Piero della Francesca and the use of geometry in his art This paper takes a look at the art work of Piero della Francesca and, in particular, the clever use of geometry in his work; there will be a diagram illustrating this feature of his work at the end of this essay. To begin, the paper will explore one of the geometric proofs worked out in art by Piero and, in the process of doing so, will capture his exquisite command of geometry as geometry is expressed – or can be expressed – in art.
By looking at some of Piero’s most noteworthy works, we also can see the skilful geometry behind them. For instance, the Flagellation of Christ is characterized by the fact that the frame is a root-two rectangle; significantly, Piero manages to ensure that Christ’s head is at the center of the original square, which requires a considerable amount of geometric know-how, as we shall see. In another great work, Piero uses the central vertical and horizontal zones to symbolically reference the resurrection of Christ and also his masterful place in the hierarchy that distinguishes God from Man.
Finally, Bussagli presents a sophisticated analysis of Piero’s, Baptism of Christ that reveals the extent to which the man employed different axes in order to create works that reinforced the Trinitarian message of the scriptures. Overall, his work is a compelling display of how the best painting inevitably requires more than a little mathematics.
Piero is noteworthy for us today because he was keen to use perspective painting in his artwork.
He offered the world his treatise on perspective painting entitled, De Prospectiva Pingendi (On the perspective for painting). The series of perspective problems posed and solved builds from the simple to the complex: in Book I, Piero introduces the idea that the apparent size of the object is its angle subtended at the eye; he refers to Euclid’s Elements Books I and VI (and to Euclid’s Optics) and, in Proposition 13, he explores the representation of a square lying flat on the ground before the viewer. To put a complex matter simply, a horizontal square with side BC is to be viewed from point A, which is above the ground plane and in front of the square, over point D. The square is supposed to be horizontal, but it is shown as if it had been raised up and standing vertically; the construction lines AC and AG cut the vertical side BF in points E and H, respectively.
BE, subtending the same angle at A as the horizontal side BC, represents the height occupied by the square in the drawing. EH, subtending the same angle at A as the far side of the square (CG) constitutes the length of that side of the square drawn. According to Piero, the artist can then draw parallels to BC through A and E and locate a point A on the first of these to represent the viewer’s position with respect to the edge of the square designated BC. Finally, the aspiring artist reading Piero’s treatise can draw A’B and A’C, cutting the parallel through E at D’ and E’. Piero gives the following proof in illustrating his work: Theorem: E’D’ = EH. This simple theorem is described as the first new European theorem in geometry since Fibonacci (Petersen, para.8-12). It is not for nothing that some scholars have described Piero as being an early champion of, and innovator in, primary geometry (Evans, 385).
The Flagellation of Christ is a classic instance of Piero’s wonderful command of geometry at work. Those who have looked at this scrupulously detailed and planned work note that the dimensions of the painting are as follows: 58.4 cm by 81.5 cm; this means that the ratio of the sides stands at 1.40 ~ 21/2. If one were to swing arc EB from A, one ends up with a square (this will all be illustrated at the very end of this paper in the appendices). Thus, to cut to the core of the matter, the width of the painting equals the diagonal of the square, thereby verifying that the frame is a root-two rectangle. Scholars further note that the diagonal, AE, of the square mentioned above passes through the V, which happens to be the vanishing point of perspective. Additionally, in square ATVK we find that the arc KT from A cuts the diagonal at Christ’s head, F, halfway up the painting; this essentially means that Christ’s head is at the center of the original square, (Calter, slide 14.2). A visual depiction of the geometry of the Flagellation of Christ is located in the appendices of this paper.
Paul Calter has provided us with some of the best descriptions of how Piero cleverly uses geometry to create works of enduring beauty, symmetry and subtlety. He takes a great deal of time elaborating upon Piero’s Resurrection of Christ (created between 1460-1463) in which Piero employs the square format to great effect. Chiefly stated, the painting is constructed as a square and the square format gives a mood of overall stillness to the finished product. Christies located exactly on center and this, too, gives the final good a sense of overall stillness.
The central vertical divides the scene with winter on left and summer on the right; clearly, the demarcation is intended to correlate the rebirth of nature with the rebirth of Christ. Finally, Calter notes that horizontal zones are manifest in the work: the painting is actually divided into three horizontal bands and Christ occupies the middle band, with his head and shoulders reaching into the upper band of sky. The guards are in the zone below the line marked by Christ’s foot (Calter, slide 14.3). In the appendix of this paper one can bear witness to the quiet geometry at play in the work by looking at the finished product.
One other work of Piero’s that calls attention to his use of geometry is the Baptism of Christ. In a sophisticated analysis, Bussagli writes that there are two ideal axes that shape the entire composition: the first axis is central, paradigmatic and vertical; the second axis is horizontal and perspective oriented. The first one, according to Bussagli coordinates the characters related to the Gospel episode and thus to the Trinitarian epiphany; the second axis indicates the human dimension – where the story takes place – and intersects with the divine, as represented by the figure of Christ. To elaborate on the specifics of the complex first axis, Bussagli writes that Piero placed the angels that represent the trinity, the catechumen about to receive the sacrament, and the Pharisees on the perspective directed horizontal axis (Bussagli, 12). The end result is that the Trinitarian message is reinforced in a way that never distracts or detracts from the majesty of the actual composition.
To end, this paper has looked at some of Piero Della Francesca’s most impressive works and at the astounding way in which Piero uses geometry to impress his religious vision and sensibilities upon those fortunate enough to gaze upon his works. Piero had a subtle understanding of geometry and geometry, in his hands, becomes a means of telling a story that might otherwise escape the notice of the casual observer. In this gentleman’s work, the aesthetic beauty of great art, the penetrating logic of exact mathematics, and the devotion of the truly committed all come together as one.
Source: Calter, Paul. “Polyhedra and plagiarism in the Renaissance.” 1998. 25 Oct. 2011 http://www.dartmouth.edu/~matc/math5.geometry/unit13/unit13.html#Francesca
Appendix B: visual illustration of the Resurrection of Christ [pic]
Source: Source: Calter, Paul. “Polyhedra and plagiarism in the Renaissance.” 1998. 25 Oct. 2011 http://www.dartmouth.edu/~matc/math5.geometry/unit13/unit13.html#Francesca
Bussagli, Marco. Piero Della Francesca. Italy: Giunti Editore, 1998. Calter, Paul. “Polyhedra and plagiarism in the Renaissance.” 1998. 25 Oct. 2011 http://www.dartmouth.edu/~matc/math5.geometry/unit13/unit13.html#Francesca Evans, Robin. The Projective Cast: Architecture and its three geometries. USA: MIT Press, 1995. Petersen, Mark. “The Geometry of Piero Della Francesca.” Math across the Curriculum. 1999. 25 Oct. 2011 http://www.mtholyoke.edu/courses/rschwart/mac/Italian/geometry.shtml
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