Story of Three Similar Triangles

first published on 24 May 2012,

Figure 1 Robin Heath’s original set of three right angled triangles that exploited 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.

Interpreting Lochmariaquer in 2012, an early discovery was of a near-Pythagorean triangle with sides 18, 19 and 6. This year (2018) I found that triangle as between the start of the Erdevan Alignments near Carnac. But how did our work on cosmic N:N+1 triangles get started?

Robin Heath’s earliest work, A Key to Stonehenge (1993) placed his Lunation Triangle within a sequence of three right-angled triangles which could easily be constructed using one megalithic yard per lunar month. These would then have been useful in generating some key lengths proportional to the lunar year:  

  • the number of lunar months in the solar year,
  • the number of lunar orbits in the solar year and 
  • the length of the eclipse year in 30-day months. 

all in lunar months. These triangles are to be constructed using the number series 11, 12, 13, 14 so as to form N:N+1 triangles (see figure 1).

n.b. In the 1990s the primary geometry used to explore megalithic astronomy was N:N+1 triangles, where N could be non-integer, since the lunation triangle was just such whilst easily set out using the 12:13:5 Pythagorean triangle and forming the intermediate hypotenuse to the 3 point of the 5 side. In the 11:12 and 13:14 triangles, the short side is not equal to 5.

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 lunar 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

Robin’s triplet of triangles was a fantastic early result in geometry that really did not prove to be useful or likely. There is no need to have linked triangles and the megalithic appears to have found other ways to geometrically link and compare different counted lengths. It is interesting also that day-inch counting was not seen as the initial technique leading to such geometries in the megalithic, until around 2008. Whilst counted lengths were frequently interpreted within monuments, the idea that they were relics of actual time counts using constant lengths was overlooked and very hard for me or Robin to see, the Le Manio Quadrilateral proving a decisive confirmation.

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, if the triangle is to be right angled. This number (18.027) was familiar to me as being the length in solar years of the Saros eclipse period, of 18.030 solar years. This 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. This is to be found in the angle of the Erdevan Alignments near Carnac see [post2post id=”402″]).

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 itself became eclipsed by new progress. One Christmas (1993 or 1994), Robin and I found the single unit that divides into both the eclipse and solar years, later revealing that the moon’s nodal period, the solar year and the eclipse year are normalised to the tropical day, through the rate by which the eclipse nodes move slowly retrograde. (the nodes are where the orbit of the moon crosses the sun’s path in the year, due to which eclipses can only occur when the sun is on one of these two nodes.)

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, exactly because the difference between these two types of year is 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, or see Figure 7 for an early summary) Because of this the eclipse year is the square of 18.618 or more properly, the square root of the eclipse year is 18.6177. 

Out of this relationship comes the VERY important triangle found at Le Menec (the angle of its Alignments) and Mane Lud/Locmariaquer, with longer sides 18.618 and 19.618 and a third side 6.184.

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 less, 18.36 instead of 18.40 degrees.

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

Figure 4 How the four-square 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 that, implicit within the Type B design, an 18.618:19.618 triangle can be inferred, 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.

Within the last decade, Howard Crowhurst  discovered a whole system of megalithic alignments, using multiple squares, between sites near Carnac (see his Megalithes book, 2007). I could see this probably also applied to time measures, as lengths within megalithic constructions directly using triple squares (figure 5), most notably at Locmariaquer (figure 6).

The above three triangles of figures 2, 3, and 5, are similar and near congruent and yet their smallest angles are slightly different and this may be measurable to say which geometry was used.

Figure 5 The diagonal of the Triple Square again provides another 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.

If the Tumulus d’Er Grah ended after a Saros period of day-inch counting from Er Grah this would mean that a point 19 years of day-inch counting lay directly north of Er Grah and it would be 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 my Sacred Number and the Lords of Time (pages 56-63), the north end of the Tumulus d’Er Grah (the broken grand menhir) corresponds with the end of the west passageway.

Figure 6 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 angular bearing

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, we become capable of glimpsing, for the first time, the earliest use of geometry as a tool.

Figure 7 A 1995 list of 18.618 factored data sent as a postcard by Robin Heath.
The silence is still deafening. 

2 thoughts on “Story of Three Similar Triangles”

    1. I thought you meant the 6 side of the near-Pythagorean {6 18 19} triangle since the anomoly divides into 18 years to give the Saros of very similar eclipses.

      However, you are pointing elsewhere.
      To the anomalistic month (A) of 27.554551 days and draconic month (D) of 27.21222 days.
      Six years is 2191.4532 days. Divided by A = 79.53144 and divided by B = 80.5319, one different so that, as a triangle, it is normalized with a ratio between the months if 1.01258, or 81/80 (the syntonic comma) to one part in 12,649!
      An excellent dehydrated tip-off.

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