A Science-and-Art Interstellar Message: The Self-Similar Sierpinski Gasket

If extraterrestrial intelligence (ETI) exists, whether carbon-based or not, it is nothing but a particular type of complex system existing in the universe. Being a complex system, it must share the common traits of all (or almost all) known complex systems. For example, an ETI, like a human, is very likely an active walker [1] that changes its surrounding landscape (physical or virtual) through its every action and is in turn influenced and limited by the changed landscape.

Like us, being a complex system, an ETI will recognize immediately the property of self-similarity (the principle behind fractals), which is a property shared by many complex systems. Being self-similar is not the same as being identical or similar. An object is self-similar if a small part of it (when blown up in scale) looks similar or identical to the whole. Trees are such an example; in fact, the existence of bonsai (called "plate scenery" by the Chinese) is an example of self-similarity, in which a small branch of a tree may look like the whole tree. In Africa, Ba-ila settlements established before 1944 are arranged in a self-similar pattern [2], even though the settlers themselves may not be aware of fractals. A self-similar mosaic (a Sierpinski gasket in fact), an art object, actually lies on the floor of a cathedral built in the year 1304 in Agnani, Italy [3]. Finally, the foam-like material structure of our universe is a fractal. Self-similar objects thus exist ubiquitously over many scales on Earth and elsewhere in the universe.

Fig. 4.

Photo of Tuareg leatherwork with cotton embroidery, Bourem (Mali), 1947. Some of the fractal qualities of the Sierpinski gasket are reflected in this piece of African art.

Photo © Jean Gabus. Courtesy Musée d'Ethnographie, Neuchatel, Switzerland

The simplest example of fractals is the Sierpinski gasket (SG), named by B.B. Mandelbrot after the Polish mathematician Waclaw Sierpinski [4]. The SG is obtained by first dividing an equilateral triangle into four equal, smaller equilateral triangles, and then removing the middle one. The process is repeated for the three existing triangles ad infinitum. SG designs are at once simple and beautiful, already existing throughout human civilization (see Fig. 4), and perhaps also existing in the history of ETI, or is at least recognizable by ETI. The SG is both a science and an art object; its human-made nature cannot be mistaken by ETI because of its neat design. (In fact, it is more than self-similar; it is "self-identical.") Moreover, the SG is extremely easy to draw graphically, and it can be computer-generated in at least four different ways, all with only a few lines of code. In short, the SG is the ideal science-and-art object to be beamed to any ETI.