It’s not unusual to think of the world as number, many mathematicians and physicists do. But it would be unusual to routinely encode numbers derived from the natural world and astronomical cycles into the proportions and measurements of important buildings, as if our lives should harmonise with the cosmos. This is probably because we do not think of natural or astronomical cycles as particularly relevant to our lives, nor of measure as of any consequence. It wouldn’t occur to most of us to live according to a mathematical ideal derived from observing the patterns of nature and the rhythm of the heavenly bodies, or from numbers with interesting properties or factors. But this was the case with some ancient philosophers, notably Plato, who suggested the ideal city should have no fewer or more than 7 x 8 x 9 x 10 = 5040 inhabitants, for example. Any surplus could go off and found their own city. This might be somewhat easier to understand as a form of utilitarianism, according to which the well-being of the entire cosmos depends on the actions of all its elements, and so the organising of a city will affect the well-being of the world and the individual within it. Plato didn’t think in terms of utility as we might today, or as Bentham or Mill understood it, but he did believe that when the parts of a society are proportioned according to number and harmony, reflecting the order of the cosmos, both the individual and the collective thrive. In this sense, the health of the city and the well-being of its citizens were seen as structurally and spiritually aligned with the greater order of nature.

Plato is clear that this is not his, or Socrates's invention, but comes from a tradition, based on authority from prophetic places in Greece and Egypt, and must be respected:

What is the basis of this tradition, and how far back in time does it go? How widespread was this idea that number should govern lives?  The idea that the good of the whole depends on the harmonious order of its parts, and that this order reflects the greater cosmos, could be understood as a kind of cosmic-functionalism. It echoes the idea that a well-ordered soul, or city, contributes to the universal harmony.

The Pythagoreans believed that 'number rules the universe' (arithmos kyrei tou pantos), and tried to uncover the hidden numerical relationships governing nature. They drew parallels between the musical octave, fifth, and fourth, corresponding to the simple ratios 2:1, 3:2, and 4:3 respectively, and the motions of the celestial bodies, which produced an inaudible harmony, a 'music of the spheres'. This was a concept that Plato would later echo in Timaeus. But were the Pythagoreans the first to dance to this tune? Did the dimensions and proportions of megalithic monuments and ancient pyramids also respond to these harmonies? Interpretations of data derived from the dimensions, locations, and the behaviour of the sun, and occasionally the moon, at important megalithic sites such as Stonehenge and pyramids such as Giza reveal the possibility of a philosophy rooted in number at play. This can tell us something of the mathematical and scientific nature of the world view of the builders of these structures. 

   Plato famously said that none should enter his academy without knowledge of geometry, a curious statement which today we could almost interpret as just another form of elitism. However, there may have been good reason to promote geometry in a school of philosophy. It’s interesting to remember, in this context, that number, geometry and astronomy were the three “branches of learning” for Plato:

   What if geometry, number and astronomy were in fact a central part of a very old world view, much older even than Plato’s, whereby the many structures and cycles of nature were respected and incorporated into a numerical pattern or structure, and which formed the basis of a belief system? Looking at almost every ancient culture allows us to see a respect for number, proportion, nature or astronomy. In ancient Egypt, the concept of Maat, the divine principle of truth, balance, and cosmic order, shaped not only legal and moral codes but also architecture. The Great Pyramid of Giza, aligned to within a fraction of a degree of true north, and employing a base perimeter to height ratio approximating 2π, can be seen as a monumental embodiment of Maat, order imposed on the earth according to cosmic principles. It is accepted that by at least the second millennium BC, Babylonian astronomers had identified the 19-year Metonic cycle, and were recording the periodicities of Venus, Jupiter, and the moon in systematic tables. Their massive ziggurats, such as Etemenanki, the 'temple tower' of Babylon, may have symbolised the cosmic mountain, linking earth to the heavens and serving as a microcosm of the ordered sky. In early Chinese cosmology, the emperor maintained cosmic harmony by aligning the human realm with the heavens. The imperial capital Chang'an (under the Han and Tang dynasties) was laid out on a strict north-south axis, with the city's gates, palaces, and temples carefully positioned according to principles outlined in the Zhouli (Rites of Zhou), reflecting the belief that earthly order must mirror celestial patterns. The Vedic Śulbasūtras, ancient Sanskrit texts on altar construction, show a profound mathematical understanding used to create ritual spaces embodying cosmic principles. One instruction, for instance, requires constructing a square altar of the same area as a given rectangle, necessitating a method equivalent to the Pythagorean theorem centuries before Pythagoras. Altars like the Agnicayana, built of precisely arranged bricks in shapes symbolising cosmic forces, were believed to replicate the universe in miniature.

   Philosophy is about interpreting the world around us, exploring questions of existence, knowledge, values, reason, ethics, and reality, what we think is beautiful or worthwhile, and what we can know. There’s also something more fundamental, primordial even, in a philosophy, some core belief or guiding principle that underlies the kinds or answers we might give to these questions, internalised to such an extent that it is difficult to put into words. It might result in a deep respect for authority, as when people chose to live by the rules of a religion, and are keen for others to do the same; or in a deep respect for courtesy, putting the needs of others before their own, as when Saint Francis of Assisi vowed to make poverty his lady. It might result in a deep respect for the natural world, and today this might mean trying to protect creatures and habitats from humans, and the consequences of our collective actions, past and present. It might also result in a deep respect for an ideal of order and beauty, a love of patterns, being in awe of the inner workings of the natural world, and of the universe more generally. This might make a person an artist or scientist. If geometry is important to a philosopher, it could be that the world itself is being interpreted as somehow geometric.

   It’s probable that the designers and builders of the megalithic monuments and the pyramids shared a view of the world in which the perceived underlying order to the universe was key, a harmony shaped by reason and necessity, and that this order should guide the way people live. The natural world was something that could be understood through mathematical principles, where even the motions of celestial bodies and natural patterns could be expressed in terms of harmony and proportion. This is evident in Plato’s account of the demiurge, who created the universe according to an ideal of perfection, expressed through proportion and numbers. This suggests an awareness of the regularity of certain phenomena, from the seasons, to the motion of the moon, the planets and the stars, to the lives and patterns in nature, and our place within this larger framework. On one hand, we can relate to such a view today, perhaps as artists, scientists, naturalists, or mathematicians. For anyone who cares to look, the natural world is awesome. On the other hand, it might seem strange to choose to live by rules derived from the workings of the cosmos, and the living world. Today we are familiar with the idea of trying to understand the architecture of the universe, but less so with the idea of incorporating the universe into architecture.

   If we grant that our ancestors of 5,000 or even 10,000 years ago were as intelligent and observant as we are today, then it is not unreasonable to imagine that they amassed a sophisticated body of knowledge about the natural world. Their understanding, encoded perhaps since in stone and myth, may have been sophisticated. It is clear from various astronomical tablets and devices that their ability to observe, compute, and find meaning in the regularities of the cosmos was great. But what can ancient megalithic sites, considered in relation to their alignments, proportions, and metrological foundations, tell us about the beliefs of their creators? Can we find in them a philosophy which celebrated the patterns and cycles in the natural world, intertwining them with cultural and perhaps even spiritual practices?

   In the first part of this article we will consider the ancient megalithic monuments and pyramids as astro-geometric creations, that is, monuments built with the cycles of astronomy and geometry in mind. In the second part, we will look at how it’s possible to think of the natural world, and in particular the cycles of the sun, moon and planets in terms of geometry. Lastly, we will consider a possible worldview that would have brought about the need to encode the passage of time and the natural rhythms of the world into stone, and in particular, consider the cosmos, as a living being. What hidden knowledge, what ancient poetry of number and form, might still be speaking to us across the millennia?

1. From Star to Stone: Astro-geometric Creation

   It is well known that many megalithic sites are aligned with the solstices and equinoxes, and this seems to reveal a worldview in which tracking the movement of the sun was of profound importance. Often, this is interpreted as evidence that the builders of these monuments were farmers, reliant on precise knowledge of the agricultural calendar. However, farming communities have always had access to a wide range of natural indicators, such as the behaviour of birds, and other animals, the presence of frost or snow, and the flowering of certain plants, as well as the length of days and the rising of stars to guide them in their work. As for marking the turning points of the year, observing the shadows of a sundial is sufficient. In any case, these temples may have been important to non-farming communities, from the inhabitants of cities, to wandering traders, and nomads.

Farmers may well have benefited from the marking of the solstices and equinoxes. However, the emphasis in some archaeological accounts on interpreting the astronomical properties of ancient monuments mostly in terms of practicalities linked to survival serves to uphold a particular ideology, according to which our ancestors were primitive, often perilously close to some grim death, and understandably preoccupied with cheating it. According to this view, any thought of a grand philosophy of the ancestors, according to which life might best be lived in accordance with the rhythms of nature and the properties of number, is fanciful and should not be encouraged. It might be called a sort of Marxist-light approach: religion and rite exist but the less said about them the better, what really matters is understanding the means of production, social hierarchies and conflict.

  The writer Simon Ferandou coined the term “astrocreation”, to describe ancient monuments which seem to have astronomical properties, be it through their position in relation to an important sunrise and sunset, very often at the solstices and equinoxes, or to some other feature of the design which suggests a connection to astronomy. If many ancient monuments are considered as astrocreations, interesting observations can be made which allow us to identify connections to the study of the cosmos, and consider aspects of their design (location, dimensions, proportions) in relation to the sun, the moon, or the stars. Many of these ancient sites might in fact be called astro-geometric, because geometry underpins the astronomy. 

  In the very first episode of The Ascent of Man, a BBC television series from the 1970s (4), Jacob Bronowski begins on a Californian beach, at night, marvelling at the grunion. He tells us how the native American people used to describe this species of fish as dancing on the beach at full moon. And of course this is exactly what happens: at the full moon, the fish arrive on to the sand, the females lay their eggs just above the shoreline, and the males dance about, fertilising the eggs. Then, ten days later, at the next high tide, the hatchlings can swim into the sea for their maiden journey. All of this depends on the cycle of the moon, which controls the cycle of the tides. When we consider the seasons, and the cycles of the moon as elements of the design of a building, or stone circle or alignment, or a passage tomb or even a pyramid, we are connecting with the cycles of all of the living world too.

   Perhaps if we watched nature more closely, as a society, we would generate, or require, a philosophies and beliefs that reflected the wonders that exist within it. By contrast, many religions today encourage us to exploit nature. It is difficult for us to get used to the idea of a mindset in which we are actually encouraged to marvel at the wondrous ingenuity and beauty of the natural world, at least beyond an hour of it on a screen. If we pursue this idea a little further, we might find ourselves becoming slightly obsessed with the cycles and geometry that underpin this architecture of the universe. The natural world offers countless examples of such connections. Consider the synchronised spawning of grunions with the phases of the moon, or any of the countless plants and animals whose lives are shaped by the rhythm of the seasons, driven by the apparent motion of the sun across the sky. It seems likely that ancient observers, curious and methodical, would have sought to understand the cycles governing these phenomena, from the intricate relationship between the solar and lunar calendars to the periods of the planets. They may even have recognised mathematical patterns like the Phi ratio, which appears in everything from the branching of trees to the proportions of flowers and human bodies.

   If the megalithic monuments and pyramids were considered sacred, then we can perhaps speculate that the structure of the cosmos, the language of mathematics and the wondrous inventiveness and beauty of the natural world were being celebrated in their design, perhaps from a purely aesthetic point of view, or perhaps as the basis of an ethical, religious or political point of view. It is linked to the idea that by discovering the workings of nature, it is possible to connect to the divine, a view which is very old and has never really gone away.

The alignment of ancient monuments to celestial phenomena reveals a deep connection between human cultures and the rhythms of the sky. The practice of aligning structures to the movements of the sun, moon, and stars may have belonged to a worldview in which timekeeping, spatial orientation, and cosmic order were intricately linked. One striking example is Newgrange, the famous passage tomb in Co. Meath, in Ireland, which demonstrates a sophisticated understanding of solar movements. At sunrise on the winter solstice, a beam of light travels through a narrow corridor to illuminate the inner chamber, creating a spectacular and exacting alignment. In fact, this article was prompted by watching a live stream of the winter solstice beam of light entering the chamber at Newgrange, and wondering about the reason for building such a sophisticated way of channelling the winter solstice sunrise light in a monument. Such phenomena are not isolated occurrences; they echo across cultures and landscapes, as seen in the equinox serpent shadow at Teotihuacan, or the solar and lunar alignments at Stonehenge.

  The precise orientation of Egypt’s Great Pyramid of Giza to true north demonstrates this concept on a grand scale. In ancient Egypt, the cardinal directions and the pole star seem to have been crucial elements of religious and cosmological symbolism. Similarly, Indian temples often align with the cardinal points or significant astronomical events, highlighting a shared philosophical emphasis on the celestial order. The alignment of structures to celestial phenomena often extended beyond individual monuments to entire landscapes. Traditional Christian churches, especially medieval ones in Europe, are often aligned with particular sunrise azimuths, some aligned "ad orientem", meaning toward the east, symbolising the resurrection, Christ as the "light of the world," and the rising sun, and others to sunrise on the feast day of the patron saint to whom the church is dedicated. The alignment of structures to celestial phenomena can often be extended beyond individual monuments to entire landscapes. 

2. Two rectangles: Stonehenge and Carnac

  Perhaps the most famous of all the different types of megalithic stone constructions are the stone circles, found most frequently in western Europe. However, in this article we will look at two examples of stone rectangles. In both cases, these rectangles can only be understood as part of the broader context, meaning other nearby stones, the landscape, and the latitude, but we’ll focus just on the rectangles themselves. Both examples are linked to Pythagorean triangles, which are right-angled triangles with integer values for all three sides, such as 3:4:5, or 5:12:13. The first example, in France, is a 3 x 4 rectangle, with a diagonal of 5, and the second, in England, is a 5 x 12 rectangle, with a diameter of 13. As Professor Thom noted:

1. Carnac  

   

     At Carnac, in France, the Crucuno rectangle encapsulates summer, winter and equinox sunrises and sunsets in a 3 x 4 rectangle. The rectangle’s width and length lie on the north-south and east-west axis. The angles in a 3:4:5 triangle are 53.13° (53°7'48"), 36.87° (36°52'12") and 90°. According to the website sunearthtools.com, the summer solstice sunrise, for the first moments of dawn, is at azimuth 52.72°, and sunset at 307.27°, for our epoch. According to the astronomy software Stellarium, in 3000 BC summer solstice sunrise was at azimuth 52° and sunset at 308°, and in 5000 BC, at 51.8° and 308.4°, to give an idea of the progression over time. The winter sunrise and sunset azimuths are 125.01° and 234.99° respectively, for our epoch, 126° and 234° for 3000 BC. As the sun rises in the morning, it not only travels upwards but also slightly south, as it starts on its daily journey back south and then west, where it sets. So the difference between the azimuths given by these websites and the angles given by the rectangle can be interpreted as the sun being a little higher in the sky, not the first moment of dawn. Likewise, just before sunset, the sun will be slightly to the south of its ultimate setting point on the horizon. The landscape itself also needs to be taken into account, with its altitude and features along the skyline. With this in mind, the geometry of the rectangular layouts also ties directly to the celestial phenomena. 

At Crucuno, the sun rises on the morning of the summer solstice, and sets that evening, in such a way as to draw out the same angles as would be found in a rectangle with sides of 3 and 4, and this rectangle is precisely aligned with the cardinal points. This type of rectangle is significant in that it has a diagonal of 5, so is made up of two Pythagorean triangles, which are triangles which have all three sides measuring some value in a whole number, or integer. The winter solstices sunrise and sunset then echo these angles to the south. The observation of the sun's movements at the solstices at the latitude of Crucuno was tied into a geometric interpretation, and the rectangle that survives to this day was built to commemorate this marriage of astronomy and geometry. At Crucuno we can see the importance of thinking about a natural pattern in terms of geometry, and the interpretation of the solstice sun at that place as particularly significant. It's magical that the sun would trace out these lines there, in such a way. Also magical is the human reaction to this wondrous coincidence in the natural world, that intelligent minds were able to grasp and become creative with long ago.

2. Stonehenge

   
   

  The astronomer Gerald Hawkins was among the first to analyse Stonehenge systematically from an astronomical perspective. In Stonehenge Decoded (1965), he highlighted how many of the standing stones, and the spaces between them, create alignments to significant solar and lunar events. Hawkins’s findings still remain controversial, after all this time, though his evidence has not been refuted. Some detractors have argued that for Stonehenge to function as an analogue computer, its stones would have to have been erected simultaneously, yet they claim archaeological evidence suggests they were placed at different times. However, since dating the placing of individual stones is inherently imprecise, and since it is impossible to know what kind of markers might have preceded the ones that are there now, this argument does not work to undermine the Stonehenge as computer theory. The same markers would have worked, for a time, just as well with wooden posts. Stonehenge, as it is now, works as an apparatus for counting days and months in relation to important celestial cycles, monitoring sun and moon rising and setting points and that evidence is clearly there.

   One way in which the stones of Stonehenge function as markers is by creating lines to the points of sunrise on quarter and cross-quarter days. Amazingly, and this may provide a clue as to why the stones were placed where they were, some of these markers form a precise rectangle, dictated by the monument's latitude. The Station Stones, which lie on the Aubrey circle, perfectly capture this, as part of a 5 x 12 rectangle. In fact, only two of these stones remain, but the other two’s positions are clear. As at Crucuno, the rectangle defined by the Station Stones marks a naturally occurring rectangle for the observer watching the sun, and also the moon, over the course of a year. Unlike the Crucuno rectangle, the length and width of the Station Stone rectangle at Stonehenge are not aligned with the east-west and north-south axis. Instead, the Station Stone rectangle width is oriented towards the summer solstice, so that the length is at a right angle to this line. The alignments are more complicated as they feature the moon’s movements also, but as at Crucuno, everything is based around a rectangle, itself based on a Pythagorean triangle. At Crucuno the rectangle is the double of a 3:4:5 triangle, and at Stonehenge is the double of a 5:12:13 triangle. 

  Gerald Hawkins was indeed the first person in modern times to study the astronomy at Stonehenge.  It is widely accepted that the summer solstice sunrise coincides with the sun rising in the direction of the heel stone as viewed from the centre of Stonehenge, and has probably been observed for many centuries. But there is more to the placing of the stones than just the marking of the solstice.

Figure 3, the Station Stone rectangle at Stonehenge, from Gerald Hawkins’s “Sun, Moon, Men and Stones”, American Scientist, vol. 53, no. 4, 1965, pp. 460A – 408. JSTOR, http://www.jstor.org/stable/27836207.

Figure 4, the Station Stone rectangle at Stonehenge, from Gerald Hawkins’s “Sun, Moon, Men and Stones”, American Scientist, vol. 53, no. 4, 1965, pp. 460A – 408. JSTOR, http://www.jstor.org/stable/27836207.

  As Euan MacKie observes, 

With the astronomy software Stellarium, it’s fairly straightforward to check this, though it does not give the lay of the land around Stonehenge with it. At Stonehenge’s latitude, the Station Stone rectangle captures key points on the horizon where the sun rises and sets, such as the solstices and cross quarter days. Furthermore, the diagonal of the rectangle marks the moon’s maximum and minimum declination cycles. The current value for the winter solstice sunrise azimuth is 128.05°, and for the summer solstice sunrise is 49.26°. In 3000 BC according to Stellarium, the summer solstice sunrise azimuth would have been 48.3333° (Stellarium, for date: - 3000 14 July (longest day)) and the winter solstice sunrise 311.933°(Stellarium for date: - 3000 9 January (shortest day)). For 5000 BC, the summer solstice sunrise would have been 48.1° and the winter solstice sunrise would have been 129.55°. These values are for the first moment of sunrise. However, it may be that the moment when the partial or full disc of the sun was first visible was being marked out.

The orientation of the widths of the rectangle can be found in the azimuths between stones 93 and 94, and stones 92 and 91. In his table “Astronomical alignments at Stonehenge determined photogrammetrically”(9), Hawkins gives 50.6° and 51.13° respectively. While these azimuths are not quite aligned with the azimuths of the first moments of sunrise for the summer and winter solstices, they do fit with azimuths several minutes after this moment, and so, also taking into account the lie of the land around Stonehenge, the azimuths between the stones can indeed be said to coincide with these key sunrise azimuths. If these sunrise alignments are designed to accommodate the full disc of the rising sun on the horizon at Stonehenge, that would seem to match what is happening at Crucuno.

   The diagonal of the Station Stone rectangle, the 13 side of the 5:12:13 triangle, from stone 91 to 93, heading north-west, according to Hawkins, corresponds to the “moonset + 18.8”, which is the major standstill moonset. Euan MacKie has noted that this is also the azimuth of the setting sun of the first of May, an important date in the calendar when taken as one of eight points in the year, two being the solstice, two being the equinoxes, and the remaining four being the midpoints between the solstices and equinoxes. The first of May is an important cross quarter day. In the opposite direction, as MacKie has noted, heading south-east, from stone 93 to stone 91, corresponds according to MacKie to the sunrise on the 1st of February, another important cross quarter day. The other two cross quarter days (1st November and 1st August) are also included, being the equivalents between the summer and winter solstices.

   One aspect of these days at Stonehenge is of particular interest. The first of May would have been especially important because the ratio of day to night, both to each other, and to the full 24 hour period of a turn upon the earth’s axis, is Phi, the golden ratio. With 2025 values, the length of a day on May 1st is 14 hours and 50 minutes, which is 890 minutes. A 24 hour period divided by Phi (1.61803) is 889.97 minutes. The azimuth of sunset on that day, in our epoch, is around 296.1°, and the azimuth of stone 93 from stone 91 is given by Hawkins as 297.29°. The 3000 BC value for sunset for 1st May is 295.867° (Stellarium, -3000, 27 May, day with closest to 14 hours and 50 minutes of daylight, being 14 hours and 49 minutes), and for 5000 BC 295.417°(Stellarium, - 5000, 13 June, day with closest to 14 hours and 50 minutes of daylight). These values correspond to just a few minutes after the moment of sunset. 

   The diagram below shows the azimuths between the station stones as per the table by Gerald Hawkins (see above), with a winter Phi day sunrise line, the angle of sunrise at Stonehenge on the 10th November, when there is a Phi ratio between darkness and light.  

The screenshots below are taken from Stellarium and illustrate the sunrise data for key historical dates. 

117° 17' = 117.29

297° 17.4' = 297.29

   Another intriguing possibility at Stonehenge is that the angle of the diagonal of the station stone rectangle, between stones 94 and 92, in relation to the north-south line, and the line between stones 93 and 91 in relation to an east-west line, parallel to our equator on earth, reflect very closely the angle of the galactic equator in relation to the celestial equator. And so, it may be possible to make a comparison between the diagonals of this Station Stone rectangle and the earth's place within the Milky Way.

   We can surmise that the builders of Stonehenge, and the astronomers of the day were keen to harmonise the cycles of several heavenly bodies into a single system, with this rectangle at Stonehenge reflecting an integration of both solar and lunar observations into a single coherent design. Furthermore, this design has a sophisticated geometry of its own, with the Station Stones being placed in such a way as to trace two Pythgorean 5:12:13 triangles, within a circle. Indeed, with the circle being composed of 56 holes, such a triangle can be traced within the circle joining up these points eight times. 13 Venusian orbits (8 x 224.701 days) are very nearly equal to eight Earth years. The number eight is significant, marking two cycles, one linked to Venus, and the other the octaeteris, a sophisticated reconciliation of sun and moon, a period of 8 years, or 99 synodic lunar months, after which the moon phase will occur on the same day of the year plus one or two days. If Venus is visible beside the moon, after eight years the two will be again close together near the same date. In astronomy, there are two periods of time that last roughly eight years, one is to do with the Moon, the other is with Venus, two heavenly bodies associated with the goddess Venus / Ishtar / Inanna. A Great Year of 100 lunations is almost exactly eight years. The figure is in fact 29.53059 x 100 /365.242199 = 8.08521, or eight years and a month of 31.09 days. Venus also connects with this 5:12:13 Pythagorean triangle as it has a synodic period of 5 and a sidereal period of 13 years. It’s also interesting to note the Fibonacci sequence in the Venusian numbers: 5, 8 and 13. Perhaps there is an octagon at the centre of the design, though what we can actually see is a circle of 56 postholes, and two of the once four Station Stones which also sat on this circle. There is a slight problem with both an octagon and a 5:12:13 triangle both fitting within the same structure.  If the four Station Stones which make up the large rectangle made of two 5-12-13 Pythagorean triangles, then the angles aren't quite right for this rectangle to also form the centre part of a great octagon. This was pointed out by Jim Alison on the GHMB website. A 5-12-13 triangle has angles of 90°, 67.38° and 22.62°. A similar triangle within an octagon would have dimensions of 5, 12.07 and 13.066, with angles of  90.017°, 67.484°, and 22.499°. Some researchers are inclined to think that as there is clearly a 5-12-13 triangle at work, the octagon option is not part of the intended design. However, others believe that due to the large size of the Station Stones, both the Pythagorean triangle and the octagon are possible. It's a matter of interpretation. The Pythagorean triangle is just as likely an explanation as the octagon, in my view, both being compatible with the angles recorded by Hawkins, within a fairly small margin of error. 

   Geometrically, the octagon has been symbolic in later times of mother earth (Roman mosaics show the face of Gaia in an octagon made of two concentric squares), and of birth and death, with many baptistries in the Christian world being designed round an octagon, and mausoleums worldwide also, such as the Taj Mahal. As Euan MacKie has described, “another 5:12:13 triangle seems to connect the Heel stone to the site axis and to one long side of the Rectangle”, which supports the idea of Pythagorean geometry being used in the site design. 

   According to Euan MacKie, “astronomical practices were intimately bound up with the geometric and metrological knowledge of the time, to such an extent that the two branches of knowledge are really inseparable.” This is an important point, and one that is sometimes overlooked: to try to understand megalithic monuments it is essential to take into account the most precise measurements possible.

     It was Alexander Thom who first showed that builders of stone circles often used  Pythagorean triangles in the geometry of the construction of many stone circles. The dimensions of the rectangles at both sites reveal an intentional use of geometric principles to encode astronomical knowledge. These rectangles often feature proportions that reflect key astronomical ratios, such as the relationship between the lunar month and the solar year, or the periodicities of lunar standstills. For instance, the ratio of the sides of the Station Stone rectangle at Stonehenge has been shown to approximate the ratio of the lunar year to the solar year, indicating that the builders were keenly aware of these cycles.

   The diagonals of the rectangles further emphasise their astronomical function. In both Stonehenge and Carnac, the diagonals align with critical points on the horizon where the sun or moon rises or sets at specific times of the year. These alignments allow for the precise tracking of events like the summer and winter solstices, as well as the 18.6-year lunar nodal cycle. The builders’ ability to encode such complex astronomical relationships into simple geometric forms reflects their deep understanding of how the cycles of the heavens interacted with the specific latitude of their monuments.

   The alignment of structures to celestial phenomena often extended beyond individual monuments to entire landscapes. The precise orientation of Egypt’s Great Pyramid to true north demonstrates this concept on a grand scale. In ancient Egypt, the cardinal directions and the pole star seem to have been crucial elements of religious and cosmological symbolism. Similarly, Indian temples often align with the cardinal points or significant astronomical events, highlighting a shared philosophical emphasis on the celestial order.

  The 56 Aubrey holes are usually interpreted as part of a circle, but there is historical evidence of a 56 sided polygon related to Typhon which may be relevant at Stonehenge, though the text relates to the Greek gods. Typhon, symbolising chaos, is associated with the number 56 by the Pythagoreans.

These 56 holes are of value to astronomers. As Hawkins explains, 

Most archaeologists studying Stonehenge and megalithic monuments are not keen on the idea of megalithic constructions being the product of astronomy and geometry. The evidence that these constructions are indeed rooted in geometry and astronomy is often ignored. It’s difficult to understand what the motivation for this cognitive dissonance is. 

However, thanks to Hawkins, Thom, Hoyle, and independent researchers, the theory of the megalithic monuments being astronomical markers and calculating devices has been demonstrated. The implications of the astro-geometric nature of the Pythagorean triangle in the Station Stones, on the Aubrey circle are key. As MacKie observed:

3. Geometry and Latitude: A Precise Connection

   The specific latitudes of Stonehenge and Carnac play a crucial role in the geometry of their rectangles. At these latitudes, the angles of the sun’s and moon’s paths across the sky create distinctive declination ranges that are captured in the monuments’ designs. For example, the ratio of the diagonals to the sides of the rectangles reflects the maximum and minimum angles of the moon’s declination, as observed from these precise geographic locations. This suggests that the builders carefully chose these latitudes not only for their geographic significance but also for some property of their astronomical observations.

  The geometry of the rectangles also demonstrates an understanding of how latitude affects the behaviour of the sun and moon. At Carnac, for instance, the rectangles incorporate angles and proportions that correspond to the solstice sunrise and sunset angles at 47°N. Similarly, at Stonehenge, the geometry aligns with the extremes of solar and lunar movement as seen from 51°N. These alignments reveal a sophisticated interplay between local geography, celestial cycles, and architectural design. The rectangles at Stonehenge and Carnac are not isolated features but integral components of a broader system of observation and measurement. Their geometry, orientation, and proportions reflect a unified approach to integrating astronomy and geometry into the cultural and ritual lives of their builders. These rectangles served as tools for marking time, predicting celestial events, and harmonising human activity with the cosmic order.

   By linking the precise geometry of the rectangles to the specific latitudes of the sites, the builders demonstrated an awareness of the Earth’s curvature and its effect on celestial observation. This integration of geometry, astronomy, and latitude underscores the sophistication of ancient metrology and highlights the deep connection between these monuments and the natural world. The rectangles at Stonehenge and Carnac stand as enduring testaments to the ingenuity of their creators and their profound understanding of the interplay between the heavens and the Earth.

   The precision with which ancient structures align to celestial events speaks to the technological and philosophical achievements of these cultures. From the axial alignments of Giza to the carefully calibrated angles of Carnac in Brittany, latitude plays a significant role in shaping the relationship between monuments and the sky. These alignments suggest that the ancients viewed latitude as a key determinant of selecting a location to build a site.

   These two rectangles, at Carnac and at Stonehenge, are impressive feats of engineering and astronomy. Are they also reflections of a worldview that placed great importance in order, and cycles of the natural world? Can we guess at a philosophy encoded in stone, which invites us to explore the interconnectedness of human culture and the celestial sphere? It seems that the position, size and proportion of an ancient monument should be considered in order to gain some sort of understanding of the people who created it. It seems also that, on examining many such monuments, there was indeed a commitment to understanding the structure of the cosmos, to exploring and understanding our world through reason, observation, and the language of mathematics. If these monuments were for scientific research only, or to mark the passage of time only, for some purely practical purpose, then we can’t say much about the principles of their designers other than that they were scientists and good time-keepers. Indeed, many ancient sites that do mark the seasons in some way, such as the winter solstice, are often explained in terms of the practicality of time-keeping for a farming community. But is that the whole story?

Notes

1. Plato, Laws, 7.817e, from Plato in Twelve Volumes, Vols. 10 & 11 translated by R.G. Bury. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1967 & 1968. https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0166%3Abook%3D7%3Asection%3D817e

2. Plato, Laws, translated by David Horan, https://www.platonicfoundation.org/translation/laws/

3. Ibid.

4.  https://archive.org/details/theascentofman1lowerthantheangels. Thank you to Milo Gardner for this link.

5. Thom, Alexander, Megalithic Lunar Observatories, Oxford University Press 1971, reprinted 2002, page 9, https://books.google.ie/books?id=tIUdytA-K_oC&lpg=PA8&lr&pg=PA4#v=onepage&q&f=false

6. Richard Heath & Robin Heath, “The Origins of Megalithic Astronomy as found at Le Manio”, https://www.academia.edu/5384545/The_Origins_of_Megalithic_Astronomy_as_found_at_Le_Manio4

7. Euan MacKie, A New Look at the Astronomy and Geometry of Stonehenge, https://www.academia.edu/10789056/A_new_look_at_the_astronomy_and_geometry_of_Stonehenge

8. Ibid.

9. In Hawkins, Gerald, Stonehenge Astronomy - an Update, in Petrie, W. M. Flinders, Stonehenge: Plans, Decriptions and Theories, Histories & Mysteries of Man Ltd, London, 1989, p. 55. 

10. Euan MacKie, A New Look at the Astronomy and Geometry of Stonehenge, https://www.academia.edu/10789056/A_new_look_at_the_astronomy_and_geometry_of_Stonehenge

11. Ibid

12. G.R.S. Mead, Thrice-Greatest Hermes - Volume 2, “CONCERNING TYPHON”

13. Gerald S. Hawkins; Stonehenge physics. Physics Today 1 April 1966; 19 (4): 38–42. https://doi.org/10.1063/1.3048177

14.  Hawkins, G. S. (1973). Astro-Archaeology—The Unwritten Evidence. Bulletin of the Atomic Scientists, 29(8), 58–64. https://doi.org/10.1080/00963402.1973.11455521

15. Chapter 2: Later Prehistoric Europe Clive Ruggles, in Heritage Sites of Astronomy and Archaeoastronomy in the context of the UNESCO World Heritage Convention A Thematic Study Clive Ruggles and Michel Cotte, Published by ICOMOS Office: International Secretariat of ICOMOS, 49–51 rue de la Fédération, F–75015 Paris, France and the International Astronomical Union IAU–UAI Secretariat, 98-bis Blvd Arago, F–75014 Paris, France ISBN 978-2-918086-01-7 (e-book) © The individual authors, 2010 https://openarchive.icomos.org/id/eprint/267/1/ICOMOS_IAU_Thematic_Study_Heritage_Sites_Astronomy_2010.pdf

16. Euan MacKie, A New Look at the Astronomy and Geometry of Stonehenge, https://www.academia.edu/10789056/A_new_look_at_the_astronomy_and_geometry_of_Stonehenge