Virgil and Arnaut, for Adalinda Gasparini





Adalinda Gasparini wrote a fine paper on the French troubadour Arnaut, for an international Dante conference held in Florence in early November 2012, online in Italian and French:


    Tan m’abellis vostre cortes deman…

    Formazione amorosa come peregrinatio & periclitatio


She quotes the first sestina (sixtain) of Arnaut, an intricate form of a poem invented by himself, and explains the cledisat or framework – returning permutations and shifting rings of rime words – via an elegant algorithm in form of an Archimedean spiral. Arnaut may well have influenced Dante who mentions him toward the end of Purgatorio 26, where the troubadour announces himself in Provençal


    Ieu sui Arnaut   (I am Arnaut)


in perfect symmetry to an earlier Italian announcement


    Io son Virgilio   (I am Virgil)


at the beginning of Purgatorio 7. The Divina Commedia has 100 canti:


    34 (Inferno)   33 (Purgatorio)   33 (Paradiso)


Virgil and Arnaut mark another partition of the one hundred canti:


    40  20  40


    40 canti    1-40    (Inferno 1-34, Purgatorio 1-6)

    20 canti    41-60    (Purgatorio 7-26)

    40 canti    61-100    (Purgatorio 27-33, Paradiso 1-33)


    Io son Virgilio    seven lines into canto 41

    Ieu sui Arnaut    seven lines before the end of canto 60


The partition 40 20 40 generates musical proportions:


    40 / 20   or  2/1  octave


    40 + 20 = 60    60 / 40  or  3/2  quint


    40 + 40 = 80    80 / 60  or 4/3  quart


    4/3 times 3/2 equals 2/1


    quart and quint combined yield an octave


A similar number game leads from the partition 40 20 40 to the so-called Sacred Triangle 3-4-5 that played an important role in early geometry:


    40 + 20 = 60    40 + 40 = 80    40 + 20 + 40 = 100


    60-80-100  being a multiple of  3-4-5


The Sacred Triangle 3-4-5 allowed the first systematic method for the calculation of the circle, along a sequence of triples (obtainable with a linear algorithm)


    3-4-5    7-24-25    44-117-125    336-527-625   


This triples generate a sequence of ever rounder polygons of 12 20 28 36 … sides (of two or three different lengths per polygon, multiples of the square roots of 2 and 5 or 2 times 5 that can be approximated by additive number columns). The slowly rounding polygons approximate the circle of radius 5 25 125 625 ... in the ever finer grid 10 by 10, 50 by 50, 250 by 250, 1250 by 1250 …


Let us imagine a circle of radius 50 and diameter 100 in the grid 100 by 100, according to the number of canti in the Divina Commedia. The crossing horizontal and vertical axes mark the center of the circle, while their ends provide four points of the circumference. The Sacred Triangle 3-4-5 in the form of 30-40-50 defines eight more points, and so does the next triple 7-24-25 in the form of 14-48-50. Now you have twenty rational points of the circumference. If you connect them one by one with straight lines you get an already fairly round polygon of eight longer sides (given by the square root of eighty) and a dozen shorter sides (given by the square root of fifty).


Let us imagine an ellipse defined by the partition 40 20 40 as marked by Virgil and Arnaut




This ellipse has remarkable numbers. The long horizontal axis measures 100 units and the slightly shorter vertical one practically 98 units (49 plus 49). The side of the inscribed rhomb, nearly a square, measures exactly 70 units, and the side of the narrow vertical rhomb confined by the foci (focal points) held by Virgil and Arnaut measures exactly 50 units. Finally, the circumference of the round ellipse measures practically 311 units. From all the ellipses of the long axis 100 only this one has such interesting numbers, integers and near integers. (In mathematical terms, the ellipse visualizes a whole number solution of the equation aa minus bb equals bb minus cc, namely, 70x70 minus 50x50 equals 50x50 minus 10x10.)


The medieval saying deus est sphaera means that God is present in the perfect form of the circle. A sphere on paper, reduced by one dimension, is a circle, still a perfect form. Let us regard the imaginary circle of diameter 100 as the circle of divine perfection and knowledge and truth, while the ellipse given by the partition 40 20 40 may symbolize the Divina Commedia revolving around Virgil, whom Dante owns his style (Inferno 1:85-87), and Arnaut who would have provided the seed of inspiration, so to say. The full circle stands for divine perfection, all embracing knowledge and absolute truth, whereas the ellipse, a round one almost filling the circle, represents the best of human work that comes close to perfection and truth, but only in way of better and better approximations, always leaving a gap that can’t be bridged, not even by the most perfect work of art and most elegant scientific theory, a principal gap holding surprises for the future, new ideas and theories to be found and developed by coming generations. We can only approach the truth, never really reach it. If you look at a disk from an angle, the circle is reduced to an ellipse. One dimension of the circle is preserved, the other reduced by a factor between one (full circle) and zero (line). The word ellipse goes back to the Greek and means wanting, standing behind; something is missing, left out, as in the elliptic way of speaking. Our human knowledge is elliptic: adequate in one dimension, wanting in the other (or another) dimension.


The ellipse defined by the 100 canti of the Divina Commedia and the partition 40 20 40 marked by Virgil and Arnaut conveys a philosophical message that goes along with my cosmological interpretation of the Divina Commedia. The long poem has 14,233 lines, one line short of the cosmological number 14,234 in Dante’s model universe. One line is missing, the line of the divine messenger that would complete the epic, as each canto closes on a beautiful line of poetic power. Only that the one line to be delivered by the divine messenger would transcend all human wisdom. It is that single line we are working on, generation for generation.


December 2, 2012,

Franz Gnaedinger



Postscript. An example of symbolic geometry in an Italian painting, Baptism of Christ, by Piero della Francesca. The drawings may speak for themselves