About the geometry of a marble intarsia

On 25 August 2015, the Journal of Cultural Heritage published online the article The medieval roots of modern scientific thought. A Fibonacci abacus on the facade of the church of San Nicola in Pisa (alternative link).
The author, Professor Pietro Armienti, discusses some geometric properties of a beautiful marble intarsia on the facade of the church of San Nicola in Pisa which was recently restored.

In particular, the author claims that the intarsia hides a recursive geometrical construction which allows to approximate the length of the side of any regular polygon inscribed in the largest circle of the intarsia. I invite the interested reader to consult the paper for a description of this construction. This construction leads to the following formula (as it appears in the article):

where phi=(1+sqrt(5))/2=1.618033988... is the golden ratio and tau=phi-1=0.618033988...
Note that above formula can be written in a more compact form as

Now, given an integer number m, the length of the side of a regular m-sided polygon inscribed in a circle of radius r is M2=2*r*sin(pi/m). In the paper the radius r is assumed to be 55 units. To show how the formula works, the author selects the values of R and n which minimize the difference between the exact value M2 and the approximation M1 given by the formula. For example in Table 1 (see the paper), we find that for m=5 the best choice is R=55 and n=2. One can verify the approximation by typing the values of R, n and m below and then pressing the button.
R: n: m:

M1=, M2=, |M2-M1|=

In my opinion the most serious objection against this method is the following:

IF WE HAVE THE FREEDOM TO CHOOSE THE TWO PARAMETERS R AND n, THEN WE ARE ABLE TO APPROXIMATE ANY NUMERIC VALUE IN A WIDE RANGE!!!

Indeed, since tau^n decreases to zero as n becomes larger, it follows that the formula gives well-distributed values between R*0.66 and R*1.41. Therefore that formula is not so peculiar. You can for example replace tau with another positive number less than 1 and obtain similar results!!

The table below gives the values of the formula by varying R (rows) and n (columns).

 R\n 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 0.66 0.66 0.66 0.66 0.66 0.67 0.67 0.69 0.7 0.74 0.78 0.86 0.98 1.16 1.41 2 1.31 1.31 1.31 1.32 1.32 1.33 1.35 1.37 1.41 1.47 1.57 1.72 1.96 2.32 2.83 3 1.97 1.97 1.97 1.98 1.98 2 2.02 2.06 2.11 2.21 2.35 2.58 2.94 3.47 4.24 4 2.62 2.62 2.63 2.63 2.65 2.66 2.69 2.74 2.82 2.94 3.14 3.44 3.92 4.63 5.66 5 3.28 3.28 3.28 3.29 3.31 3.33 3.37 3.43 3.52 3.68 3.92 4.3 4.9 5.79 7.07 6 3.93 3.93 3.94 3.95 3.97 4 4.04 4.11 4.23 4.41 4.7 5.17 5.88 6.95 8.49 7 4.59 4.59 4.6 4.61 4.63 4.66 4.71 4.8 4.93 5.15 5.49 6.03 6.86 8.11 9.9 8 5.24 5.25 5.25 5.27 5.29 5.33 5.39 5.48 5.64 5.88 6.27 6.89 7.84 9.27 11.31 9 5.9 5.9 5.91 5.93 5.95 5.99 6.06 6.17 6.34 6.62 7.06 7.75 8.82 10.42 12.73 10 6.55 6.56 6.57 6.59 6.61 6.66 6.73 6.85 7.05 7.35 7.84 8.61 9.8 11.58 14.14 11 7.21 7.21 7.23 7.24 7.28 7.33 7.41 7.54 7.75 8.09 8.63 9.47 10.78 12.74 15.56 12 7.86 7.87 7.88 7.9 7.94 7.99 8.08 8.22 8.46 8.82 9.41 10.33 11.76 13.9 16.97 13 8.52 8.52 8.54 8.56 8.6 8.66 8.75 8.91 9.16 9.56 10.19 11.19 12.74 15.06 18.38 14 9.17 9.18 9.2 9.22 9.26 9.32 9.43 9.6 9.86 10.29 10.98 12.05 13.72 16.22 19.8 15 9.83 9.84 9.85 9.88 9.92 9.99 10.1 10.28 10.57 11.03 11.76 12.91 14.7 17.37 21.21 16 10.48 10.49 10.51 10.54 10.58 10.66 10.78 10.97 11.27 11.76 12.55 13.78 15.68 18.53 22.63 17 11.14 11.15 11.17 11.2 11.24 11.32 11.45 11.65 11.98 12.5 13.33 14.64 16.65 19.69 24.04 18 11.79 11.8 11.82 11.85 11.91 11.99 12.12 12.34 12.68 13.24 14.11 15.5 17.63 20.85 25.46 19 12.45 12.46 12.48 12.51 12.57 12.65 12.8 13.02 13.39 13.97 14.9 16.36 18.61 22.01 26.87 20 13.1 13.11 13.14 13.17 13.23 13.32 13.47 13.71 14.09 14.71 15.68 17.22 19.59 23.17 28.28 21 13.76 13.77 13.79 13.83 13.89 13.99 14.14 14.39 14.8 15.44 16.47 18.08 20.57 24.32 29.7 22 14.41 14.43 14.45 14.49 14.55 14.65 14.82 15.08 15.5 16.18 17.25 18.94 21.55 25.48 31.11 23 15.07 15.08 15.11 15.15 15.21 15.32 15.49 15.76 16.21 16.91 18.04 19.8 22.53 26.64 32.53 24 15.72 15.74 15.76 15.81 15.87 15.98 16.16 16.45 16.91 17.65 18.82 20.66 23.51 27.8 33.94 25 16.38 16.39 16.42 16.46 16.54 16.65 16.84 17.13 17.61 18.38 19.6 21.52 24.49 28.96 35.36 26 17.03 17.05 17.08 17.12 17.2 17.32 17.51 17.82 18.32 19.12 20.39 22.38 25.47 30.12 36.77 27 17.69 17.71 17.73 17.78 17.86 17.98 18.18 18.51 19.02 19.85 21.17 23.25 26.45 31.27 38.18 28 18.34 18.36 18.39 18.44 18.52 18.65 18.86 19.19 19.73 20.59 21.96 24.11 27.43 32.43 39.6 29 19 19.02 19.05 19.1 19.18 19.32 19.53 19.88 20.43 21.32 22.74 24.97 28.41 33.59 41.01 30 19.65 19.67 19.71 19.76 19.84 19.98 20.2 20.56 21.14 22.06 23.52 25.83 29.39 34.75 42.43 31 20.31 20.33 20.36 20.42 20.5 20.65 20.88 21.25 21.84 22.79 24.31 26.69 30.37 35.91 43.84 32 20.96 20.98 21.02 21.08 21.17 21.31 21.55 21.93 22.55 23.53 25.09 27.55 31.35 37.07 45.25 33 21.62 21.64 21.68 21.73 21.83 21.98 22.22 22.62 23.25 24.27 25.88 28.41 32.33 38.22 46.67 34 22.27 22.3 22.33 22.39 22.49 22.65 22.9 23.3 23.96 25 26.66 29.27 33.31 39.38 48.08 35 22.93 22.95 22.99 23.05 23.15 23.31 23.57 23.99 24.66 25.74 27.44 30.13 34.29 40.54 49.5 36 23.58 23.61 23.65 23.71 23.81 23.98 24.24 24.67 25.37 26.47 28.23 30.99 35.27 41.7 50.91 37 24.24 24.26 24.3 24.37 24.47 24.64 24.92 25.36 26.07 27.21 29.01 31.85 36.25 42.86 52.33 38 24.89 24.92 24.96 25.03 25.13 25.31 25.59 26.05 26.77 27.94 29.8 32.72 37.23 44.02 53.74 39 25.55 25.57 25.62 25.69 25.8 25.98 26.26 26.73 27.48 28.68 30.58 33.58 38.21 45.17 55.15 40 26.2 26.23 26.27 26.34 26.46 26.64 26.94 27.42 28.18 29.41 31.37 34.44 39.19 46.33 56.57 41 26.86 26.89 26.93 27 27.12 27.31 27.61 28.1 28.89 30.15 32.15 35.3 40.17 47.49 57.98 42 27.51 27.54 27.59 27.66 27.78 27.97 28.28 28.79 29.59 30.88 32.93 36.16 41.15 48.65 59.4 43 28.17 28.2 28.24 28.32 28.44 28.64 28.96 29.47 30.3 31.62 33.72 37.02 42.13 49.81 60.81 44 28.82 28.85 28.9 28.98 29.1 29.31 29.63 30.16 31 32.35 34.5 37.88 43.11 50.97 62.23 45 29.48 29.51 29.56 29.64 29.76 29.97 30.31 30.84 31.71 33.09 35.29 38.74 44.09 52.12 63.64 46 30.13 30.16 30.21 30.3 30.43 30.64 30.98 31.53 32.41 33.82 36.07 39.6 45.07 53.28 65.05 47 30.79 30.82 30.87 30.95 31.09 31.3 31.65 32.21 33.12 34.56 36.85 40.46 46.05 54.44 66.47 48 31.44 31.48 31.53 31.61 31.75 31.97 32.33 32.9 33.82 35.29 37.64 41.33 47.03 55.6 67.88 49 32.1 32.13 32.18 32.27 32.41 32.64 33 33.58 34.53 36.03 38.42 42.19 48 56.76 69.3 50 32.75 32.79 32.84 32.93 33.07 33.3 33.67 34.27 35.23 36.77 39.21 43.05 48.98 57.92 70.71 51 33.41 33.44 33.5 33.59 33.73 33.97 34.35 34.96 35.93 37.5 39.99 43.91 49.96 59.07 72.12 52 34.06 34.1 34.16 34.25 34.39 34.63 35.02 35.64 36.64 38.24 40.78 44.77 50.94 60.23 73.54 53 34.72 34.75 34.81 34.91 35.06 35.3 35.69 36.33 37.34 38.97 41.56 45.63 51.92 61.39 74.95 54 35.37 35.41 35.47 35.56 35.72 35.97 36.37 37.01 38.05 39.71 42.34 46.49 52.9 62.55 76.37 55 36.03 36.07 36.13 36.22 36.38 36.63 37.04 37.7 38.75 40.44 43.13 47.35 53.88 63.71 77.78

Roberto Tauraso
(Last update September 24, 2015)