above: the dolmenA chamber made of vertical megaliths upon which a roof or ceiling slab was balanced. of Pentre Ifan (wiki tab)
In previous lessons, fixed lengths have been divided into any number of equal parts, to serve the notion of integer fractions in which the same length can then be reinterpreted as to its units or as a numerically different measurement. This allows all sorts of rescaling and exploitation of the properties of integer numbers.
Here we present a megalithic method which extended two or more fixed bearings (or alignmentsIn general, to the sun and moon on the horizon, rising in the east or setting in the west.
Also, a name special to Carnac's groups of parallel rows of stones, called Le Menec, Kermario, Kerlescan, and Erdevan.), usually based upon a simple geometrical form such as a triangle or a rectangle. This can be how the larger geometries came to be drawn on the landscape (here called landforms) of separated megalithsStructures built out of large little-altered stones in the new stone age or neolithic between 5,000-2,500 (bronze age), in the pursuit of astronomical knowledge. and natural features which appear to belong together. For example,
Outliers: Alexander ThomScottish engineer 1894-1985. Discovered, through surveying, that Britain's megalithic circles expressed astronomy using exact measures, geometrical forms and, where possible, whole numbers. found that British stone circle were often associated with single outliers (standing stones) on a bearing that may correspond to horizon event but equally, appears to give clues to the metrologyThe application of units of length to problems of measurement, design, comparison or calculation. of the circle in the itinerary length to the outlier from the circle’s centre.
Figure 1 Stone circle plans often indicate nearby outliers and stone circles
Stone circles were also placed a significant distance and bearing away (figure 1), according to geometry or horizon events. This can be seen between Castle Rigg and Long Meg, two large flattened circles – the first Thom’s Type-A and the second his Type-B.
Figure 2 Two large megalithic circles appear linked in design and relative placement according to the geometry of the double squareA unit rectangle of 1 by 2, with important use for alignment (Carnac), cosmology (Egypt) and tuning theory (Honnecourt Man)..
Expanding geometrically
The site plan of Castle Rigg (bottom left, fig. 1) can have the diagonal of a double square (in red) emerging between two stones which then bracket the chosen direction. This bearing could be maintained by expanding the double square so that west-to-east and south to north expand as the double and single length of a triangle while the hypotenuse then grows towards the desired spot according to a criteria such as, a latitude different to that of Castle Rigg. That is, at any expansion the eastings and northings are known as well as the distance between the two circles while the alignments, east and northeast in this example, are kept true by alignment to previous established points. Indeed, one sees that the small outlier circle of Long Meg, to Little Meg beyond, was again on the same diagonal bearing, according to the slope angle of the cardinal double square.*** One can call this a type of projective geometry.
***This extensive double square relation between megalithic sites was first developed by Howard Crowhurst, in Ireland between Newgrange and Douth in same orientation as figure 2, and by Robin HeathEngineer, teacher and author, who discovered the Lunation Triangle (c. 1990), that enabled the lunar year to be rationally related to the solar year. During the 1990s we collaborated to further understand the astronomical and numerical discoveries of the megalithic astronomers. at https://robinheath.info/the-english-lake-district-stone-circles/.
It seems impossible for such arrangements to have been achieved without modern equipment and so the preference is to call these landforms co-incidental. But, by embracing their intentionality, one can see a natural order between Castle Rigg and, only then, Long Meg’s outlying Little Meg circle, and through this find otherwise hidden evidence of the working methods in the form of erratics or outliers, whose purpose is otherwise unclear.
Equilateral Expansion
The work of Robin Heath in West Wales can be an interesting challenge since not all the key points on his Preseli Vesica are clearly megalithic, perhaps because megaliths can be displaced by settlements or be subsumed by churches, castles and so on. (see Bluestone Magic, chapter 8). First, for completeness, how is a vesica defined today? In his classic Sacred Geometry, Robert Lawlor explains the usual construction and properties of the vesica :
Draw the major and minor axes CD and AB. Draw CA, AD, DB and BC. By swinging arcs of our given radius from either centre A or B we trace along the vesica to points C and D, thus verifying that lines AB, BC, CA, BD and AD are equal to one another and to the radius common to both circles.
We now have two identical equilateral triangles emerging from within the Vesica Piscis. Extend lines CA and CB to intersect circles A and B at points G and F. Lines CG and CF are diameters of the two circles and thus twice the length of any of the sides of the triangles ABC and ABD. Draw FG passing through point D.
Sacred Geometry by Robert Lawlor
Primitive versus later geometry
Lawlor’s presentation have the triangles appearing as the conjuction of two circles and their centers. However, the points and lines of modern geometry translate, when interpreting the megalithic, into built structures or significant features, and the alignments which may join them. The alignments are environmental and in the sky or landscape.
- A is Pentre Ifan, a dolmen dating from around 3500 BC.
- B is located in the Carningli Hillfort, a mess of boulders below the peak Carningli (meaning angel mountain). Directly East,
- C is the ancient village, church and castle of Nevern.
- D is a recently excavated stone circle, third largest in Britain at around 360 feet diameter, but now ruinous, call Waun Mawn.
The two equilateral triangles have an average side length around 11,760 feet but, as drawn, each line is an alignment of azimuththe standard angular measurement of angles on the horizon, measured clockwise from zero degrees true North. 330, 0, 30, and 90 degrees and their antipodes.
The Constructional Order
Relevant here is how one would lay out such a large landformThose geometrically connected buildings, natural features or horizon events, necessary for quantifying astronomy, measuring the landscape or, in later times, connecting communities. and we will illustrate how this would be done using the method of expansion.
North can be deduced from the extreme elongation of the circumpolar stars in the north, since no pole star existed in 3200BC. At the same time it is possible to align to plus and minus 30 degrees using Ursa Major. This would give the geometry without the geometry so to speak, since ropes 11760 feet long are unfeasible. It seems likely that the Waun Mawn could function as a circumpolar observatoryA geometrical arrangement suited to the tracking the motion of circumpolar stars, around the north pole, during the night. (as appears the case at Le Menec in Brittany, see my Lords of Time).
If the work was to start at Carningli fort, then the two alignments (a) east and (b) to Waun Mawn could be expanded in tandem until the sides were 11760 feet long, ending at the circle to the south and dolmen to the east. The third side between these sites should then be correct.
The vesica has been formed to run alongside the mountain. The new eastern point is a dolmen that points north to another dolmen Llech-y-Drybedd on the raised horizon, itself a waypoint to Bardsey Island.
The reason for building the vesica appears wrapped up in the fact that its alignments are only three, tightly held within a fan of 60 degrees pointing north and back to the south. But the building of the double equilateral cannot be assumed to be related to the circular means of its construction given by Lawlor above. That is, megalithic geometry did not have the same roots as sacred geometry which has evolved over millennia since.