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.
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.
What really happens when Earth turns? The rotation of Earth describes periods that are measured in days. The solar year is 365.242 days long, the lunation period 29.53 days long, and so forth.
Extracted from Matrix of Creation, page 42.
Earth orbits the Sun and, from Earth, the Sun appears to move through the stars. But the stars are lost in the brightness of the daytime skies and this obscures the Sun’s progress from human view. However, through observation of the inexorable seasonal changes in the positions of the constellations, the Sun’s motion can be determined.
The sidereal day is defined by the rotation of Earth relative to the stars. But this is different from what we commonly call a day, the full title of which is a tropical day. Our day includes extra time for Earth to catch up with the Sun before another sunrise. Our clocks are synchronized to this tropical day of twenty-four hours (1,440 minutes).
The Sun circumnavigates the zodiac in 365 tropical days, within which 366 sidereal days have occurred. There is one full Earth rotation more than there are sunrises within a year. This hidden oneness within the year is recapitulated in the one-unit difference between the number of sidereal days and the number of tropical days in a practical year.
The small catch-up time in every day is about three minutes and fifty-six seconds long. This unit defines not only a sidereal day with 365 such units but also the practical year of 365 tropical days. The catchup unit is the difference between the duration of a sidereal day and that of a tropical day. It relates the Sun’s daily motion to the rotation of Earth and is a fundamental unit of Earth time (figure 3.6).
Figure 3.6. A polar view of Earth’s equator showing sunrises for two consecutive days. Compared with clock time, the stars rise three minutes and fifty-six seconds earlier each evening. (Drawn by Robin Heath)
THE MOON GATHERS THE TEN THOUSAND WATERS
The sidereal day (the duration of one rotation of Earth) is a very significant cosmic unit. The Jupiter synodic period of 398.88 tropical days is within 99.993% of four hundred sidereal days long. Therefore, twenty-five Jupiter synods (365 lunar orbital periods) equal 10,000 sidereal days since four hundred times twenty-five is 10,000.
A sidereal day differs from a tropical day due to the motion of the Sun during one tropical day. The three-minute-and-fifty-six-second time difference between these two days, the aforementioned catch-up unit, is quite useful when applied as the unit to measure the length of these days. A tropical day has 366 of these units while the sidereal day has 365 of the same units. The difference between the two is one unit.
Since 365 lunar orbits equal 10,000 sidereal days, it follows that a single lunar orbit has a duration of 10000/365 sidereal days. There are 365 units in a sidereal day, and therefore 10,000 units in a lunar orbit, so this new unit of time is 1/10000 of a lunar orbit. One ten-thousandth of a lunar orbit coincidentally is three minutes and fifty-six seconds in duration. The proportions in the Jupiter cycle combine with the lunar orbit, solar year, and Earth’s rotation to generate a parallel number system involving the numbers 25, 40, 365, 366, 400, and 10,000.
This daily catch-up unit I shall a chronon. Its existence means that the rotation of Earth is synchronized with both the lunar orbit and the Jupiter synodic period using a time unit of about three minutes and fifty-six seconds.
The sidereal day of 365 chronons is the equivalent of the 365-day practical year, the chronon itself is equivalent to the sidereal day, and so on. The creation of equivalents through exact scaling enables a larger structure to be modeled within itself on a smaller scale. This is a recipe for the integration of sympathetic vibratory rhythms between the greater and the lesser structures, a planetary law of subsumption.
The exemplar of the chronon was found at Le Menec: It’s egg-shaped western cromlech has a circumference of 10,000 inches and, if inches were chronons (1/365th of the earth’s rotation), then the egg’s circumference would be the number of chronons in the lunar orbit of 10,000. Dividing 10,000 by 366 (the chronons in the tropical day) gives a lunar orbit of 27.3224 – accurate to one part in 36704! The forming circle of Le Menec’s egg geometry provided a circumpolar observatory of circumference 365 x 24 inches, which is two feet per chronon versus the chronon per inch of the egg as lunar orbit.
The quantified form of the Le Menec cromlech was therefore chosen by the builders to be a unified lunar orbital egg, with a forming circle represented the rotation of the Earth at a scaling of 1:24 between orbital and rotational time.
The form of Le Mence’s cromlech unified the 10,000 chronon orbit of the Moon and 365 chronon circle of the Earth rotation because Thom’s Type 1 geometry naturally achieved the desired ratio. When the circle’s circumference (light blue) was 24 x 365 inches there were 10,000 inches on the egg’s. Underlying site plan by Thom, MRBB.
This design is further considered in Sacred Number and the Lords of Time, chapter 4: The Framework of Change on Earth, from the point of view of the cromlech’s purpose of providing a working model of the lunar orbit relative to the rotation of the circumpolar sky, leading to the placement of stones in rows according to the moon’s late or early rising to the East.
Capturing Sidereal Time
We can now complete our treatment of Carnac’s astronomical monuments by returning to Le Menec where the challenge was to measure time accurately in units less than a single day. This is done today at every astronomical observatory using a clock that keeps pace with the stars rather than the sun.
The 24 hours of a sidereal clock, roughly four minutes short of a normal day, are actually tracking the rotation of the Earth since Earth rotation is what makes all the stars move. Even the sun during the day moves through the sky because the Earth moves. Therefore, in all sidereal astronomy, the Earth is actually the prime mover. The geometry of a circumpolar observatory can reveal not only which particular circumpolar star was used to build the observatory but also the relatively short period of time in which the observatory was designed. Each bright circumpolar star is recognizable by its unique elongation on the horizon in azimuth and its correspondingly unique and representative circumpolar orbital radius in azimuth. …
The knowledge that was discovered due to the Le Menec observatory is awe inspiring when the perimeter of the egg shape is taken into account. It is close to 10,000 inches, the number of units of sidereal time the moon takes to orbit the Earth. The egg was enlarged in order to quantify the orbit of the moon as follows: every 82 days (three lunar orbits) the moon appears over the same part of the ecliptic. Dividing the ecliptic into sidereal days we arrive at 366 units of time per solar day.*
*These units are each the time required for an observer on the surface of the Earth to catch up with a sun that has moved within the last 24 hours, on the ecliptic, a time difference of just less than four minutes.
82 days times 366 divided by the three lunar orbits gives the moon’s sidereal orbit as 122 times 82 day-inches. Instead of dividing 82 by three as we might today to find the moon’s orbit, the pre-arithmetic of metrology enabled the solar day (of 366 units) to be divided into three lengths of 122. If a rope 122 inches long is then used 82 times (a whole number), to lay out a longer length, a length of 10,004 inches results. If 10,004 is divided by 366 units per day then the moon’s orbit emerges as 82/3 or 27⅓ days.
If a moon marker is placed upon the Le Menec perimeter and moved 122 inches per day, the perimeter becomes a simulator of the moon. … Knowing the moon’s position on the western cromlech’s model of ecliptic and knowing which parts of the ecliptic are currently rising from the circumpolar stars enabled the astronomers to measure the moon’s ecliptic latitude.
Hence the phenomena related to the retrograde motion of the lunar orbit’s nodal period could be studied and its 6800 day length.
about how the cardinal directions of north, south, east and west were determined, from Sacred Number and the Lords of Time, chapter 4, pages 84-86.
Away from the tropics there is always a circle of the sky whose circumpolar stars never set and that can be used for observational astronomy. As latitude increases the pole gets higher in the north and the disk of the circumpolar region, set at the angular height of the pole, ascends so as to dominate the northern sky at night.
Northern circumpolar stars appearing to revolve around the north celestial pole. Note that Polaris, the bright star near the center, remains almost stationary in the sky. The north pole star is constantly above the horizon throughout the year, viewed from the Northern Hemisphere. (The graphic shows how the apparent positions of the stars move over a 24-hour period, but in practice, they are invisible in daylight, in which sunlight outshines them.) [courtesy Wikipedia on “circumpolar star”, animation by user:Mjchael CC-ASA2.5]
Therefore, the angular height of the pole at any latitude is the same angle we use to define that latitude, and this equals the half angle between the outer circumpolar stars and the pole itself. For example, Carnac has a latitude of 47.5 degrees north so that the pole will be raised by 47.5 degrees above a flat horizon, while the circumpolar region will then be 95 degrees in angular extent.
It is perhaps no accident that the pole is called a pole since to visualize the polar axis one can imagine a physical pole with a star on top, like a toy angel’s wand. The circumpolar region is “suspended” around the pole like a plate “held up” by the pole. Therefore, a physical pole, set into the ground, can be used to view the north pole from a suitable distance south (i.e., with the pole’s top as a foresight for the observer’s backsight). Such an observing pole would probably have been set at the center of a circle drawn on the ground, representing the circumpolar region around the north pole. This arrangement, a gnomon,* existed throughout history but usually presented as part of a sun dial.
*According to the testimony of Herodotus, the gnomon was originally an astronomical instrument invented in Mesopotamia and introduced to Greece by Anaximander. It was innovated even earlier, in the megalithic period, because structures that could operate one still exist within megalithic monuments.
It now appears a gnomic pole was also used in prehistory to locate the north pole in the middle of circumpolar skies. The north pole is opposite the shadow of the equinoctal sun at midday. The gnomic pole could also be used to find “true north,” as located halfway between the extremes of the same circumpolar star above the northern horizon. This can make use of the fact that when the sun is at equinox, it lies on the celestial equator and therefore is at a right angle to the north pole (see figure 4.4). This right angle is expressed at the top of the gnomic pole used and hence can enable the alignment of the pole through the similarity (or congruence) of all the right-angled triangles within the arrangement.
Figure 4.4. From pole to pole. It is possible to determine the angle of the north pole using a gnomic pole as shadow stick, but only at noon on the equinox. The laddie on the left cannot do this to a possible few minutes of a degree, but the geometry of the stick and shadow length can, providing true north and equinox alignments of east and west can be determined. Illustration on left from Robin Heath, Sun, Moon and Stonehenge, fig. 9.3.
Figure 4.4. From pole to pole. It is possible to determine the angle of the north pole using a gnomic pole as shadow stick, but only at noon on the equinox. The laddie on the left cannot do this to a possible few minutes of a degree, but the geometry of the stick and shadow length can, providing true north and equinox can be determined. Illustration on left from Robin Heath, Sun, Moon and Stonehenge, fig. 9.3.
To achieve an accurate bearing to true north, a circumpolar observatory can use the gnomic pole method, not just at noon on the equinox but every night, by dividing the angular range of circumpolar star, in azimuth. The north pole’s altitude, known to us as latitude on the Earth, can then be identified by dividing the angular range in altitude of a circumpolar star, a task achievable through geometry and metrology, so as to create a metrological model of the latitude upon the Earth.
It would seem obvious today that the pole star Polaris could have been used, but this is a persistent and widespread misunderstanding of the role of pole stars within the ancient and prehistoric world. Epochs in which there is a star within one degree of the pole are very rare and shortlived. Our pole star, Polaris (alpha Ursa Minor), is currently placed two thirds of a degree from a pole that is moving through the northern sky (figure 4.3 on p. 83) in a circle around the ecliptic pole of the solar system. Polaris will be nearest the pole (about ½ degree) at the end of this century.
The last time there was a star at all near the north pole was prior to the construction of the Great Pyramid in 2540 BCE . That pole star was Thuban (Alpha Draconis), and it was just one-fifteenth of a degree from the pole in 2800 BCE . The Great Pyramid has a narrow air shaft that pointed to Thuban, at which time it was already departing the pole and nearly one-third of a degree from it. Therefore, the megalithic people at Carnac, as well as almost all cultures throughout time, did not have the convenience of a pole star in approximately locating the north pole.
From 5000 to 4000 BCE , the time of megalithic building at Carnac, the north pole was a dark region surrounded by many bright stars. The inability to locate the north pole using a pole star challenged the people of the megalithic to develop a more sophisticated and accurate method. In any case, true north and latitude needed to be located more accurately than by using a pole star, which can only ever approximate the position of the north pole. Also, through the circumpolar observatory, sidereal time and even longitude between sites could be measured once the movement of the circumpolar stars could be exploited. True north, based upon these stars around the pole, can give the cardinal directions to an observatory.
The equinoctial sunrise in the east and sunset in the west can give a mean azimuth (horizon angle) to obtain south and north but only if the horizon is dead flat to each of these alignments. Far better then to observe the extremes of motion of a single circumpolar star, to the east and the west, to then find the North pole in between those two alignments.
Defining North by bringing the “clock in the sky” down to earth
One can therefore see that the circumpolar stars and sighting techniques, involving a gnomon, allowed north and the cardinal directions in a more reliable way than recording sunrise and sunsets since the sun on the horizon is variable between years due to the solar year having nearly ¼ day more than 365 days. The circumpolar stars enabled buildings and long sights to be built to true north-south-east-west, and by ignoring this, a building such as the Great Pyramid of Giza surprises us with its accurate placement relative to the cardinal directions.
This paper proposes that an unfamiliar type of circumpolar astronomy was practiced by the time Le Menec was built, around 4000 BCE. This observatory enabled the rotation of the earth and ecliptic location of eastern and western horizons to be known in real time, by observing stellar motion by night and solar motion by day. This method avoided stellar extinction angles by measuring the circular motion of a circumpolar marker star as a range in azimuth, which could then be equated with the diameter of a suitably calibrated observatory circle. The advent of day-inch counting and simple geometrical calculators, already found at Le Manio’s Quadrilateral, enabled the articulation of large time periods within Carnac’s megalithic monuments, the Western Alignments being revealed to be a study of moonrises during half of the moon’s nodal period. Le Menec’s Type 1 egg is found to be a time-factored model of the moon’s orbit relative to the earth’s rotation. This interpretation of Le Menec finds that key stones have survived and that the gaps seen in the cromlech’s walls were an essential part of its symbolic language, guiding contemporary visitors as to how its purpose was to be interpreted within the pre-literate megalithic culture.
Two key lengths are found at Le Manio and Le Menec: The first, of 4 eclipse years is a day-inch count of the Octon eclipse cycle; the second is a four solar year count that, with the first, forms a triangle, marked clearly by stones at Le Menec. The principles worked out at Le Manio appear fully developed in Le Menec’s western cromlech, including the use of an 8 eclipse year day-inch count, consequently forming a diameter of 3400 megalithic inches which equals in number the days in half a nodal period. The scaling of the Western Alignments is found to be 17 days per metre, a scaling naturally produced by the diagonal of a triple square geometrical construction. A single sloping length on the top of the central stone initiating row 9, indicates a single lunar orbit at 17 days per metre, a length of 1.607 metres. This control of time counting within geometrical structures reveals that almost all of Le Menec’s western cromlech and alignments express a necessary form, so as to represent a megalithic study of (a) circumpolar time as having 365 time units, (b) the moon’s orbit as having 82 times 122 of those units and (c) the variations of successive moonrises over most of a lunar nodal period of 18.6 solar years.
