Ad Quadratum is a convenient and profound technique in which continuous scaling of size can be given to square shapes, either from a centre or periphery. The differences in scale are multiples of the square root

of two [sqrt(2)] between two types of square: cardinal (flat) and diamond (pointed).

When a square’s diagonal touches opposite sides sides of a larger square, the two squares differ by sqrt(2) and a square becomes a diamond or visa versa. By repeating this effect a continuum can be generated to form the large patterns found in sacred buildings such as the Vedic Angkor Wat, Christian Gothic cathedrals, Islamic Dome of the Rock and even Stonehenge. The technique is therefore very ancient yet still used. Subjectively such buildings express a high degree of visual order by employing what is called a geometrical progression. Objectively, such patterns appear to convey cosmic principles known to ancient builders but largely forgotten recently.

It is easiest, when maintaining a common centre, to define the outer square and then scale inwards, as was probably done at Ankor Wat (and at Stonehenge, where the Aubrey Holes existed before the Sarsen Circle).

Within the Indian traditions there are many texts defining *yantra *patterns in which square temples have their sides divided by small numbers, and their internal areas are then the squares of those side numbers. Typically 3 (9 in area), 4 (16), 5 (25), 6 (36), etc., also give a design able to define local units of measure used to measure them. Having identified an ad quadratum structure, it is important to find out how the local units of length correspond to its nested squares.

Whilst the inner core of Angkor Wat was square, employing ad quadratum in its design, the outer square was extended to the west so that the overall building was then rectangular, as below.

Some rectangles can be seen to express a rational ratio between their sides. The rectangle of Angkor Wat shares one side length with the outer square and, measuring the longer side results in the ratio 1.125 or 9 to 8 (9/8) which is a whole tone. As many of my other articles and books show, 9/8 is the counted time-length relationship between the lunar year and the synod of the planet Jupiter. This might be the reason why Angkor Wat was given this rectangular form, in which case the outer square form of the ad quadratum would symbolize the lunar year.

A following post (link to come) will investigate further the meaning of this ratio in Angkor Wat.

#### see also

STONEHENGE: ASPECTS OF AD QUADRATUM GEOMETRY

Author(s): Rory Fonseca

Source: Journal of Architectural and Planning Research, Vol. 12, No. 4 (Winter, 1995), pp.357-365