Rabatment Plus
Toward the end of the Medieval period rabatment armatures became more complex. One development was the use of a compass to determine the height of an altarpiece's gable. Cimabue's Virgin and Child Enthroned, with Prophets, below, was so well designed and executed that it can serve as a template for this type of rabatment armature.
Cimabue, Virgin and Child Enthroned, with Prophets (Santa Trinita Maesta), c. 1290-1300, tempera and gold leaf on wood, 384 x 223 cm, Uffizi.
The overlap between the two squares of the rabatment (identified by the diagonals) is exactly one-half of their height. As a result, the rabatment divides the format into three equal sections — one at the top, one in the middle, and one on the bottom. Since these sections are exactly twice as wide as they are high, bisecting them vertically produces a grid of six perfect squares. Bisecting each of these secondary squares vertically and horizontally produces a grid of twenty-four tertiary squares. To create the gable, the compass point is placed on the outer edge of the format at this tertiary grid's topmost horizontal division. The compass arm is extended across the format in a horizontal line and its point is placed on the opposite edge. Swinging an arc from both of these positions establishes the height of the gable, as shown.
A few years earlier Duccio created a similar armature for one of his enthroned Madonnas. Because of the format's proportions, the tertiary grid has only twenty squares and the rabatment doesn't extend to the bottom. But the height of the altarpiece's gable height has been determined in the same way as Cimabue determined his a few years later.
Duccio, Rucellai Madonna, c. 1285, tempera and gold leaf on wood, 450 x 290 cm, Uffizi.
A Spanish altarpiece, below, from the same period deviates from the standard rabatment formula. Its armature has two secondary rabatments, one at the top of the rectangular format and one at the bottom. There is no primary rabatment. (Overlapping squares the width of the format don't produce the divisions seen in the altarpiece.) The two secondary rabatments divide the format horizontally into three equal sections. These sections are approximately a third as high as they are wide, so they cannot be used to generate a grid of squares. Instead, the altarpiece's creator has divided them unequally to make a column for Saint Christopher that is wider than the two columns that flank him.
Saint Christopher altarpiece, late 13th century, tempera on wood, 275 x 189 cm, Prado.
The gable height of the Saint Christopher altarpiece may have been determined by extending the center column into the gable. Or, it could have been derived from a vesica piscis. (To see the entire vesica piscis, enlarge the armature by clicking on it.)
Giotto's The Death of the Virgin, below, was painted early in the fourteenth century, ten or twenty years after Cimabue's Virgin and Child Enthroned. The format is horizontal and extremely wide, too wide for a standard rabatment — four squares are needed. The center squares overlap.
Giotto, The Death of the Virgin, c. 1310, tempera on poplar panel, 76 x 180 cm, Gemaldegalerie, Berlin.
As in the previous armatures, the gable height was probably determined by a compass, as shown.
At the same time Giotto was painting The Death of the Virgin in Florence, Duccio was working on the Maesta altarpiece in Siena. The surviving portion of the front,6 shown below, is twice as wide as it is high.
Duccio, Maesta, 1308-11, tempera and gold on wood, 213 x 396 cm, Museo dell'Opera Metropolitana del Duomo, Siena.
This is the only armature that I've been able to discover that accounts for all of the painting's major spatial elements. To account for the line that divides the prophets' portraits at the top from the saints and angels below them, it is necessary to increase the height of the format.7
To place the throne, Duccio appears to have placed three squares across the bottom of the format. Their area of overlap is shown in orange.
The angels leaning on the throne have been arranged in a circular band. The outer circle is half the width of the format and centered in the portion of the altarpiece that has survived. The inner circle's radius is the distance between the circle's midpoint and the point that establishes the inside top of the throne.
To plan and execute a composition that is this symmetrical, complex, and methodically organized, an armature like this is necessary.

6 You probably know that the Maesta altarpiece was sawn apart in 1711. Pieces of it are in museums and private collections in Europe and the U.S., and some of the pictures are missing. It has only been reconstructed as photographic montages. The main portion of the front, shown in the photograph above, is still in Siena Cathedral.
7 To define the column in which the Madonna appears, a standard rabatment of the format shown in the armature seems to be necessary. The addition is the same size as the band of prophets and immediately above it. It changes the format's width-to-height ratio from 2:1 to 1⅔:1. Of course, it's possible that this line is unrelated to the rest of the painting's geometry — that Duccio placed it where he did for aesthetic or practical reasons. But since everything else in this painting has a geometric underpinning, I think it's reasonable to assume that this line has one, too.