Before, during and after Sacred Geometry

above: Carreg Coetan Arthur portal dolmen in Newport, Pembrokeshire.

The prehistory of sacred geometry was the late stone age, when the stone circles, dolmens, and long alignments to astronomical events on the horizon, used megaliths (large stones) in geometrical ways. Their geometries served their quest to understand the heavens, without telescopes or arithmetic, by using counted time periods as geometrical lines, squares and circles. Geometry, supplemented by the days counted between alignment events, was therefore a prelude to sacred and then secular geometry.

By developing early geometrical methods, they forged an enduring cultural norm lasting millennia, as part (or not) of the more-familiar aspect of the neolithic, innovating an agricultural pastoralism, that could support settlements, cities and, only then, the great civilizations of the middle and far east. It was civilization that generated our earliest written histories; these still powering our historical context and leading the basic notion of economic progress and territorial expansion, as superior to all that went before.

Our surviving megaliths are hence deeply enigmatic, a mysterious and mute presence in a world far less mysterious. The megaliths may have something we have forgotten in a collective way, something pushed out by millennia of later ideas and now relatively recent ones too.

There seems little trace of the megalithic astronomers themselves, their geometricized landscape overlaid by our notions of a primitive Stone Age.  And, as the prelude to world history, their geometry gave birth to sacred geometry and sacred buildings; pyramids, ziggurats, temples and religious complexes. In some way, therefore, geometry obtained its sacredness from the skies or the earth itself, as if these had been built from the harmonious organization of the solar system seen from Earth and given to it by one or more gods or angels.

Sacred geometry the became a secular and analytical geometry, which would become an encyclopedic exploration of all that geometry could do, rather than a set of techniques dreamt up by a band of roaming astronomers. In our schools, many lose interest in having to learn geometry in the abstract and so, in this, the megalithic had an advantage. They could learn geometry as and when they needed it, as their astronomy brought up new questions to solve, learning by finding methods to answer questions.

If one truly travels backwards in time, to discover what the megalithic astronomers had understood, I believe one has to decide which bits of your own skills have to be applied to solve the riddles of the megalithic mind. Each modern researcher must not assume the megalithic could calculate using numbers, use trigonometry, knew Pythagoras’ theorum, and so on. And yet, one can employ modern equipment to help investigate the megalithic. Google Earth, for example, can allow megalithic alignments to be studied, their azimuth, length and interrelation, whilst the context of sites can be seen that may provide clues not available in site plans, written descriptions and so on, which are sometimes difficult to obtain or require a personal expedition. The most basic tool for me has been the Casio scientific calculators, since the megalithic interaction with space (geometry) was blended with the interaction of numerical time counting, numbers which exist in the geocentric world of time.

Finally, one must realise the past is only in the present through our attention to it and, in the absence of much official interest in applied geometry, dimensionality and astronomical intent of the sites, it is left to non-specialists to become new specialists in the sense of recovering and conserving the true achievements of the megalithic, for our present age, while the monuments still exist as living mysteries. In this I advocate the path leading to what this website is about.

from Book 5: Harmonic Origins of the World

Intelligent Star Systems

The harmony of the spheres can only be found in our world of time, where it is a strong and compelling phenomenon. Such a harmony was no prescientific fantasy. Pythagoras, who coined the term, probably did so based on the geocentric time world, a view lost to history apart from cryptic references that can no longer be interpreted.

In our age of system science, musical harmony is not thought relevant to the design of dynamic systems such as the planets, yet they appear adapted to just intonation seen from the exclusive perspective of our planet. Why should our planet have a harmonious view of time, and what difference does time’s harmoniousness make to life on Earth? Is there some other purpose to this harmony or none at all? To answer such questions one has to recognize just intonation as being a holistic system that demands human insight into the nature of whole phenomena (a so-called gestalt). Such gestalts flow from the need to see higher-level relationships rather than the raw complexity of their parts. All higher structures of meaning subsume lower levels of meaning.  For example, microclimates are a structuring of meaning higher than  trees, water, weather, and topography, usefully integrating these parts within a newly perceived whole. Such insights reveal a higher idea that indicates new potentials within a system. The new level of conceptual order has not changed in the phenomenon but how we relate to it. This profound faculty is the basis of what we call understanding rather than knowing, and it enlarges our “world.” The world is already structured, and a sensory insight re-creates that structure as a simplifying aspect, already present, to expand the intelligibility of the sensory world and with it, our present moment. Insight and the world’s creation were considered similar acts within ancient cosmologies, in that an insight about the world resembles the structure of the world as it would be conceived by any god in the act of creating it. Such a vision involves a special effort but provides a creative view of the world, in which simplicity and relatedness replace functional complexity with a new appreciation of the sensory world. The celestial behavior in Earth’s skies is a prime example of such an action: the rotation of Earth, its orbit around the sun, the moon’s orbit, and its illumination by the sun complicate the observed orbital periods of the other planets and yet, that added complexity has produced harmonic simplicity between synodic periods!

Chapter 1 showed how Late Stone Age astronomers used geometrical counts of synodic periods to discover this harmony of the spheres, which modern astronomers have not seen because scientific calculation methods deal instead with planetary dynamics modeled by equations. Simplicity has somehow adapted our solar system without breaking physical laws. At the level of gravitational dynamics, many complexities were required to achieve just intonation seen only from Earth, especially the lengthening of the lunar month as an intermediary to the planetary synods seen from Earth. Any demiurgic preference for harmony (seen from Earth) resembles the human gestalt that revealed the harmony of the spheres to human sensory intelligence in the Late Stone Age, and it must be noted, humanity has become demiurgic since the Stone Age, creating man-made worlds.

Demiurgic intelligences are probably part of each star system and, if our star has a demiurgic intelligence, this action seems to have used the moon to establish a justly intoned time world for the third planet. It adapted the unchanging orbital pitches of an n-body planetary system to present harmonic synodic systems that planetary orbital periods alone could never express. Our geocentric system is harmonically founded between 1, the zeroth power of 2 (the Saturn synod) and the fifth power of 60 (YHWH, as 365-day year), which is the smallest numerical resolution to contain just intonation of both inner and outer planets, as in the implied holy mountains of our ancient texts.

Harmonic Origins of the World
Contents (272 pages, 100 b&w illustrations)
Preface
Introduction: The Significance of Planetary Harmony (5)
PART 1: RECOVERING LOST KNOWLEDGE OF THE WORLD SOUL
1 Climbing the Harmonic Mountain (20)
2 Heroic Gods of the Tritone (19)
3 YHWH Rejects the Gods (15)
4 Plato’s Dilemma (22)
PART 2: A COSMICALLY CREATIVE HARMONY
5 The Quest for Apollo’s Lyre (25)
6 Life on the Mountain (23)
PART 3 THE WAR IN HEAVEN
7 Gilgamesh Kills the Stone Men (16)
8 Quetzalcoatl’s Brave New World (31)
9 YHWH’s Matrix of Creation (19)
10 The Abrahamic Incarnation (15)
Postscript: Intelligent Star Systems
APPENDIX 1: Astronomical Periods and Their Matrix Equivalents
APPENDIX 2: Ancient Use of Tone Circles (11)
Notes
Bibliography
Index

Gavrinis R8: Diagram of the Saros-Metonic Cycle

The Saros cycle is made up of 19 eclipse years of 364.62 days whilst the Metonic cycle is made up of 19 solar years of 365.2422 days. This unusually small number of years, NINETEEN, arises because of a close coupling of most of the major parameters of the Earth-Sun-Moon system which acts as a discrete system, a system also commensurate with Jupiter, Saturn, Uranus and Venus. It is this type of coherent cyclicity which lies at the centre of what the megalithic were able to achieve through day-inch or similar counting of visible time periods and comparing of counts using geometric means. [see my books, especially Sacred Number and the Lords of Time, for a fuller discussion].

It would have been relatively easy for megaithic astronomy to notice that eclipses occur in slots separated by eclipse seasons of 173.3 days and also to see that the difference between lunar and solar years resolves over the 19 year of the Metonic so that lunar orbits, lunar months, the starry sky and the rotation of the earth provide a close repetition of alignments over 19 solar years which equal 235 lunar months and 254 lunar orbits. The Saros period is 223 lunar months long and is therefore one lunar year of 12 months short of the Metonic of 235 lunar months.

The situation in the last year of the Metonic is therefore identical but (symmetrically) in-reverse to the first year, on a continuous but discrete basis [that is, providing you start counting on an eclipse]. The Saros then ends12 months before the Metonic so that the Saros is 18 solar years long plus, quite closely, the 10.8 day difference between the lunar and solar years. This phenomenon is clearly presented on Gavrinis’ stone R8, in the middle “register”, such engraved art at Gavrinis dividing their stone pallettes into different elements of a related summary of astronomical phenomena seen through the tools of a megalithic science involving counting, alignment, geometry, and metrology.

Central section of Gavrinis stone R8 clearly shows the Saros and Metonic Cycles as ending between 18 and 19 years less the difference between the lunar and solar years

 In the figure above, the right shows the four-square geometry whose diagonal is the length of the solar year relative to the length of the base (=4) as being the length of the lunar year of 12 lunar months. The difference in length of these two years is shown “centre stage” and is accurately 10.8 inches long, numerically representing the difference in terms of day-inch counting. The curvilinear lines around the vertical are emblematic of counting as fundamental to this type of art. The diagonal actually shown here is continued into the representation of a series of solar years, here numbered so that, in the 19th year, something new happens: the year rises up but is bent leftwards in what is one of the most distinctive patterns in Gavrinis’ art.

We know, as stated above, that the Metonic is 19 years long and that the Saros is a year less, plus the 10.8 day difference between lunar and solar years, so that the 10.8 day-inches is shown centrally above on R8 both refers to the initial four-square relating the solar and lunar year, by the excess then found over 18 years, of 10.8 day-inches.

This is a very compact and intuitive diagramming language which communicates, without words but with an implicit familarity of day-inch counting, an inter-related cyclicity of crucial importance discoverable using this megalithic science. Similar components are to be found on other stones and astronomy appears to be the purpose of this notational art, designed to educate and explain important facts, within an oral megalithic culture.

What stone L9 might teach us

image of stone L9, left of corridor of Gavrinis Cairn,
4Km east of Carnac complex. [image: neolithiqueblog]

This article was first published in 2012.

One test of validity for any interpretation of a megalithic monument, as an astronomically inspired work, is whether the act of interpretation has revealed something true but unknown about astronomical time periods. The Gavrinis stone L9, now digitally scanned, indicates a way of counting the 18 year Saros period using triangular counters  founded on the three solar year relationship of just over 37 lunar months, a major subject (around 4000 BC) of the Le Manio Quadrilateral, 4 Km west of Gavrinis. The Saros period is a whole number, 223, of lunar months because the moon must be in the same phase (full or new) as the earlier eclipse for an eclipse to be possible. 

On the roof with Anthony Blake (left) on the DuVersity Albion Tour, in August 2004.

Handling the Saros Period

223 is a prime number not divisible by any lower number of lunar months, such as 12 in the lunar year. 18 lunar years equates to 216 lunar months, requiring seven further months to reach the Saros condition where not only is the lunar phase the same but also, the sun is sitting upon the same lunar node, after 19 eclipse years of 346.62 days.

However, astronomers at Carnac already had a number of 37 lunar months (just less than three solar years) in their minds and, it appears, they could apply this as a length 37 units long, as if each unit was a lunar month. We also know that the unit they used for counting lunar months was originally 29.53 inches (3/4 metre) or later, the megalithic yard. Visualising a rope of length 37 megalithic yards, the length can be multiplied by repeating the rope end-to-end. After six lengths, 222 or 6*37 lunar months were represented, one lunar month less than the 223 lunar months which define the Saros period.

Figure 1 The near-integer Anniversary of Lunar Months over Three Years

This six-fold use of the number 37 appears to be used within the graphic design of Gavrinis stone L9 (see figure 2), as the triangular shape which has an apex angle of 14 degrees and which refers to the triangle formed at Le Manio between day-inch counts over three solar and three lunar years. It appears that this triangular shape was used to refer to the counting of solar years relative to a stone age lunar calendar (see 2nd register of stone R8) but it could also have the numerical meaning of 37 because three solar years contained 37 whole lunar months just as a single solar year contains 12 whole lunar months (the lunar year).

I believe this triangle, already symbolic of 37, appears in pairs within stone L9, as a single counter showing two axe heads, their points adjacent so that they have one side also adjacent. The two triangles are found to be held accurately within the apex angle of another triangle, known to be in use at Carnac, the triangle with side lengths 5-12-13, with apex angle 22.6 degrees. These pairs would then effect the notion of addition so that each is valued at 37 + 37 = 74 lunar months.

Figure 2. The use of two three-year triangles, made to fit within the 5-12-13 triangle to form a single counter worth 74 lunar months. (MegalithicScience.org eventually became this website)

All of the three pairs have this same apex angle, of the 5-12-13 triangle, chosen perhaps because 12+12+13 = 37 whilst the 14 degree triangle was known to be rationally held within it when the 12 side is seen as the lunar year of 12 months. The third side is then 3 lunar months long (¼ lunar year) forming an intermediate hypotenuse within a 5-12-13 triangle, which is equal to the 12.368 months of the solar year. Robin Heath first identified the smaller triangle when studying the properties of the 5 by 12 rectangle of Stonehenge’s Station Rectangle, arguably made up of two 5-12-13 triangles joined by their 13 sides. Three solar years then seems to have become associated with the pattern 12+12+13 (= 37) by the historical period, since Arab and medieval astronomers came to organize their intercalary months within the Callippic cycle of 4 Metonic periods (= 4 x 19 years equaling 76 solar years).

Figure 3. The quantification of the Saros as 18 solar years and 11 days equal to 223 lunar months. The language of days and years at Gavrinis might well have been the primary perception of light and dark periods.

The Saros period of 223 lunar months then also appears indicated on stone L9, below these triangles, within the main feature of this stone, a near-square Quadrilateral having one right angle. It has a rounded top, containing a wavy engraved design emanating from a central vertical, not unlike a menhir. The waves proceed upwards but then narrow to a vestigial extent after the 18th, which would be one way to symbolise the Saros period as 18 years and eleven days in duration. A different graphical allusion was used on stone R8, again showing lines as years but giving the 19th year as a shortened “hockey stick”.

Conclusions

In Gavrinis stone L9, a “primitive” numerical and phenomenological symbolism appears to have expressed a useful computational fact: that the Saros period was one lunar month more than six periods of 37 lunar months. These three periods of 37 months were shown as blade shapes, each symbolising three solar years, but shown as pairs within three 5-12-13 triangles above a quadrilateral shape indicating 18 wavy lines plus a smallest period, this symbolising the 11 days over 18 years of the Saros Period, defined by 223 lunar months. This allowed the Saros to be seen as six periods of 37 lunar months, equal to 222, plus one lunar month. Once the count reached 222, attention to the end of the next lunar month would be key. This enabled a pre-arithmetic culture to approach prime number 223 from another large prime (37) which was nearly expressed by 3 solar years, then repeated six times yo become 222 lunar months. This same counting regime appears to have been employed elsewhere:

  1. Astronomical Rock Art at Stoupe Brow, Fylingdales.
  2. Eleven Questions on Sacred Numbers.
  3. Counting lunar eclipses using the Phaistos Disk.

Many thanks to Laurent Lescop of Nantes University Architecture Dept,
for providing the scan on which this work is based.

Astronomy 3: Understanding Time Cycles

above: a 21-petal object in the Heraklion Museum which could represent the 21 seven-day weeks in the 399 days of the Jupiter synod. [2004, Richard Heath]

One of the unfortunate aspects of adopting the number 360 for calibrating the Ecliptic in degrees is that the megalithic counted time in days and instead saw the ecliptic as divided by the 365¼ days. In transferring to the number 360, with all of its easy factors, 8 x 9 x 5, moderns cannot exploit a key advantage of 365¼ days.

If the lunar orbit takes 27.32166 days then each day the moon moves by 1/27.32166 of the ecliptic every day. For this reason, after 27.32166 days the orbit completes because the Moon’s “year” then equals one as the angular motion has been 27.32166/ 27.32166 = 1.

The same is true of the lunar nodes, which retrograde to the east along the ecliptic in 18.618 years. For this reason one can say, the lunar nodes move by 1/18.618 DAYS (in angle) every day and to travel one DAY in angle, the nodes take 18.618 DAYS per day (needing the new term “node day” equal the 18.618 days.*** A solar year takes 19.618 node days (since 365¼ equals 18.618 x 19.618) and an eclipse year takes 18.618 x 18.618 – 346.62 days

*** These are average figures since the moon comes under variable gravitational influences that are episodic.

A general rule emerges in which the larger, whole cycles, lead to reciprocals which can be numerically characterized by knowing the number of the days in the larger period.

For instance, Jupiter has a synodic excess over the solar year of 398.88 days and this means its angular motion is 1/ 398.88 DAYS per day while Saturn’s synod is 378.09 days and its angular motion is 1/ 378.09 DAYS per day. These synods are, by definition, differential to the Sun at 1/ 365.2422 DAYS per day.

Without seeing astronomy as calibrated to day and year cycles, one is robbed of much chance to appreciate the megalithic view of time and the time-factored buildings that came to be built in pursuit of quite advanced knowledge.

Looking from the relatively large cycles to the extremely small, daily angular changes of celestial bodies seen from Earth, reveals a further obscuration created, in this case, by the heliocentric view of the solar system, rather than the geocentric view which is obviously founded on days and years seen from the surface of the planet.

The largest cycle the megalithic could see using their techniques, reverses the direction from large-to-small to small-to-large since the precessional cycle (of the equinoctal nodes of the earth’s obliquity) is around 25,800 ± 100 years long. A star or constellation on the ecliptic appears to move east, like the lunar nodes, and using the angular measure of DAYS, it is possible to estimate that the equinoctal points move by a single DAY, in a given epoch, something like 71 years. The precessional cycle is therefore 71 years multiplied by the 365.2422 DAYS of the whole ecliptic.

The most important benefit of using DAY angles is that knowledge of a few celestial periods opens up a realm in which different scales of time can be derived from first principles. And added to that, the celestial periods appear related to one another so that so-called sacred numbers emerge such as the seven day week which divides into both the Saturn synod (54 weeks), Jupiter synod (57 weeks), the 364 day saturnian year (52 weeks) and others.

To understand the full scope of megalithic astronomy requires a geocentric calibration of the ecliptic as having 365¼ angular DAYS.

Geometry 7: Geometrical Expansion

above: the dolmen 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 alignments), 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 megaliths and natural features which appear to belong together. For example,

Outliers: Alexander Thom 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 metrology 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 square.

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 Heath 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 :

Drawing 2.3. Geometric proof of the √3 proportion within the Vesica Piscis. from Sacred Geometry by Robert Lawlor.

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 azimuth 330, 0, 30, and 90 degrees and their antipodes. 

The Constructional Order

Relevant here is how one would lay out such a large landform 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 observatory (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.

Figure 3 Proposed use of equilateral expansion from Carnigli fort to both what would become the dolmen of Pentre Ifan (az. 90 degrees) and Waun Mawn (azimuth 150 degrees).

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.