are practically, but not absolutely identical. Figure 4 follows 3 meticulously, but there are tiny deviations and does not follow the other figure perfectly. The Koch Island thus demonstrates its strong self-simlar structures: 3 is similar to 2; 4 is similar to 3, and so forth on to infinity. The Island's self-similarity is characteristic of fractal forms.
Let us run quickly over the words written according to the geometry of the Koch Island, the words that form the insula smaragdina poem. To see the poem and generate different versions of it, use the Poem generation software (soon in english) (click here for installation instructions). Different aspects of the poem can be seen more clearly by clicking on the gallery. Given the difficulty of faithfully reproducing the poems generated by the programme here, let's look at a small section of one of them:
Figure 8. A section of poem Reading the text in a clockwise direction, the following sentence can be isolated: "QUODESTINFERIUSESTSICUTQUODESTSUPERIUSET We can, in turn, separate it into words that form the maxim: "QUOD EST INFERIUS EST SICUT QUOD EST SUPERIUS ET QUOD EST SUPERIUS EST SICUT QUOD EST INFERIUS ET" If the poem is read patiently, the reader sees that the first sentence is repeated again and again. During this reading, reader sees that the last word of one sentence, "ET", is followed by the first word, "QUOD", of the next. The method is to read until we find ourselves back at the original starting point. We begin to read again and so on, ad infinitum. The following diagram outlines the reading procedure (click on the image to enlarge):
Figure 9. How the text is repeated to infinity Finally, there is no one single interpretation of the poem. In other words, a definitive version of the poem does not exist. Thus anyone who wishes to may create his or her own version by using the insula computer programme. As such, each individual possesses the means to construct his or her own poem, and to print it (or not).
|
|||||||||||||||||||||||||||||
