One of the key objections for the megalithic building flattened circles concerned their use of ropes and a knowledge of geometry in managing radii to achieve a lesser circumference than a circle would have. If instead Thom's Type A or Type B flattened circles were constructed using a grid of squares, then some of the all-important key points where a flattened circle's radius of curvature changes (of which there are only four) should be points of intersection within the grid intersections. It became clear that this was a possible alternative means to their production when considering the Type A geometry and specifically its implicit pair of triple-square triangles, as right triangles available within such a grid.
Robin Heath has already noted [in Sun Moon and Earth, p52-55] that these triangles are close to the invariant ratio, in their longest sides, of
- the eclipse year and solar year, and
- this same ratio is also to be found between the solar year and the thirteen lunar month year.
The baseline of such a right triangle is found to be 6/7 of the diameter MN of the Type A flattened circle and this implies, given the left-right symmetry of this form, that the key point at the end of the hypotenuse (where the radius of curvature changes) would sit on the corner of a grid point of 14 by 14 squares, as a length then equal to twelve grid units. The forming circle used by Thom, of diameter MN, would then inscribe the grid square.
Figure 1 Type A drawn on a 14 square grid
We also know, from Carnac, that the astronomers used a triple square to frame this right triangle so as to relate the periods of eclipse and solar year. Since the vertical position of the key point is 12 units, then to left and right the key points either end of the central flattened arc are 4 units, either side of the central axis. Therefore, to right and left of these triple squares can be found two four-square rectangles, whose diagonals express (with an accuracy better than a day count could could) the relationship of the lunar year (side length = 4) to the solar year (as hypotenuse/diagonal). These four squares (each 3 by 3 = 9 grid squares) have a baseline of twelve grid squares which exactly matches the number of lunar months within the lunar year.
One therefore sees useful megalithic "resources" within such a 14-square grid in that many multiple squares can be formed; such as these triple squares either side of the vertical centreline have two four-square rectangles to the right and left (shown in red below, the ripple-squares being blue). These leave a row of 14 by 2 squares at the top which can be seen as a seven-square, the rectangle whose diagonal to side alignment is found between a double and a triple square: These include triple and four square rectangles which give good approximations in their ratios, between diagonals and longer side lengths, which can be used as calendric devices for lunar to solar year, eclipse to solar year and solar to 13 month year.***
***This habit, around megalithic Carnac, of "finding" right triangles within multiple squares corresponds to the astronomical reality whilst enabling accurate generation of these counted lengths without any day counting of periods; once the triple square and four square were discovered to be "cosmograms"
The other two points at which a Type A's radius of curvature changes, lies a further two grid squares from the central axis, but falls exactly half way; along the vertical edge of a grid square. To achieve a grid in which these two key points would also be commensurate, the number of squares in the grid needs to be doubled to 28, or so it would seem. But in practice the metrology of a grid's side lengths would have ready made subdivisions, especially by half, and so one comes to the question of how early metrology defined units of length.
Figure 2 Some of the multiple squares present within the grid.