First Published on in Thursday, 24 May 2012 14:22 where it was read 362 times

In working at Lochmariaquer, an early discovery has returned in the form of a near-Pythagorean triangle with sides 18, 19 and 6. But first, how did this work on cosmic N:N+1 triangles get started?

Robin's earliest work couched the Lunation Triangle within three right angled triangles that could easily be constructred yet describe the number of lunar months and orbits in the solar year and the length of the eclipse year, using the number series 11, 12, 13, 14 to form N:N+1 triangles. Each triangle could then have an intermediate hypotenuse set at the 3:2 point of the shortest side, so as to form the eclipse year (11.37 mean solar months) and solar year (12.368 in lunar months), plus the orbits in a solar year (13.368 orbits). The 12 length is the lunar months in a lunar year but also the mean solar months in a solar year and the length 13 is the length of another type of lunar year (in lunar months) and the number of orbits in a 12 month lunar year. A bit of a mouthful so I have made a diagram of them below in figure 1.

Fig1 Copy of RobinsThreeTriangles

Figure 1 Robin Heath's original set of three right angled triangles that exploit the 3:2 points to make intermediate hypotenuses so as to achieve numerically accurate time lengths in units of lunar or solar months and lunar orbits.

It came to me that the 18-19-6 triple was interesting in that the amount by which it deviated from the Pythagorean rule (whole numbers for all three sides) causes the 18 side to be 18.027 long, so that the triangle is then still a right angled triangle. This number (18.027) was familiar to me as being the length in years of the Saros eclipse period, of 18.030 solar years. It seemed wonderful, as if nature was somehow shaping reality to suit a near-Pythagorean triangle. It was also interesting because the Saros is 19 eclipse years, by definition, so the base of the triangle can be metrological shifted to units in the ratio solar year to eclipse year, so that both the longer sides are then, each 19 (different) units long, eclipse years on the base and solar years on the hypotenuse. 

Fig2 Copy of TRIANGLE 19 6 18 027

Figure 2 The near Pythagorean triple 6-18-19 can have the 18 side equal 18.027 so as to restore the triangle as right angles. This length is very close to the Saros period in solar years, since 19 eclipse years and 223 lunar months long are 18.030 solar years long.

This triangle soon became "eclipsed" by new progress, when Robin and I, one Christmas (1993 or 4), found the single unit that divides into both the eclipse and solar years, revealing that the moon's nodal period, the solar year and the eclipse year are normalised to the tropical day, through be rate the eclipse nodes move per day.

The number of days it takes for the lunar nodes to move one DAY, in angle, (the angle the sun moves in a single solar day) is 18.618 days. This makes the eclipse year equal 18.618 x 18.618 days, the solar year 19.618 x 18.618 days, the difference between the years then being 18.618 days and the nodal period being 18.618 solar years and 19.618 eclipse years long. (This was further explored in Robin Heath's Wooden Book called Sun, Moon and Earth)

Out of this relationship comes a VERY important triangle found at Le Menec and Mane Lud/Locmariaquer, with longer sides 18.618 and 19.618 and third side 6.18.

Fig3 triangle 18 19 618

Figure 3 The Node Day of 18.618 days provides a normalising unit for the Solar and Eclipse year, meaning that 18.618 days equals the one difference between the two types of year.

Obviously, this is just larger than the 18-19-6 triangle mentioned above and its sharp angle is 18.36 instead of 18.40 degrees.

The third triangle that is similar is that produced by the diagonal of a triple square and this similarity was understood by Robin through his work on that geometry known through Alexander Thom as a Type B flattened circle - a circle whose perimeter has been reduced in length through using arcs or variable radius.

Fig4 Copy of Figure 2 7 TypeB

Figure 4 How the geometry of Le Manio could be linked to the triple square and the flattened Type B ring design. The green dotted lines show the Vesica of overlapping circles, to be found in the design which would then be two eclipse years long in day-inch counting.

Robin noted the proximity to an 18.618:19.618 triangle, implicit within these designs, though it is actually the angle of the diagonal of a triple square, which is 18.43 degrees and only 1 1/2 minutes of a degree different to the angle of an 18.027:19:6 triangle. In Carnac, a study group in Carnac area called AAK published site plans in which multiple squares were found to define many monuments and in the last decade, a past member of that group Howard Crowhurst has published further developments of the idea. For me, this connected the lengths of time as lengths with the megalithic construction of some triple squares most notably at Locmariaquer and four-squares at Le Manio (and Locmariaquer).

Fig5 triplesquare Saros Metonic

Figure 5 The diagonal of the Triple Square again provides a similar triangle relating the time periods of the solar and eclipse year, for example between the Metonic period of 19 years and the Saros period of 19 eclipse years. The subsequent apparent use of triple squares around Carnac can then be interpreted unsing day-inch counting as being  practiced there.

The above three triangles of figures 2, 3, and 5, being similar and near congruent, the fact that the Tumulus d'Er Grah ended after a Saros period of day-inch counting from Er Grah means that a point 19 years of day-inch counting directly north of Er Grah would be 6 years of day-inch counting away from the (original but now sadly altered) northern tip of the Tumulus d'Er Grah. The significance of this is brought out in the dolmen of Mane Lud where, observing the possible parallelism noted in Sacred Number and the Lords of Time, the north end of the Tumulus d'Er Grah corresponds with the end of the west passageway.

Fig6 ManeLud dolmen TimeMap SAROS

Figure 5 The Tumulus of Er Grah can be seen as either of the three triangles because these are very similar. They might all have been known (to the builders) and each can provide an different way of solving problems in construction or in interpretation. For example, the 19-6 triangle enables the length of the eclipse year to be estimated as 346.55 days using a geometrical method, the eclipse year being  346.62 days. However, three square construction can be "grown" along the 18.4 degree vector to maintain a similar anglular bearing. The civilization that built Locmariaquer probably had at least a thousand years to have developed their practices and understanding. 

There are obviously further questions about the whole matter that await some practical measurements. However, by  exploring what the astronomical and geometrical facts are and by accepting day-inch counting as the first feasible means, for the monuments around Locmariaquer to have been built the way they are, we become capable of glimpsing, for the first time, our own intellectual birth as connected to the cosmos that gave birth to us.