The Creation of Time

Introducing Neolithic (Stone-) Circle Mathematics

Measuring circles

It will be difficult to convince anybody of what I am going to put forward here, I suppose, because it is so completely contrary to the image we are bound to have of the people of the late Stone Age, and of their state of mind. In general we tend to think that peoples that could not write cannot be very sophisticated whatsoever. The term Stone Age itself has a ring of primitiveness about it.
This whole website is an exercise in trying to overcome prejudices regarding the state of peoples knowledge and insights (also regarding my own).
For some my speculations may be outlandish or very unlikely. But I usually have good, often original, arguments, I think.

What follows is a reconstruction of the state of Stone Age people’s knowledge based on the ratios they expressed in their megalithic works and architecture, and especially those of the Maeshowe Grooved Ware culture in Orkney, roughly 5000 years ago. (3000BCE, Before Common Era)
Although the Unstan Ware is in engravings similar to Gr.Ware it has a rounded bottom. It is possible that the autochtonous Unstan ware was the inspiration for the new migrated Grooved Ware people, avant la lettre, who though invented the revolutionary flatbottomed pots and vessels, possibly because they started to use tables of Orkney flagstones to put their pottery on, more hygenic, away from the roaming animals, the rats etc. etc.

It will be shown that given a unit of measurement based on a bone of the male forearm, the ulna, the designs of Stenness, Brodgar and Maes Howe, can be translated into specific mathematical number ratios that fit into a natural number calculus relating the geometric interaction of circles and squares.
The geometric ‘square’ and its relation to the ‘circle’ may have been a revolutionary new insight in those days in Orkney. The excavation at the Ness of Brodgar has revealed that the building nr.10 is typically oriented on Maeshowe, a mile away. In my view The Ness was a Stone Age university where the many roaming wizards flocked to learn about the stars, may be the tides. The lay-out of Structure 1 at the Ness reveals several intriguing squares and rectangles, I’ve named it the Lecture Hall. I made the measurements with laser clandestinely in the evening, and the result was astounding. I wrote a Page on it, that also there a pendulum was possible in the hall

Both the Stones of Stenness and the Ring(s) of Brodgar deal with the problem of the ‘irrationality’ of square root 2, the fact that the diagonal of the square did not relate to its side in a simple whole number ratio. This ‘practical’ problem Stone Age man seems to have solved brilliantly in practice by creating two material units of length which had the ratio √2 : 1 (=1,4142135…), besides also finding the really big numbers where the approximation is virtually correct (198:140=1.41428…). This is similar to what the Egyptians did at about the same time in their well established units of the Royal Cubit RC and the Royal Remen RR. This ‘invention’ of two incompatible units made it possible to express all their measures in natural numbers, and in our analysis it does the same.

In the chapter ‘the Bone Measures’ (Metrology zer0) it is explained, but here it may suffice to state, that a standard unit, the Megalithic Ell (ME= 0.5236m), was 2 (male) Megalithic Ulna (MU=0.2618m, ulna = underarm bone) and that a (female) Megalithic Fibula (MF= 0.37025, fibula = calf bone)) was 1 Ulna x √2 (0.740482  = 2MF)
I don’t mean to say they had a concept of square root, but in practice they could make a right angle with ulna bones and connect them with a fibula bone to make a triangle (see extensive use of this in design Maeshowe), as they could make a right angle with fibula bones and connect them with 2 ulna bones, and create the standard length of 1 Megalithic Ell = 0.5236m. (Maybe they never had the concept of the double ulna as Ell, but that does not really matter, it is more convenient as length, Thom’s megalithic yard was also the double of an original  length he had found in his statistics)

At the Stones of Stenness we see the equation 70 Megalithic Ell = 99 Megalithic Fibula, at the Ring of Brodgar it is 198 Megalithic Ell = 280 Megalithic Fibula.

These two ratios approximate square root 2 as: 99 : 70 = 1.4 14 28 57 14… and as 280 :198 (140:99) = 1.4 14 14 14……., whereas the irrational √2 = 1.41421356 2 37… , which is practically ‘exactly’ in the middle of these two ratios (99/70+140/99)/2= 1.41421356 4 213564 21…, a difference of 0.00000000184 which in a world of megaliths, or in any world except mathematics or high technology for that matter, such difference is trivial and of no consequence. Nevertheless the inconvenient big numbers 70, 99, 140 and 198 give every reason to use such a handy unit as the ‘remen’ and to conserve its derivation in huge stone settings for posterity .

When we analyse these two ratios, which are better expressed as 99 : 70 and 140 : 99, then we see which factors they are made of and that is, indeed again, 9×11 and 7×10, and these numbers are in the two basic ratios that their whole geometry was based on, that is-> 10 : 9 and 11 : 7 ; and it is easy to see why.
What follows is an example of an imagination of how emperical mathematics came about

By setting out circles in the field, as we very well know they did, they had as a matter of course discovered that if you divided a circle’s diameter by 7 you got a unit length which exactly fitted the arc over this basis, the half circle, in a whole number, that is: 11 times 1/7.

 

The next step they took, and here is where the Maeshowe culture comes in, is that they also found that, if you put a square in the circle, there was no ready unit that could be found to fit both diagonal and sides of squares. What they did find though is that if you divided the side of the square inscribed in the circle by 9 and you added 1 (=10), that then you get exactly the length of the arc over the side of the square, the quarter circle. So from these two empirical facts: 11 over 7 for half circle and diameter and 10 over 9 for circle and square circumference, they eventually found out that these two ratios would meet where they merge, which is at 99 : 70 (9×11 : 7×10) (because 70√2 = 98.998…., which is virtually 99), so these two are the pivotal ratios of the approximation calculus and the geometry we found, which are based on the first 5 primes, since 10 = 2×5 and 9 = 3×3, plus 7 and 11. (primes: 2, 3, 5, 7, 11)

We shall see that we will find the quintessential numbers of this ‘natural number arithmetic cum geometry’ back in the ratios of the circles, because from now on it must be clear that what distinguishes a henge from a stone circle, mathematically, is that a henge has in principle rather more circles and symbolises more ratios (platform diameter, ditch width, embankment width, maybe middle ditch and embankment width + a stone circle proper, if present).

Before we can start to apply our geometry to the plans, which are drawn by the archaeologists who conducted excavations at the sites, we have to get an idea of how to deal with the data. Archaeologists differ in their assessments of diameters because it usually is an average of, in general, not-perfect circles, and there is the perennial question of the inner face or the centre of the stone as marker. One has to keep this in mind in appreciating the approximations in the calculations, absolute precision here is not attainable, there is always a kind of arbitrariness, but this may have been true also for the circle-setters themselves and it is, in a sense, inherent in ‘the model, the material and the methods.’

The differences in the data of the monuments we focus on here, are often not more than a few centimetres, but still these can provide ammunition for great debate when a serious mathematical or other claim is made. This is notoriously the case with the data of the Pyramids in Egypt, and will inevitably play a role in the judgement of the model that is proposed here. The question is what margin of precision is reasonable to demand and in character with the material, before the theorem is judged as: impossible, unlikely, possible, highly probable, or: proven.

Elements of this mathematical model have been successfully applied to the geometries of Giza (see impressive work by John Legon) but it has not been applied by anyone to the other geometries, that of the stone circles and the megalithic chambers, as far as I know.
Well-founded claims as these made here are bound not to be taken seriously by the scientific community, partly because of the notorious nonsense that has gathered around the pyramids over time, but mostly because that level of sophistication in antiquity is just not contemplated by most scientists (they just don’t believe it, period).

This is the same with the Pyramids; serious surveyors as Flinders Petrie have from the beginning pointed out that the 11:7 ratio was involved and later also the 10:9 ratio was highlighted by Lauer, but it is still not accepted by the archaeological, let alone the wider scientific, establishment.
On a public lecture I attended on the history of Pi, the Pyramids were not even mentioned, and, on questioning, treated as a possible footnote, whereas probably one of the main incentives to build the pyramids as they are built was to preserve the knowledge of Pi as 22/7 in the Great Pyramid, the Pythagorean theorem 3:4:5 in the Second Pyramid and the ratio 280: 198 (140:99) as square root 2 in the Third Pyramid.
When people in antiquity have gone to such great lengths to preserve such precious mathematical knowledge and you choose to ignore it with your ‘sceptical-critical-modern-mindset’ as a scientist, then, in my view, you are in an important sense not up to your job, because you willfully disregard the evident.

It is ridiculous, but any proof in this archaeological area can be rejected, although the present day obsession with precision cannot be demanded; and then, when there is precision, it is rejected as accidental. The point here of course is, and I say this again and again, that what matters is what they intended, not the precision of the execution. When we know as fact that the Egyptians had standard units of measurement, the Royal Cubit (RC) and the Remen, and when we know that the sides of the Great Pyramid come to within centimetres of 440RC and the height to 280RC so a ratio of 11:7 is evident, and you know that the perimeter is equal to a circle of circumference of 1760RC (4×440) which has a radius of 280RC, then you must be an incurable blockhead not wanting to admit that to all probability this mathematical theorem is meant by such a design, sorry. Do you want it in black on white, then? Not in chiseled stone? This is even beyond unreasonable doubt.

I would like to emphasize here again that, whether this model has indeed been used by megalithic man, as I tend to believe and think to having proven in several places, this is of no consequence to the validity of the model in itself, because it is a logically rigorous mathematics in its own right, whether this is accepted by established mathematics or not.

There is nothing fabricated about the model, once the ‘natural number rule’ is accepted, because then everything follows clearly from the numbers themselves, that is the great thing about it and that is why I believe they were capable of finding this as well, out in the field, experimentally, empirically, with numbers, some set in stone and that they started to put things together. I think this is how mathematics began in the Stone Age, with whole, that is, natural numbers, of course, and as an abstraction from the empirical data, and, maybe more importantly, I believe that they could find this because I could; because when I can do it, they could do it. They put in a collective endeavour over generations, I did it on my own with internet. They had a fabulous unclouded memory that could store information in songs or mantras and combine it for practical use or guidance. It was initially a collective endeavour and in many ways it still is, we are always standing on the shoulders of those who went before.

After all, given the age of developing agriculture, it is about reasoning on the climate and finding ways to calculate to keep the calendar, for one, or anything useful. subsequently they evolved so far they could with pride and great inventivity preserve their knowledge to posterity in the hidden ratios of their buildings, because when you have 11:7 and 10:9 as basis the rest follows inevitably when you adapt the ‘rules of engagement’ along the way.

The same mathematics applies to Egypt and the British Isles concerning their megalithic building and architecture. The Maeshowe culture which I deem to have come forward from the Boyne valley in Ireland seem proven to have overlapped but to what extent is unclear. Egypt we can pin at 2600 BCE when Imhotep started building Djosers step pyramid and Maeshowe 2700BCE and forebears is earlier, about 3000-2700BCE, but is unclear how long it lasted.

Aware of the fact that this is speculation, (but no phantasy, when it was possible), I try to paint a picture of the real struggle for life that this age characterized. In archaeologists view this occupation with huge stones was some spiritual endeavour and highly ceremonial. In my view working with the megaliths was pure necessity to survive, at least at first. The huge capstones were the storages of environmental warmth and a secure roof that could withstand the pressure of the mount above. These stones are essentially the skeleton of the function of the mount, that is to provide a temporary refuge place in cold and stormy winters, for families or whole communities. The average temperature in a cellar is about 12C, so that is cool but not cold. The point is that no fires could be made because of the already scarce oxygen around. The roof openings of the Maeshowe/Grooved Ware culture chambers, the hall 2-3m high would usually be elongated and the roof opening covered with big flat slates, abundant in Orkney. This flatness of the stones was used as tables, like the dressers in the houses of Skarabrae, flagstone desk, with backwards another cabinet on top against the wall, to shield food from the dogs or goats in the house. this is the reason the Grooved Ware culture, 3200BCE, were the first to give pottery a flat bottom, before pottery stood on a sandy floor with rounded bottoms, like Unstan, its inspiration. This innovation spread all over the British Isles, but started by the special geology of Orkney and the genius of its culture.
When they could make tables of flagstone, then benches would follow. This meant that people started to sit at tables instead of sitting on the ground.
Because Grooved Ware spread so fast over Britain owed to the many people that made a ‘pilgrimage’ to Orkney, because of its growing fame. So potters from all over the British isles would congregate, learn how to make flat bottoms and flat tables and spread the word.
The Grooved Ware culture started a kind of cultural revolution in Britain and Ireland, a trend of no longer sitting on the ground but seated at a table.
This may have been the root of the class system and it was by all means more hygienic, so less weakness and illness.

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Brodgar and Stenness

Here we set out to describe how the model works in practice, because it is a model originating in physical practice and well suited to the work within the field.
The crucial data in this approach are diameters, because they define the circles and may also define the numbers of stones, given a supposed mathematical theorem.

One of the best spaced stone circles ever is the Stones of Stenness in Orkney, but its circle is flawed, but one of the most enigmatic is the Ring of Brodgar, a stone throw away, where the spacing is irregular, but I believe that in the specific returning width of different spacings there is mathematical regularity, anyway here the circularity of the circle is perfect.

The pivotal diameter on which the whole argument is based is the diameter of Brodgar and on that basis we lay down the rules of how to apply and interpret the data.

The Ring of Brodgar has been measured by many, but the most authoritative surveyors are Aubrey Burl (103.60m), Colin Renfrew (103.70m) and Alexander Thom (103.625m = 125 Megalithic Yard)

We see that these experts agree within 10cm (103.60 – 103.70) and this means for us that any value in between is acceptable. In our model the value hovers between 103.65 and 103.68 , which depends on the type of factor that is chosen, but because there are no uses of √2 and π, everything is calculated with 22/7 for Pi and the other rational numbers that govern this model, as 10/9 – 11/7 – 14/11 and the specific unit-values. This entails that there are a few centimetres difference in our calculations, but in reality they are of no consequence because of the dimensions of the large, to very large, rough stones.

When for whatever reason a huge stone got set deeper or not exactly at centre, there was little they could do to correct it, what counted was the precision of the mathematical principle and the intended design, its order and its practicality, not the kind of precision of execution demanded or expected by some unreasonable critics 5000 years later. For them mathematics was not something of absolute precision as it is today, to its own peril as will be shown elsewhere, no, it was in the first place a tool and a wonderment, a gift from Heaven or as some say a ‘sacred geometry’.

There seems circumstantial evidence that large ‘imprecision’ of the settings means it was done on purpose and would have mathematical or acoustic reasons, as is probably the case at both the Stenness and Brodgar circles and the standing stones inside Maeshowe.

The point is that when more and more features sustain the proposed model, then there comes a moment that one must infer that mathematically something can be expected in terms of recurring ratios and it can be shown that indeed the mathematical model fits the most important of the megalithic structures once we have accepted that minor approximations are part of the heart of the model.

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See for a further discussion and explanation ‘The Neolithic Numbers’ and ‘Neolithic Numbers Explained’ and ‘Neolithic Bone Measures’.