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The Square
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Egyptian grid
"Wooden drawing board: this object consists of a rectangular wooden board covered with a thin layer of plaster. Originally a squared grid, ruled in red, covered the whole of one side of the board, the other side is blank. The grid still remains on the left half of the inscribed surface, where a seated figure of a king has been drawn. At some time, the grid was erased from the right side." British Museum
"Egyptian linear measuring systems were based on idealized human proportions . . ." Burrows, 3D Thinking in Design and Architecture, pp. 75-76. Even so, they used a grid of squares to map those proportions.
Wooden drawing board with a figure of Thutmose III, 1479-1425 BC, gesso and paint(?) on wood, 38 x 53.4 cm, British Museum.
Rabatment
"Rabatment has been proposed as an explanation and origin for several Renaissance [perspective] techniques. In each case I believe that the invocation of rabatment is anachronistic and that the nature of the explanation is demonstrably at odds with the methods themselves." Elkins, The Poetics of Perspective, 1994, p. 276.
Sacred cut
"This name was coined by the Danish Engineer Tons Brunés, in his book The Secrets of Ancient Geometry and Its Use. In that book he claims the sacred cut is found in the layout of many ancient building, including the Parthenon. . . Brunés calls this construction sacred because it contains both square and circle, uniting the earthly and the divine as in the Vitruvian man. Furthermore, it squares the circle. The length of the four arcs equal the four diagonals of the half-square. And, as mentioned, it gives the octagon, the shape universally used for baptistries and baptismal fonts." (Calter, “Ad Quadratum, the Sacred Cut, and Roman Archtitecture,”, but cite Brunés.)
"There is evidence that the Sacred Cut has been used as a foundation for architectural designs. Brunés myriad examples include the Great Pyramid of Khufu, the Parthenon, and the Pantheon (Brunés 1967: vol 1, 123-147; vol 1, 301-310; vol 2, 38-56); more recent analyses involve the layout of a Roman housing complex in Ostia, and the Baptistery of San Giovanni in Florence (Watts and Watts 1986; Williams 1994)." (Wassel, “Art and Mathematics Before the Quattrocento: A Context for Understanding Renaissance Architecture,” 2015: 68.)
Left: Construction of the sacred cut. Right: Extension of the sacred cut.
In the construction, there are two proportionally related squares. In the extension, there is an additional proportional square – the outer square – established by the outermost intersections of the circles and the diagonals.
The construction of the sacred cut is done with a straightedge and compass as follows: Draw a square1 and its diagonals, then place the compass leg on each of the square's four corners and draw an arc through its center. Draw lines through the square where the arcs intersect with its sides – two horizontal and two vertical. The small square that is thereby created in the center (shown here in pink) is called the sacred cut square.
If the side length of the original (or reference) square is 1, then the side length of the sacred cut square is √2 - 1, or approximately .414. "The construction can be extended inward, by repeating the construction on the sacred cut square. It can also be extended outward [the figure on the right], joining the intersections of the circles and the diagonals, to form a square of which the original square is the sacred cut square." (Calter, “Ad Quadratum, the Sacred Cut, and Roman Archtitecture.”) By means of these extensions and subdivisions, a series of proportionally related squares can be created.
"The Sacred Cut is perhaps the first component in the long history of using ad quadratum relationships in architectural design." (Wassel, “Art and Mathematics Before the Quattrocento: A Context for Understanding Renaissance Architecture,” 2015: 68.) Unless the Egyptians were using ad quadratum proportions – check on this.

1 The construction of a square is shown in the Constructions appendix.

Sections
The Square
The Square
Introduction