Architecture
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by David Speiser 1. Symmetries of the Battistero and the Torre Pendente The symmetries of the Battistero and of the Torre are closely connected. But while the symmetries of the Battistero present a complicated pattern, the symmetry of the Torre is a simple if also a very surprising one. Therefore I will begin with the latter. 1.1. The Symmetry of the Torre Pendente By "symmetry of a tower" I mean here simply the symmetry of the ground plan (Grundriss). The ground plan of the tower of a Church is usually a square. Not always but often its covering has the same fourfold symmetry. Sometimes the tower has only an ordinary roof, the 4-fold symmetry then is lost and only a twofold symmetry survives; we then say that the fourfold symmetry is broken. Sometimes, to the contrary, the covering shows an eightfold, more refined symmetry. The most famous example is provided by the tower of the Cathedral of Freiburg i. Br. and in Pisa the tower of San Nicola is round below, but its upper levels are octagonal. But there are many other, smaller towers, whose plan follows this idea. Occasionally, we find a tower with a sixfold symmetry. But these cases are very rare: this square and possibly the square that becomes an octagonal, or is roofed, are the rule. R. Krautheimer has listed a few exceptions. Of course, there are also round towers, e.g. the ones in Ravenna. And such a round tower is the Torre Pendente. The symmetry of a circle is an infinite one, however the architectural elements of the tower can break this infinite symmetry, so that only a finite one survives. Before we show how this happens in the case of the Torre, I must summarize its structure in a few words. The ground floor consists of a massive wall with exterior arcs which rest on half pillars. Above this ground level, six more, open, and but for the last one which is a bit higher, equal levels follow. But all have twice as many arcs as the ground level. These surround the inner shaft of the tower as a colonnade. On the top there is a somewhat smaller drum (tambour): thus all counted there are eight levels. Obviously the finite symmetry of the tower is determined by the number of pillars and arcs. What is this number? If one counts them one is stunned to find, that there are neither 16, as one would have guessed at first, nor 12 or maybe 20: there are 15 arcs on the ground level and 30 on the following ones! The drum finally has six major and six minor arcs, in such a manner that the minor ones are spanned above one, the major ones above four arcs of the seventh level. This 15-fold, resp.-30-fold symmetry is probably unique; at least it must be extremely rare. Therefore it demands an explanation, the more so, as it is used on such an outstanding building. Before we explain what motivated the old Pisans to pick or at least to support this surprising and audacious choice, we add a few other facts about the tower, and we shall then analyze the symmetries of the Battistero. This will be a much more involved task. In a recent book P. Pierotti has demonstrated the importance which the Pisans attached to exact numerical measures: the height of the Torre is exactly 100 bracci Pisani (Pisan arms), the equivalent of 20 pertiche Pisane (Pisan staff), while the circumference measures exactly 100 piedi Pisani (Pisan feet). These numbers underline strongly how eagerly the Pisans of these days took delight in mathematical relations and used them for designing a building: so they knew why they did construct a pentadecagonal campanile (tav. CXXXIV)! But it is of fundamental importance to distinguish between significant numerical values such as the ones just mentioned, and between geometrical relations, in our case geometrical symmetries. While the former are numerology, whatever one may think of it, the symmetries are partitions of space or maybe partitions of the plane. What partition of a plane means may be illustrated by the following example. Into a square the diagonals are drawn, and thus the (horizontal) line through this intersection halves the square. Again the diagonals in the upper rectangle are drawn and this rectangle too is halved by a horizontal line and once more diagonals are drawn (fig. 1). FIG.1 What is the use of this construction? Just insert with a colour pencil the following profile which follows the patterns in an obvious way (fig. 2). That this construction is not a mere curiosity noticed post festum, but that it is the root of the design of the Duomo's facade, is attested by the exactness of the lines seen in the plate CXXXV. Obviously numbers, except for the 2 (since we always divided in two equal parts) do not play any role here: the construction is a purely geometrical one: Thus in the case of the Torre and also of the Battistero it is not sufficient to speak of twelve or of fifteen arcs, windows, etc.; these elements express, respectively, a twelve-fold, fifteen fold or a twenty-fold symmetry partition of space. There is one other partition of space to be found in the Torre. According to Pierotti it is divided into seven levels that comprise resp. 5, 2, 2, 2, 2, 2, 5 pertiche; it is a translational together with a mirror symmetry. We shall come back to the translational symmetry. Here it suffices to point out the tight coherency which these partitions of space confer to the Torre: it presents itself as being all of one piece; only its leaning diminishes its magnificent solemnity. I cannot resist the temptation to add here a guess, although at this moment I cannot prove it. The question arises why does the Torre have exactly eight levels? Many answers have been proposed, in part theological ones, but they do not seem convincing. We must remember, that when the Torre was begun it was the highest building in Italy, the highest perhaps in Europe. As such it was an expression of the immense orgoglio of the Pisans of that period. Now: what was the highest tower, which western mankind always could recall? Of course the tower of Babel, the often reproduced and celebrated example of a high building. And what was known of it? The Bible does not tell us much, it speaks only of its destruction. But the more informative Herodotus states explicitely that it had eight levels. In the first book of his history he writes:
The missing link for proving this guess is: did the Pisans know this passage of Herodotus? Herodot's work was not known in toto to the Western world, nor was it to the Arabs. On the other hand it cannot be excluded that certain pieces of information did reach the West, for instance through Constantinople or Salerno. If this guess should be correct, then the exact measures discovered by Pierotti acquire an additional meaning. In his book Europaische Literatur und Lateinisches Mittelalter E. R. Curtius writes:
David Speiser
(rif. annali vol. 24 n. 2-3 /1994, Scuola Normale Superiore)
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