A series of articles have built up aspects of Locmariaquer's megaliths with an eye to three interpretive themes,

1. The use of day-inch counting as a natural proto-metrology.
2. The possible use of the northern horizon to align with circumpolar stars.
3. The megalithic technique exploiting the properties of multiple squares.

The last article, about the 3, 4 and 13 squares relationship, strayed upon a staggering possibility: that the Locmariaquer monuments Er Grah, its unusual tumulus, and the Tumulus of Mane Lud, could represent a day-inch count of all the astronomical cycles discovered by the astronomers working between 4700 and 4000 BCE, up to the solar hero time frame of thirty three years for a sunrise, on a particular day of the year, to repeat perfectly on the horizon. A total grasp of the sun and moon and eclipses appears to have been represented through this monumental complex. If so then the day-inch counting is crucial to its explanation.

The key length of 17 megalithic yards, found in the radius of Le Menec's western cromlech, is to be found three squares up of side length 4 eclipse years in day-inch counting, this side length equalling the cromlech radius, and the diagonal achieving 12 solar years and landing within the top stone of the Tumulus d'Er Grah. This implies a connections between Er Grah and Le Menec's ability to count the Octon period of 4 eclipse years and about 47 lunations, 1/5th of the Saros period's 235 lunations and also count half a nodal period of 3400 days across the cromlech's diameter, when counted in megalithic-inches. At Le Menec, 17 megalithic rods of 82 inches can create a 365 unit circumference, of units equal to 24 inches, to enable sidereal time measurement based upon a marker star, in this case Alkaid, and with Le Menec Dubhe (either end of Ursa major).

The principle of multiple squares can be found fully in use throughout most of my Locmariaquer articles and found to be more interesting when the side lengths can be associated with astronomical day-inch counting as well as the needs of megalith builders in laying out their alignments between backsights, menhirs, dolmen, horizon features and horizon events.

Many more interesting discoveries and co-incidences can be found through this work through their combination the interpretation of the monuments also becoming clearer for the non-specialist.

THIS ARTICLE IS UNDER DEVELOPMENT incl. ILLUSTRATIONS

In a previous article, some of the multiple square combinations, of lower number and found around the Carnac area, were explored. These miraculous mathematical anomolies, born within the number field, must have combined in the minds of Carnac's astronomers with the astronomical co-incidences true at their latitude, co-incidences that the sun and moon, in their extremes, follow the diagonals of a single-square, a double-square and a 3x4 rectangle, relative to east.

There are many different articulations possible for each of the totally exact multi-square relationships, but they all hinge on a relation such that the sum or difference of two inverse tangents equals another inverse tangent, that is two of the angles of these multiple square rectangles diagonals exactly equal a third diagonal angle. Sometimes this is achieved by aligning one rectangle upon another's diagonal.

The next most significant relationship occurs between a four-square rectangle, a three-square rectangle and their difference in diagonal angle, which is that of a thirteen-square rectangle. This relationship has an astronomical utility, but first of all the archetypal form of this relationship is probably as below.

Figure 1 The simple and exact relationship between four-square and three-square diagonals in the diagonal angle of thirteen-squares

It is immediately interesting to think of the four-square geometry as it is articulated at Le Manio for its baseline represents the lunar year of twelve lunar months whilst its diagonal represents the solar year as a day-inch count (and formerly, in the work of Robin Heath, his "lunation triangle" was calibrated in lunar months using megallithic yards to obtain 12.368 months for the solar year.) There is also a prior connection with thirteen for the triple square since its diagonal, relevant to the moon's nodal cycle, can represent a lunar year of 13 lunar months when the baseline is the day-inch count for the solar year. The red arrow below shows how an arc from 12 equals the baseline of the four square when its diagonal length (the solar year) equals that for the triple square (also the solar year but in the context of the nodal cycle rather than the lunation relationship).

Figure 2 It is possible to geometrically equate the four-square diagonal and three-square baseline, as being the solar year, relative to the lunar years of 12 and 13 months, so as to reveal a cosmological correspondance between this multiple square relationship and the structure of time on earth.

This literal correspondance between nodal and lunational invariances of sun, moon and lunar nodes is tantamount to a design or, forbidding such a thing, then perhaps it could be called a geometrical representation of a genuine "strange attractor" built into the sun-moon-earth triad. The diagram in figure 2 demonstrates a geometrical link between four types of year, the eclipse year, the 12 and 13 month lunar years and the solar year inbetween these lunar years. This is specifically marked in figure 3 below as length between point A and points B, C, D and E.

Figure 3 If one were to use the day-inch count for the solar year as a shared length AE, between a triple-square and a four-square, then one could automatically generate the other three types of year, eclipse and 12 and 13 lunar month years, as day-counts on the ground.

The eclipse year can be generated from a perpendicular to the main triple-square's diagonal, by growing a diagonal from E. Note that then, this diagonal is oriented like the tumulus of Mane Lud relative to a 33 year day-inch count extending from Er Grah (Grand Menhir) at point A and point E, directly north. Line AD is then 33 times 13 lunar months. The dolmen of Mane Lud lies at a bearing 360 - 14 degrees = 346 degrees, the angle from north of a four-square diagonal, whilst the dolmen includes the point thirty-three eclipse years from Er Grah as a day-inch count.

It therefore appears that
Mane Lud is a 33 solar year implementation
of figure 3, constructed over six thousand years ago

Figure 4 The Parallelism between the Er Grah - Mane Lud complex and Mane Lud's Dolmen is now even more interesting since 33 eclipse years has been arced to locate the dolmen on the diagonal of a 13-square rectangle relative to the triple-square's diagonal, relative to north to have a bearing 14 degrees west of north from Er Grah. (based upon figure 3 of Metrological Temple to the Solar Hero)

Note that the Tumulus d'Er Grah now has a meaning as representing the Saros period of 223 lunar months or just over 18 years. The Saros is the best eclipse period over which any eclipse of one character will recur in a very similar way. At this point on the dolmen's scale map of the complex, the passageway opens into the chamber where one is standing beyond the Metonic/ Saros period of 18 to 19 solar years but within the Solar Hero's 33 years.

To close this article, which started innocently as an investigation of the 3, 4, 13 squares geometry, I present an integration of the 5-12-13 triangle, the second-simplest whole-number side triangle, whose longest sides express the numbers 12 and 13 that are the two types of lunar month, either side of the solar year. It looked like another "marriage made in heaven", with 13-squares bridging the gaps between both four and three squares AND three squares to 5-12-13, but the latter arrangement is not fully perfect with tan^-1(1/13) being just 13 minutes of a degree too large.

Figure 5 Thirteen squares can almost bridge between a triple-square and a 5-12-13 triangle as it perfectly does between the four-square and the three-square.