Here is the so-called Brera Altarpiece, painted by Piero della Francesca, showing the condottiere Federico da Montefeltro in adoration of the Virgin and Child, surrounded by saints. Piero della Francesca is famous for his writings on perspective and solid geometry, but this composition was actually governed by a simple two-dimensional geometrical scheme whose logic would have been familiar to any compass-wielding Gothic architect or early medieval craftsman. To see this, start by considering a few details of the painting. First, its sides have been slightly trimmed, and the elbows of John the Baptist at left, and of Andrew at right, thus appear slightly truncated. Second, these two saints hold their attributes in ways that serve as geometrical markers, which is particularly conspicuous in the case of the Baptist’s staff. Both the staff and Andrew’s book are aligned at 22.5 degrees from the vertical.
Together they thus describe the converging sides of a triangle with an interior angle of 45 degrees, which can be understood as one eighth of a regular octagon. The baseline of this triangle likely corresponds to the original width of the panel, as the red verticals here indicate. A related construction correctly gives the height of the panel.
Here, at the top of the image, five facets of an orange octagon are centered on the apex of the red wedge and framed by the red verticals. The upper corners of the octagon’s lateral facets locate the top edge of the panel. In the bottom half of the image, an orange circle fits within the square left over beneath the octagon’s lateral facets, and a square can be inscribed within that circle. Notice that the vertical facets of the square locate the corners of the pilasters on the wall behind the saints, and that its top facet locates the bottom edge of the pilaster capitals.
Next, another quadrature step inwards can be performed by inscribing a yellow circle within the square, and a yellow square within the new circle. Note that the Virgin’s blue cloak just fills the arc of the yellow circle, that the right lower corner of the yellow square locates Federico’s elbow, and that the left yellow vertical defines the midline of Saint Jerome’s bifurcated garment.
One large green triangle, sharing its baseline with the yellow square, has its upper vertex in the Virgin’s left eye socket, since she inclines her head slightly to one side as she gazes at her child. The child’s body lies along of 30-degree slope departing from Federico’s elbow, which also locates the elbow of the angel to the left of the Virgin. The comparable point on the right locates the hand of Saint Francis. In the upper section of the panel, meanwhile, octature creates a new green horizontal just above the orange one, by inscribing arcs within the lateral wedges of the orange octagon. This new green horizontal matters because it serves as the geometrical baseline for the painting’s upper architectural structure. It coincides with the top edge of the entablature on either side of the main niche.
By constructing nested blue octagons and semicircles centered on this level and framed by the orange uprights, moreover, both the location and the thickness of the arch over the main niche can be determined. The large blue arc circumscribing the orange octagon and sweeping through the Virgin’s face also locates the eye level of most of the saints. That relationship is only approximate, but the bottom point on this arc also serves quite precisely as the point of convergence for the perspectival system defined by the converging lines of the entablatures, as the blue raking lines along the entablatures indicate.
This painting is the property of the Pinacoteca di Brera in Milan, Italy. This analysis was based on the image: