Catálogo de publicaciones - libros

“How nice and symmetrical,” we often say or think to ourselves. The associations that the word awakens in us depend on the experiences in our past that have established its meaning for us. The term ‘symmetry’ can have three separate types of meaning, as a phenomenon, a concept, or an operation. The is what we consider to be symmetrical on the basis of our experience or of knowledge we have learned. The is what circumscribes all such phenomena. The is what gives rise to the phenomenon or makes it possible.

The first chapter acquainted us with the concept of symmetry. We saw what significant substantive changes the concept itself has undergone over the millennia. We now survey the study and depiction of the phenomenon over the ages.

In the preceding chapters I showed how there are certain common elements in humanity’s cultural history which can equally be found in European, Middle and Far Eastern, African, American and Australian aboriginal cultures. Their origins are ancient. These decorative elements already appeared in periods when — we assume — these cultures existed independently, without influencing one another. The silk route, for example, only began to ‘operate’ in the 14 century BC at the earliest. We are not suggesting that neighbouring peoples did not establish contact with each other before this, but that larger items displaying decorative elements can only have been taken to far-away lands when the required means of transportation became available. There are many possible explanations for similar decorative elements in different places, one of which can be from humanity’s natural environment.

Despite all rational explanations, a number of enigmatic questions were raised by the subject of tiling a plane and filling a space, the answers to which have been pursued for centuries, just like the secret of “magic numbers”, behind which further mystical explanations were expected as to why these numbers are as special in nature as they are. The most perfect shape (that with an infinite number of symmetries) is the circle. The sphere is a body with similar characteristics. A special proportion, π, is attributed to both. Another mathematical proportion, the base number for the natural logarithm, , is attributed to the perfectly swirling line, the logarithmic spiral. These numbers are made all the more mystical by the fact that, as delineators of proportion, they are independent of the scale chosen, that is they remain the same even when measured in different units — and it is in this scale invariance that their symmetry lies.

The example of the reproduction of rabbits (or any other animal that is quick to breed) iswell-known from most people’s secondary school studies. If we begin with a pregnant rabbit, it produces another rabbit, and we have two rabbits. In the first generation, the little rabbit will not breed, so at the next stage it will only be the old rabbit that gives birth again, giving us three rabbits. In the following generation, this little rabbit will not yet breed, but we have two rabbits from the previous generation that will, which adds two rabbits to the existing three rabbits, giving us a total of five rabbits. At the next stage, the two new rabbits will not yet breed, but the older three will, which together with the existing five rabbits gives us eight rabbits in all. In the following generation, five rabbits will produce new rabbits, and there will be eight plus five, that is thirteen rabbits, of eight will breed, giving us thirteen plus eight, that is twenty-one rabbits, and so on, until this system is not upset by some external factor. The number of new-born rabbits will always equal the cumulative number of rabbits from two generations before. As we began with the second member ( = 1) of the sequence of numbers, there is no need for us to subtract the 1 ( = 1) on the right hand side of the equation .

In Chapter 3 we encountered frieze and ribbon designs that can be placed in one dimension, and wallpaper groups, suitable for filling the plane or surfaces, that can be used for tiling or tessellation. Since ancient times, the possible repeating patterns in both of these dimensions have been used to decorate our environment. We completed our train of thought by saying that we would deal with the possible repeated close filling of space without gaps in a separate chapter.

Since ancient times, people have thought the number five to have hidden mysteries. What is mystical has symbolic significance. The mystical treatment of the number five belongs to the cults that date back to ancient times. Many cultures treat it as a fairy-tale number, others as a lucky number.

What do the surface structure of viruses, the morphology of sea radiolaria, the pattern on a golf ball, the weaving of baskets, the geodesic dome structure of Buckminster Fuller, the sewing design on a soccer ball and the C fullerene molecule all have in common? The feature common to all these and many other phenomena is a family of structures displaying a characteristic symmetry: truncated icosahedra. Truncated icosahedral structures belong to the sizeable set of polyhedra invariant under spatial rotations. It is no accident that a similar structure developed. Their formation involved a common principle of structure construction, that of so-called synergy.

The symmetries of polyhedra were primarily rotational. Turned around one of their axes of symmetry, they would overlap themselves. A body displaying -fold symmetry around a given axis will, after rotations, not just overlap itself in terms of the space taken up by its shape, but really return to exactly the same position: if we mark a particular point on it (e.g. the one we use to rotate it), it will return to the same place. If we continue to repeat the rotations of angle 2π/, oneachth occasion we will return to the same position (though, unless we place a mark on the polyhedron, we will not be able to distinguish the intermediate stages from one another). During the rotation, that is, the shape will periodically repeat itself.

Periodically repeating processes form a very broad class of natural phenomena. We encountered the symmetries of periodically repeated patterns during the mathematical and crystallographical description of friezes and wallpaper motifs. They are to be found as decorative elements almost everywhere in our environment. When we were at elementary school, we decorated our notebooks with serial designs. It is periodically repeating patterns that decorate the façades of most houses, our fences, railings, and this is how lampposts and rows of trees are laid out. Periodic decorative designs could be found in all periods in all folk decorative arts. They were used as much for decorating personal items and textiles as for decorative products made specifically to be looked at. Their fashion changed from age to age and culture to culture, with their characteristic features making it possible to identify and distinguish ancient Persian designs from Greek ones, or decorative elements of the European classicism of the seventeenth and eighteenth centuries from the of the twentieth.