Italian Renaissance Armatures
“The name Renaissance is really a label for a style not for a period. In art and literature its beginnings are usually traced back to the powerful works of the painter Giotto di Bondone (c. 1266-1337) and the poet Dante Alighieri (1265-1321). In mathematics and the mathematical sciences the process of ‘rebirth’ is generally seen as starting later, perhaps in the mid-fifteenth century, with astronomers’ attempts to return directly to Greek sources. This scientific Renaissance, like that of the arts, has no definite end . . . “1
Euclid’s Optics, written in Greek in the third century BC, was well known in Latin translation throughout the Middle Ages. This is obvious to anyone who studies the works of philosophers of the time. It is not obvious at all if one looks instead at the works of artists. These regularly flout the mathematical rules that Euclid shows as establishing the elative sizes that subjects will present to the eye. Field, p. 7.
". . . I contend that the European Renaissance pictorial system, based on the principles of Euclidian solid geometry, attempted at least in the beginning to do no more than replicate what the human eye perceives according to the tenets of Euclidian geometry, which medieval Europeans understood as synonymous with the vision of God. Pictures rendered in perspective permitted human beings to see the world just as God conceived of it at Creation." Edgerton, Giotto's Geometry, p. 5.
". . . the evolving Western conception of space, both pictorially and phenomenally, as an imaginary three-dimensional latticework within which all volumes could be fitted in rectilinear relationship." NOTE: Rectilinear means consisting of or moving in straight lines. Edgerton, Giotto's Geometry, p. 149.
Alberti on proportion and beauty, quoted in Calter on p. 195. “. . . we may conclude beauty to be such a consent and agreement of the parts of a whole in which it is found, as to number, finishing, and collocation, as congruity, that is to say, the principal Law of Nature requires.”
Piero della Francesca (1416 – 1492)
"Some paintings from the Renaissance and later show a rectangular frame surmounted by a semicircle, as does the painting by Piero della Francesca [The Baptism of Christ, 1450] . . . John’s arm and Jesus’ loincloth continue the line of the circle. The square and circle overlap, as do the two circles in a vesica, but a square replaces one circle. Jesus appears in the area common to both, mediating between heaven and earth. Piero has combined the idea of squaring the circle and the idea of the vesica into one painting. NOTE: On the figure showing the painting’s armature, Calter has defined the area above the square as the heavenly realm, and the area below the circle as the earthly realm. This format may reference the vesica and squaring the circle, but the dimensions of the inscribed circle on the bottom and the circle on top aren’t the same. Calter, Squaring the Circle, p. 240.
"Finally, . . we have a representation of the root-2 rectangle. . . (p. 107) consider Piero’s Flagellation of Christ, which is a famous example of a root-2 rectangle in painting. The painting’s width is approximately √2 times its height. Calter, Squaring the Circle, pp. 106-107. NOTE: See Calter.
Leonardo da Vinci (1452 – 1519
"Amongst later generations of Renaissance painters, no one paid closer heed to Alberti than Leonardo da Vinci, whose own extensive injunctions on the theory and practice of painting contain repeated echoes of his predecessor’s ideas." Alberti, On Painting, p. 23.
"Only Leonardo among Renaissance writers could say that ‘the perspective of painters will never show an object of . . . [the same size as] the object shows itself to the eye at an equal distance.’ No other writer was as concerned with refraction, distortion, binocular vision, color perspective, the optics of the eye, the conditions for viewing, the limits of linear representation, and the proper angles and distances of view." Elkins, The Poetics of Perspective, p. 68.
"Perspective may have been developed by Brunelleschi, and codified by Alberti and Piero, but it was perfected by Leonardo. Leonardo’s notes on perspective are apparently lost, but he made great use of perspective in his paintings, such as in the study of the unfinished Adoration . . . Note the strict Albertian grid on the pavement." Calter, Squaring the Circle, p. 384. NOTE: Calter is referring to Leonardo's perspective drawing.
"The Last Supper . . . is perhaps Leonardo’s most famous perspective painting." Calter, Squaring the Circle, p. 385.
"In his notebooks, Leonardo described three kinds of perspective: size, color, and disappearance (atmospheric perspective). Together, they describe the appearance of distant objects as smaller, less distinct, paler, and bluer. For a good example of this, look for Leonardo’s The Virgin of the Rocks, which he painted around 1508." Calter, Squaring the Circle, p. 385.
1 J. V. Field, The Invention of Infinity: Mathematics and Art in the Renaissance, (Oxford, Oxford, 1999), 1.
2 Note 2.