Center of Mathematics and Design (MAyDI)
University of Buenos Aires, Argentina
In a paper published in the Journal of the Society of Architectural Historians , the Serbian Branko Mitrovic claims that in Palladio's plans there exist six unexplained ratios that are close to the ratio Ö3:1, with the closest being the four large corner rooms of the Villa Rotonda (Figure 1 and Figure 2).
Figure 1. Photo of the exterior of Palladio's Villa Rotonda. Reproduced by courtesy of Stephen R. Wassell.
Figure 2. Plan and elevation of the Villa Rotonda, from Andrea Palladio, I quattro libri dell'architettura, Book II.
Each corner room has dimensions 26 x 15, which differ from the actual ratio Ö3:1 by only 0.07%! This astonishingly close approximation could be a very good reason for suggesting that Palladio used, on some occasions, a system of proportions based on the height of the equilateral triangle, or "triangulature".
But the numbers 26 and 15 in the drawings of the Rotonda were not chosen arbitrarily. In fact, let us look for the positive solution of the quadratic equation
It can be written in the form
or, dividing by x (different from zero)
Then it is possible, by substitution of x = 4 - (1/x) to get the continued fraction equation
where we have relaxed the condition that the terms of the continued fraction have to be positive .
Let us now introduce the notation
for this purely periodic continued fraction expansion.
The positive solution of equation (1) is
from which we get:
This periodic continued fraction expansion has the following rational approximants:
s (2) = 26/15 = 1.7333...;
s (3) = 97/56 = 1.7321428...;
s (4) = 362/209 = 1.7320574...
Comparing these results with the exact value = l.7320508..., we note that the second rational approximant 26/15 is exact to two decimal places. The third one is exact to four decimal places and the fourth to five decimal places!
This continued fraction expansion of Ö3, in which the approximants are by excess, is much better than the usual one :
that has the following rational approximants:
s (2) = 5/3 = 1.666...;
s (3) = 7/4 = 1.75;
s (4) = 19/11 = 1.7222...
It is easy to notice that in this case the fourth rational approximant is only exact to one decimal place!
In the case of Ö3, we may ascertain this curious behavior of getting a much quicker convergence with a continued fraction expansion that approximates by excess than the simple continued fraction expansion. I am not sure if this happens with every square root of a natural number, but it certainly happens with the square roots you find by looking for the positive solutions of quadratic equations of the type
where n > 0 is natural number. This is the subject of the research I am now undertaking.
 Oskar Perron, Die Lehre von den Kettenbrüchen (Leipzig and Berlin: B. G. Teubner, 1929). back to text
 C. D. Olds, Continued Fractions (Washington, DC: Mathematical Association of America Books, 1963. back to text
FOR FURTHER READING
Stephen R. Wassell, "The Mathematics of Palladio's Villas", Nexus II: Architecture and Mathematics, Kim Williams, ed. (Fucecchio, Florence: Edizioni dell'Erba, 1998), 173-186.
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