Archive for the ‘Reflections on Mathematics’ Category

Menkaure, Maeshowe & Mathematical Mind

Saturday, November 1st, 2014

The Geometry of Square Root 2                           1/11/14

1 maeshowe axis
Perfection of megalithic Maeshowe design

– maeshowe corner connections 011
Original Icahm base-plan Maeshowe 1946

(This piece is a summary of the philosophy of science)

Mathematical Natural Number Theorem

This website is an effort to bridge the gap between mystical knowledge and scientific exploration of ‘existence’ or ‘reality’. This new ontology (ontos= authentic essence of being) revolves around a supposedly ancient, but simple mathematical natural number (1, 2, 3, 4, 5, … etc.) theorem with evidence and clarification given in many chapters. I’ve called it by many names, ‘Geometry of Square Root Two’,  ‘Pyramid Proportion’, ‘Maeshowe Theorem’ and ‘Rainbow Proportion’, reflecting its intriguing contents and versatility, but the  ‘ Lost Menkaure and Maeshowe Theorem’ is an appropriate historical name, I think, because the theorem went lost for over 4000 years, although its quintessence still stands in ancient buildings to this day. The crucial ratios of the theorem have been solidly recorded in buildings of at least two ancient ‘megalithic’ (big stone) cultures, that is in well-known Giza in Egypt and in Orkney, an archipelago off the northern coast of Scotland.

For Orkney it means going back into the late Stone Age some 5000 years ago, 3000 BCE, to Maeshowe and its first European squared space, for Egypt it means going back to Pharao Sneferu’s Meydum pyramid with ratios 275 : 175 (11:7) in Royal Cubits (RC) for base length and height, preceding, but like the Great Pyramid = 440RC : 280RC = 11 : 7 and the Third Pyramid Menkaure = 198RC : 126RC = also 11 : 7.
Meydum was the first ‘true’ pyramid, after having been started like Imhotep’s step-pyramid-type (Saqqara) and its design is mathematically similar to the Great & Third Pyramids at Giza, but earlier, so this means going back to at least some 4600 years ago, 2600BCE, to Pharao Sneferu at the end of his long reign. The two totally different cultures may have been partly contemporaneous.
Both the Atlantic and the Mediterranean ‘Megalithicum’ ‘eternalized’ their scientific/cosmological knowledge in big-stone/megalith buildings and both erected huge standing stones to exalt the sun and the cosmos (my definition of a ‘megalithicum’); some standing stones also functioned as sun dials (Obelisks – Egypt, Grand menhir-brise – Brittany, France).

The Lost Theorem can be seen as on a par with or even complementary to Pythagoras’ famous ‘right angled triangle and squares-theorem’ (a² + b² = c²), of which the ratios appear in Giza in the Second Pyramid (3²: 4²: 5²) Khafre, though long before Pythagoras’ time. The ‘Lost Theorem’ itself also happens to be a ‘square-theorem’ as we see it in the Great & Third Pyramid and in Maeshowe’s chamber (Orkney), but this lost theorem contains the relation of squares and circles (and eventually of torus and sphere), so you bet a more intricate one, and indeed it is different, very different; so different with its seemingly imprecise and ‘hidden’ rules, that it got lost and the lesser but apparently more precise Pythagoras-theorem (3²: 4²: 5² -> 9+16=25) became a preferred design with later pyramid-builders and survived to this day as one of the cornerstones of geometry.

maeshowe equilateral triangle

equilateral triangle Maeshowe ground plan

maeshowe Pythagoras ell-remen
Maeshowe’s plan with hidden Pythagorean triangles and parallels
The analysis of Maeshowe’s ground plan design, shows perfect Pythagorean triangles and parallels at the heart of the building, and much more. (see below and at The Measure of Maeshowe)

nine grid maeshowe
Maeshowe: Circle-Square-Circle Theorem and Rainbow Proportion
big yellow square perimeter equals red circle circumference; their areas relate 11:14
red centre square side=9Megalithic Ell (4.71m) with inner blue encompassing circle segment=10ME (5.236m), inner blue circle = 40MEll (ME=0.5236m) (MR=0.37025m)
big blue circle = 80 Megalithic Remen (MR) equals inner diameter Stonehenge!= 29.62m
radii ratio inner blue circle to red circle = (Double) Rainbow Proportion = 11:14
My ‘natural-number-logic logo’: yellow 9-grid with orange square in blue circle (orange=red here)

What the Lost Theorem does is expressing ad infinitum the ratios of ‘squares in circles in squares in circles….’ (inscribing, encompassing) in natural numbers as we see it in the picture, that is, in a specific set of fixed unchanging integer ratios, like 11:7 and 11:14 and 10:9 and 162:100, which are the quintessential ones. (To remind you: in standard maths this is only achieved through the irrational transcendental number Pi, π, which though blocks any smooth relation between natural numbers in circles and squares, and also through the irrational golden mean number 1.618… =Phi, which is here 1.620). The eventual order is achieved in the Lost Theorem by using a ‘rational’ approximation of π (ratio circle circumference and its Euclidean diameter, 3.14 159…) and by a rational value for π²-squared ( 9.8…), respectively by the rational numbers: 22/7 (3.14 2857 1428…) (also 2800/891= 3.1425 36…) and 800/81 (9.87654321.= 22/7 x 2800/891), all expressed in natural and rational numbers.
The theorem’s true significance though lies in ‘squaring the circle by circumference’ (this is different from the famous and notorious ‘area-squaring of the circle’), which we see here (picture above) realized in the yellow square perimeter being equal to the red circle circumference. The red circle diameter is 9, the big blue circle diameter is 10, as ratios. The yellow square perimeter is 36, the big blue circle perimeter 40, which is again ratio 9:10.

This ‘squaring the circle by circumference’, as mathematical operation, gave us ‘the Great Pyramid’ as the geometrically square abstraction of the (hemi-) sphere (globe), that is, of the Earth.
In Orkney though the theorem, in a peculiarly extended but more rudimentary form (8: 9: 10), lies hidden in the lay-out of Maeshowe’s megalithic chamber and comes into the open in the dimensions of the several man-made circles of nearby Ring of Brodgar ( its standing stone circle and rock-hewn moat with 2 sloping edges give 3 ratios), but there expressed in the big numbers of Giza (99 or 198 and 140 or 280). This is the full natural-number-solution for the ‘rationalisation of number-irrationality’, i.c. of ‘square root 2’ (see Rationalizing the Ir-rational)

If this number-crunching is too much for the reader just skip these parts, which are necessary for the scientific argument and its (circumstantial) proofs, but don’t go away, it’s not all like this, continue, say, at ‘Sophistication’

Megalithic Pi
What could make this website suspect for many in this respect is that I found that practically the same unit of measure as Egypt’s established Royal Cubit (RC= 0.52375m) fitted perfectly and mathematically meaningful in major Orkney megalithic works (and all over the British Isles, for that matter) and I have inferred that the Megalithic Ell (ME= 0.5236m) was used in Orkney and elsewhere in the Isles, possibly (much?) earlier (Newgrange, Ireland?), than in Egypt. On top of that I found this measure (ME) based on two male ulna bones (26.18cm, under-arm, ell(-bow), which was a gift), has, as diameter-size, an ingenious mathematical relation to a slightly revised Megalithic Yard. My MY= 0.8228m and is thus 6.2mm smaller than A.Thom’s megalithic yard =0.829m, but usually better fitting the data and always with mathematical meaning, because this revised Megalithic Yard turns out to be equal to the half-circle segment of a circle with diameter 1ME=0.5236m, and equals two male tibia bones (0.4114m, another gift, average height Neolithic male 1.75m, same as Egyptians). It results in the formula for a Megalithic Π (MPi) as 2MY/1ME (2×0.8228/0.5236= 22/7)
My only other defence for this rather ‘incredible’ but nevertheless striking situation must be that this mathematical model I found(ed) and use has connected many ‘incredible’ things in the course of my exploration, events that I personally experienced as great gifts not only because they were completely unexpected and enhance the beauty and profundity of the model, but also because their enigmatic occurrences in my personal life keep me highly motivated to pursue this long quest into the deep past; there is a treasure to be found at the end of the (double?) rainbow, as you know.

Megalithic Ell
My earlier mentioned fixed measure of 0.5236m as I came to use it for Giza’s Royal Cubit (RC), elsewhere usually given as .524m, is a gift in still another way, because as Megalithic Ell it not only fitted Orkney better than it does Giza, but it turned out to be exactly 1/6 of Pi (6x.5236=3.1416) and 1/5 of Phi^2  (5 x .5236=2.618) as I only realized during my Orkney-calculations. This means that a (stone-) circle diameter measured in metres gives multiplied by 6 the number of megalithic ells (ME or RC) in the circumference (6ME/1m = Pi ); a very handy gift this one because it may give a first indication of the possible number of stones in the original circle, the more so because also the yard and remen are at hand as measures.

To make things worse for the sceptics I show that the diameter of the stone circle of Orkney’s Brodgar is equal to the side of the Third Pyramid, and equal to the distance from the King’s Chamber floor level to the top of the Great Pyramid, both 198RC (103.67m); and that the Brodgar ring’s circumference is equal to the ground-diagonal of the Great Pyramid (880 Remen= 325.82m or 396 MYard (4×99).

Natural number Logic of Menkaure design

Moreover it shows the intrinsic relation between the Great and Third Pyramid in Giza (20:9), all this insight in relations thanks to the, possibly contemporary, measures of Brodgar and Maeshowe!
My important finding that the Third Pyramid must be the mathematical key and theoretical blueprint to the Great Pyramid and relates in volume virtually as 1 : 11, comes straight from the analysis of the Orkney ratios!

Pyramid basics

Ratio of Great Pyramid same as Menkaure  280:440 = 126:198

This is of course all a bit too much to believe in for most people and certainly for the professional archaeologist who will dismiss it as ‘lunatic fringe’ science, but, reader, I had the same problem, I was not out to get these things at all and at first not even really happy when I found it, for this very reason (this is ‘too much’), but I’ve checked it all many times and I am not in the habit of fooling myself, nor others; it’s in the data and especially in the ratios. (I compare with Lehner, Cole and Petrie data). In case of differences they are all too close to call, or just ‘controversial’ in this Megalithic-context, with differences expressed in centimetres over lengths of hundreds of metres. (Lehner’s data for Gr.P.-sides are all 230.33m,  Cole’s average = 230.36m). The inevitable bickering about the correct length does a lot of damage to a serious approach to the logic of the designs, because it is about (number-) logic integrated in design, it is about the intention of  the design, not about perfect execution, it’s about the genuine probability of a deep insight and a great Cosmic Idea, ‘Earth as a sphere’ for one thing, it is not about cynical 20yth century scientific scepticism.
It’s about possible synchronism and recurring serendipity; and about a near mystical (historical) Enigma: how two so widely differing cultures could come up with the same mathematics, the same units of measurement and the same ‘cosmic length’ at about the same time.

The high mathematical sophistication that speaks from my analysis of the  late Stone Age designs in the British Isles poses a problem; in the first place for me. Is this really possible? I am not on this quest for many years to pursue and produce an ‘impossible’ story, or trick my readers, so I had to go a long way before I could convince myself of the ‘possibility’. What convinced me most were those moments of complete recognition, where I know I could not have found any of it on my own had I not been led by the analysis of their ratios and had I not understood the mathematical logic in their designs. Those were great moments of ‘communication and revelation’, a 5000 year gap, but One mathematical Mind, it was, and is, as from the Beyond, timeless, it’s living the experience: there is no time, their mind is still here, there is a record, a communication, it is history, no longer pre-history, they have communicated  in the most universal manner possible, the language of mathematics!  So for me there are too many instances of ‘striking’ confirmation that I could reasonably doubt Orkney’s Stone Age Man’s ability and sophistication. Maybe I am tricked by fortunate coincidences (but then, so many?), but I also see it confirmed in their superb building techniques, in their involvement in cosmology and calendar making and, finally, in the calculating and recording of numbers that all this must have involved. Without having signs for numbers they realized this by incorporating ratios of numbers in the dimensions of geometrical designs and by placing them in circles and marking numbers of standing stones for counting their cosmological cycles. (See my pictorial analyses of the design in the ‘Maesure of Maeshowe’ in triangles, squares and circles, it’s fabulous)
Near Maeshowe, at the now famous Ness of Brodgar site, situated between the (huge) Stones of Stenness (ever 11 or 12 standing and with less than a third left standing some near 6 m high, it still holds some of its original echo in its centre, manmade!), itself one of the earliest stone circles and henges, if not the first (having virtually the same outside measure as Stonehenge, mind you!), these Stones of Stenness on one side and the huge Ring of Brodgar (ever 60 or more stones) on the other, there is clear evidence of paint-making at the Ness, paint which could have been used to mark the stones of the two nearby Rings with different colours for counting and recording different cosmological time cycles. Here the standing stone circle becomes a record holder, a calculator and a geometrical playground, like connecting certain stones with ropes, making a square with the four cardinal points, like in Maeshowe, knowing the diameter as 198ME and consequently the side of the square is 140 ME = 73.30m; so possibly it was an educational place, in all it was so much more scientific/cosmological than rigidly ritual in practice.
Not only the amount of stones holds numbers, these are also in the dimensions of the works, where the ratios get geometrical and arithmetical meaning. I have argued that the two standing stone circles with drumbeat together with a huge pendulum in Maeshowe (see picture),


Pendulum Maeshowe + starry sky

were instrumental in creating day-time, clock-time, at the equinoxes. This is what the Ness of Brodgar and its environs were possibly about and probably famous for: the ‘Creation of Time’, a spiritual cosmological centre with people flocking here as ‘Pilgrims of Time’ from far and wide, to hear the heartbeat of the universe in the drums that kept the rhythm through day and night at the equinoxes  and to compare it with their own heartbeat and take that home as a measure of time, but also to see the many rainbows and the Northern lights when lucky (last year!), to hear the echo of their voices at the centre of Stenness, to being at the end of the ancient world, beyond the North Winds (Hyperborea).

My explanation for the mysterious mathematical relation between Giza and Orkney has moved from Egyptian naval expeditions visiting the ‘fabled’ archipelago (Hyperborea, Atlantis) with on board the great architect and healer Imhotep in his youth, who then later built the  Saqqara pyramid, to ‘morphic resonance’, same (evolutionary) things occurring in unconnected places at the same time (Sheldrake), onto the shamanic ‘Supernatural’, of which I now think the last one is the most probable, but morphic resonance is a close second. As shamans claim they derive their knowledge about use of medicinal plants in trance from the (spirits of) the plants themselves and they climb the Rainbow, I have considered that a geometrical form of their beloved Rainbow might have revealed its numerical/rational/pictorial make-up to the shaman in a similar way. (shaman Joska Soos’ paintings are highly geometrical, as are some of his contentions, see ‘Shaman and standing stone’). Since it is very well possible that Giza and Orkney were, at least partly, contemporary, a synchronic shamanistic and supernatural revelation in both cultures must be reckoned within the  possibilities, it could be that the Megalithic Ell, as well as the equivalent Royal Cubit, do indeed represent the ideal ‘cosmic unit of length’, each occurring in some local revelation and showing again ‘man as the measure of all things’ as we see it in palms, feet, thumb and finger measures and that indeed this sacred length was found by or revealed to both scientific Megalithic cultures as they appear to use the same number-logic as well and came to the same ‘crucial length’ of 198 Megalithic Ell or Royal Cubit (198=2x99) = 103.67m.
It’s all about the ratios of 11:7 and 10:9 and their ‘interactions’. We see this length returning in the diameter of Brodgar’s Ring and in that of one of the inner circles of Avebury’s lunar clockwork  (chapter), even the (irregular) stone circle of Newgrange (103.6) is thus reported, but, more importantly, this length is also equal to the square base length of the 3rd Pyramid, 198RC.
Lehner’s data suggest 196RC = 102.63m (L=102.2) and 200RC = 104.7m (L=104.6) but this makes no difference for the relevant value of the diagonal. (I think a satellite and computer imaging should do these jobs by now).
This has consequences for the angles of the slopes, but these seem hardly different from the Great Pyramid, so I maintain that the Third is the prototype top of the First, the Great Pyramid, otherwise there must be a correlation between the different sides of the Great and Third, but Lehner gives what seem to be unwelcome rather sloppy data (230.33m) for all the sides of the GP, this is pertinently wrong considering the excellent work done by Petrie and Cole, largely in close agreement, in the past and, worse, Lehner’s measurements can never translate into the slope that he gives for the Great Pyramid; his data are intrinsically incompatible, so this makes me wonder what it is about. Lehner has given a very bad contribution to the study of the Pyramids.

gp_air[1] legon

I stick to the size 198RC because it also appears as the distance of the floor level of the King’s Chamber to the top of the Great Pyramid, 198RC (280-82). Intriguingly it is also the pyramid side lengths at the exit levels of the air-shafts (see picture John Legon), so that the top 1/11th volume of the Great Pyramid upwards from shaft-level is 126RC (280-154, Legon) high and consequently identical with the Third Pyramid. This is the reason for the angles of the shafts, they had to mark out on the outside the theoretical mathematical principle topping the building as its ‘mathematical brain’, a visible horizontal mark and a practical airshaft at the same time (there were most probably people in this building going about their business whatever it was, see gallery, no one was ever buried there), so a visibly marked top with ten times its own volume beneath it.
The number 11 (22/7) is the number of the Great Pyramid (440), 37 of the Second Pyramid (100+37; 3 x 137= 411) ,  9 and 10 and 9 and 14 of the Third Pyramid.
The height of the Great Pyramid is equal to the diagonal of the Third’s ground floor =√(198²)x2 = 280.
These differences are very small here but can be decisive in our argument when it comes to an overall picture. It is only because they had so great reverence for the building technique and genius of their ancestors (Sneferu) that even the Menkaure pyramid, the key to the mathematics, was built, where it is so much less impressive than the others, close to being an anomaly. It may even be that the foundations of all three pyramids were laid at the same time still by Sneferu or soon after. Obviously its authentic value and scaled! design was seen as crucial to the cosmological (scientific) message of the whole site. This is what I read in it. When the temple is harmonious, the universe is harmonious.

Tibetan Buddhism
The shamanic connection, has become fundamental in my understanding of the Stone Age spirit, time and again I emphasize how much I try to appreciate the world-view of shamanism as it emerges the world over in the Stone Age and with this fascination the Tibetan Buddhist connection comes to the fore. I am aware that shamanism is a sensitive topic in Tibetan Buddhist circles, because it would be seen by the outside world as not genuinely Buddhist. During the recent Kalachakra in Ladakh (’14) the Dalai Lama mentioned explicitly this perception by critics of Tibetan Buddhism, possibly within Buddhism, that Lamaism and shamanism are not true Buddhism.
But what is true Buddhism? Buddhism always takes the flavour of the culture where is flourishes, like Ch’an (Zen) in China, Nichiren and Zen in Japan and Vajrayana and Tantrism in the Himalayas. The early Buddhist monks were compelled to show the Tibetan population that their Buddhist teaching could produce the same powers as the shaman possessed in order to convert them and gain their support.
I consider the shamanic flavour of Tibetan Buddhism as one of its great attractions, which does not at all interfere with the sincerity of the Tibetan commitment to the real great Mahayana ideals of compassion and universal liberation. On the contrary Tibetan Buddhism seems designed to heal the soul and who else than the Buddha was also called the Great Physician, the Great Healer (bhisakko), the supreme surgeon ( sallakatto anuttaro) and in that respect many Tibetan Lamas are outstanding healers.

How could there be anything ’embarrassing’ about being the heirs of spiritual traditions and techniques that go back to the unfathomable depths of the Stone Age and to the first great awakenings of the ‘human’ mind in that age; after all, several Buddhas are supposed to have preceded our historical Buddha and they don’t come by every few centuries, it seems. Besides that the original Tibetan shamanic Bon-religion has now effortlessly been incorporated in the Tibetan Buddhist brotherhood, so it is all very close, but not decisive.

To me the Tibetan Buddhist and shamanistic parallels as: chosen-ones, rain-making, oracles, divination, drumming, fast travelling, form the living bridge to a ‘spiritual atmosphere’ and wisdom of the Stone Age. These techniques though are peripheral in the central teachings. The logical, dialectical and magical mindedness of the Tibetans reflect the historical situation of the tradition. It is not well-known that Tibet in its Buddhist florescence for centuries culturally dominated the whole of Central Asia to deep in Siberia. The similarity with Mongolian Buddhism is evident and that the latter also share in a shamanic tradition cannot be doubted whatsoever.
What fascinates me in shamanism is the extra-ordinary (suggestive) powers of the human mind, because those must in origin have been universal, as shamanism was and those powers must still be available to us all, in principle, I think (I hope).
Tibetan Buddhism is probably the only organized tradition which has preserved and still practices these age-old techniques, most of which are beyond scientific comprehension (at the moment) and rock the foundations of the Western rational world view; “this is not possible”, we say. This same Tibetan Buddhism is now also closest to modern scientific approaches of reality  (Kalachakra) as we will see shortly, which again shows the extraordinary intuitive insights it is based on.

Shamanism has been the spiritual development and exploration of the human mind through the Stone Age the world over in different places with different techniques but always with extra-ordinary healing mind-powers as a result. The first development of language must be thought of as within the shamanic horizon. Those who study root-languages like Proto Indo-European (PIE) know all too well how much insight and intelligence went into the formation, derivation and meaning of words, and there were sounds only, no symbolic frame work. Here again we see unfathomable wisdom and capability in the deep past.

As I have argued elsewhere: the very fact that shamans could risk their lives for the benefit of others places them on a spiritual level that is beyond any religion and at the same time expresses the highest form of spiritual attainment, that is: self-sacrifice for the right cause ( Jezus, Socrates, Buddha-teaching).
Shamanism is in this view pre-religious and pre-philosophical enlightenment expressing a rich and integrated spiritual life in which the whole world is ‘spirited’ and open to contact, even communication.

What I have stated so far is only the historical and cultural context in which I place the mathematical theorem that I found independently of these rather ‘solid historical records’. First it was ‘Giza’ that surprisingly fitted my model, then a few years later ‘Orkney’, both these  places were just other incredible ‘gifts’ to sustain my ‘improbable’ findings in ‘natural number mathematics’ and some pertinent claims in other fields, but whatever their status as scientific confirmation (at least in History), these (pre-) historical places have no bearing on the mathematical validity and importance of this model, as such. The model stands on its own. Try to grasp it, it is amazingly ‘solid and aesthetic’.


Part I

The mathematical model

Decimal nine-number system (1.111…; 2.222….)
Initially this peculiar geometry evolved from a ‘decimal nine-number system’, inspired by cuneiform clay tablets from ancient Sumer (Iraq) I saw in a newspaper and worked out.  This 9-number system, Sumerian or not, is is a rational numbersystem and still at the heart of the calculations, as it is the ratio 10:9, which produces the rational numbers (10/9  =1.11111…. to 9.99999….=10, then: 11.11111… etc., so 9 numbers in the space of 10) and it is perfectly tailored for the calculations of the derived geometry; they cannot be separated. What makes this ‘natural number-logic’ such a sure and fast calculator is that it is self-correcting and self-contained and based on  natural and rational numbers only. As soon as unsuitable input is involved the logic of the numbers crashes, moreover the system is such that the answer is already at hand and could in principle be immediate. Here we may see how ‘immediacy’ in physics actually functions, it’s a mathematical necessity, outside of time (another proof time does not exist), and, given certain specifics, the answer is logically inevitable and through timelessness becomes ‘immediate’, and easily ‘non-local’ (space-less, without distance). ‘Immediacy’ functions ‘non-local’ through the Zero-dimension, I claim, the inner border-point of space, without locality, as I will describe below.

Fine structure constant*
Our model produces ‘theoretical’ values which means that our theoretical answer is usually not precisely matched in reality; this shows nicely in the universal ‘fine structure constant’/α (alpha), alias, ‘coupling constant’/e which in my system turns out to be (repunit, R3)  111 x  (10/9)^2 = 137.0370370370…= 3700/27, α, a rational number at the heart of our model and as e it is √(27/3700) = 0.08542421…. [because it is about repunits  (numbers consisting of only ‘ones’, 111….111) and the ratio  10 : 9, squared = 9.87654321]

Richard Feynman wrote about the number e, the ‘coupling constant’: “There is a most profound and beautiful question associated with the observed coupling constant, e , the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won’t recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to Pi or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and “we don’t know how He pushed his pencil.” We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly! “

This is about a magic number, a BIG number, in short, and my system seems to show ‘how HE pushed his pencil’. I can present it in different formulas which each may show the deep-structure of the number. R3, 111,  a major repunit, is involved and our central ratio 10:9 (1.11111…….) ‘squared’ (1.234567…) is involved but the number also can be read very plainly as 3700/27 = 137.0370370370… , though scientifically defined as 137.03597….. , a difference of 0.0010……, one -thousandth) The numbers 37 and 27 have a peculiar relation in 27 x 37= 999 → 999/9= 111 (R3), the numbers 3 and 37 in R3 are firmly related to one- third of the repunits in 3 x 37 = 111. We can also produce the Feynman  coupling value of 0.08542455… as the square root of 27/3700 = 0.08542421…, with difference of 0.00000034…., that is less than 1-millionth. Not bad (and rather better than in the ‘alpha’ version of a thousandth!).
So we see that the shape of this ‘Holy Grail number of Physics’ is rather trivial in the rational numbers 3700/27 or the square root of 27/3700, whichever  way you like (the number is apparently known in different shapes). You won’t believe me now, but time will tell. You may think the prime 37 is random, but the repunits (111….111) show it isn’t:  3 x 37 = 111 (R3). Repunits are the key to the calculator and the fundamental wave-patterns as they reflect so closely the 10/9= 1.111111111….. relationship. Then there is the number 27 = 3 x 3 x 3, a basic cube of the pivotal prime 3. The number 27 appears also prominently in packing theory with circles in hexagons (Kepler density quotient pi/V18= 0.74048222… in my system is 20/27= 0.740740…).
Possibly this is a key to the  enigmatic hexagon at the pole of Saturn!
This is no longer  ‘recreational mathematics’, which repunits are supposed to be in professional circles,  this is the ‘real stuff’, however hidden and enigmatic it still may appear now.

Rainbow Proportion
As my personal contribution to the significance of this ‘pre-historic mathematics’ I regard the insights that the system’s logic describes a basic 10:9 resonance, which appears between spherical and toroidal waves, as I understand it; that this resonance is found in the Solar system data and visible in the rings of Saturn, that as other example there exists a steady 11:14 (double) Rainbow Proportion resonance (closely related to the Golden Section) and, last but not least, that the 2-D circle and square natural number relations can stand for sphere and torus (ball and ring) relations and for the necessary interaction of their wave-systems in natural number resonances (resonance is always a whole number phenomenon, keep that in mind, it is also the ultimate scientific ‘rationale’ for this peculiar mathematical model I designed and employ).
This last insight then is seen as the fundamental geometry of space down to the Planck-length. The mathematical model in my view could describe the resonances of the deep field vibration of physical reality, analogue to the manner of super-strings. Nevertheless it is simpler and more straightforward than super-strings and works on all scales, be it  not with 11 dimensions but only  5 or 6, depending on what is included and how you count. (is the zero dimension a dimension?).
Here we see what the Theorem does: it connects seemingly completely unrelated phenomena by shedding unexpected light on certain relationships, always touching on the deepest levels of reality. It’s like a magic wand sometimes. The dimensionless universal constants, for example, have no place in any existing scientific physical theory, but they find a natural translation in this model and that is a feat of great scientific importance.
Since I have never stopped thinking about the implications of these fundamental integer ratios I have naturally come to a rather complete frame for ontology and cosmology. I am aware this is pretentious, but I had no choice, it grew as of its own logic. Besides that  it was supposed to be a theory of everything, that was the challenge in those days, maybe still is, but I still wish to formulate something, find a language, that is a real bridge from science to the mystical experience and I hope it is consistent and universal and that what I do with my very limited knowledge is only set the first feeble steps in the revelation of its entire scientific and mystical scope, which might be epochal.

Part II

What reality?
Reality is an important concept in philosophy and hard to define but it could be said to be that which we experience as ‘the world’. What we perceive as ‘reality’ though, cannot be severed from our faculties. We perceive a world as it presents itself within the scope of our senses. Our, limited, senses ‘create’ the world as we ‘experience’ it. This insight is in the West usually attributed to Kant, heralding the European ‘Enlightenment’ epoch, but it was an analysis the Buddha had made already over 2000 years earlier. Today the influence of the mind on reality is dramatically shown in quantum-physics where scientists have lost their grip on their concepts, creating ‘contradictios in terminis’ and other logical monstrosities, like: ‘vacuum-energy’, ‘wave-particle duality’, ‘probability wave’…..


That there is a ‘dependent origination’ of reality is another of the Buddha’s insights which scientists now have to grapple with. The link to Buddhism is in my work most ‘material’ in the concept of the ‘space-pixall’, reminiscent of the ‘space-particle’ of the Buddhist Kalachakra philosophy, in which there are five types of particles, related to earth, water, wind and fire and the aether/space as fifth. This last one is the smallest, ‘aetherial’, but most fundamental. It is what remains when a world-system disintegrates: space-particles. The Buddhist analysis entails an emphasis on the ‘granularity of space’ as well as gives it a ‘body’ and even permanence. This characteristic makes it compatible with the concept of an ‘aether’, since this is understood the world over as ‘the stuff filling space’; the ‘body’ of space, the energy of space.

Dalai Lama
A space consisting of space-particles generates an other cosmology than the BigBang as we will see. (note the space particle in the following Buddhist quote, as the ‘fundamental con-substantial cause of the entire physical world’, so the first and most basic ontological unit).
The Dalai Lama describes it as follows: “Buddhist cosmology establishes the cycle of a universe in the following way: first there is a period of formation, then a period where the universe endures, then another during which it is destroyed, followed by a period of void before the formation of a new universe. During this void, the particles of space subsist, and from these particles the new universe will be formed. It is in these particles of space that we find the fundamental con-substantial cause of the entire physical world. If we wish to describe the formation of the universe and the physical bodies of beings, all we need do is analyse and comprehend the way in which the natural potential of different chemical and other elements constituting that universe was able to take shape from these space particles. It is on the basis of the specific potential of those particles that the structure of this universe and of the bodies of the beings present therein have come about.”
Note here the exceptionalpermanence’ of the ‘space-particles’, where nearly everything is impermanent in Buddhism, but it brings these particles on a par with the Mind, which is of great importance as we shall see. The great lama actually sums up what has become a major challenge of this website: ‘All we need do is analyse and comprehend…. how the natural potential of… elements… could take shape from these space-particles’.
Unwittingly I have for years worked according to the Dalai Lama’s ‘program’ in my efforts to give the space-particle (-pixall, -grain) the fundamental place (mathematics) and properties (resonance, field) it needs to fulfil its basic function. This is a ‘realisation’ I’d never expected when working on the ‘pixalls’ although the Dalai Lama’s talk of space-particles indeed originally had inspired me to go back to my erstwhile visions of ‘sparks filling space’, subsequently triggered by the fundamental mathematics I found when in Ireland (’98/’99).

The following is another gift, this time from Einstein, himself a bit of a mystic , where he describes exactly the characteristics of  a ‘Deep-field’ by naming ‘singularities’, extremely dense  point-forms of energy, as the fundamentals (in 1909, before GTR). He describes effectively an aether in my view ; the point here is the granularity, which disappeared in the General Theory of Relativity and with it the basic field interferences, the oscillating field.
Einstein (1909): “I imagine to myself, each such singular point surrounded by a field that has essentially the same character of a plane wave, and whose amplitude decreases with the distance between the two singular points. If many such singularities are separated by a distance small with respect to the dimensions of the field of one singular point, their fields will be super-imposed and will form in their totality an oscillating field that is only slightly different from the oscillating field of our present electromagnetic theory of light.”
Put for ‘singularity’ the term ‘space-pixall’ and it says what I could not say better, the oscillating field is the ‘deep-field’, the aether. The pixall is the ‘valve’ of the zero-dimension, which ‘carries’ the pure light.

Max Born
A  very good description of the classic ether-concept, is given by Max Born, it helps to clarify the difference between ‘somethingness’ (aether) and ‘nothingness’ (relativists)
Born (1924): “The undulatory, or wave theory, (..) sets up an analogy between the propagation of light and the motion of waves on the surface of water or sound waves in air. For this purpose it has to assume the existence of an elastic medium that permeates all transparent bodies; this is the ‘luminiferous ether’. The individual particles of this substance merely oscillate about their positions of equilibrium. That which moves on as the light wave is the state of motion of the particles and not the particles themselves.”
We see here another suitable description of the deep-field and the ‘space-pixalls’ as I propose them, only that the pixalls are not really ‘individual particles’, but interlocked singular tiny fields (down to Planck-length region), like Einstein’s description above. (It must come close to Descartes’ ‘vortexes’ which also Maxwell mentions; these are typical ‘scientific intuitions’)

These quotes of prominent proponents of mysticism and science illustrate perfectly where the two converge in the concept of discrete fundamental units and how the Big Bang-cosmology is made obsolete in one stroke (by the permanence of space-particles and by cycles).
If there were a beginning of time how could there be eternity? Eternity can only exist and be felt because there is no time, so how can there be a ‘beginning of time’, as cosmologists claim. Eternity is not material, it is space in its ‘suchness’.
We see that Einstein intuitively describes an ‘aether’, a fundamental ‘field-filling’ of space – possibly the reason he abandoned the idea (because of his adagio: no aether)- his ‘singularity’ has a field, which, I say, it can only have because it spins in absorbing  and emitting waves. In every point of space there is a spherical incoming wave-front as the sum of all the cosmic wave-field fronts. This front creates spin in every point and creates a reflex wave going outward, this creates a standing wave field (oscillating) and gives the centre stability and inertia. It is this interaction of the incoming and outgoing wave, the order of the standing wave which relates to the number 9, that our mathematical model describes or, how a spherical wave transforms into an outgoing toroidal wave, that is what this website’s peculiar mathematics seems created for.
At specific frequencies the pixalls lock into field-mode and get energized, this is the reality we see and the light it is made of. This spin is also seen as a vortex to negative infinity and, as such, a border-point of space opening to the zero-dimension, the pure light.
Space only has an inner boundary, like there is no ‘other-side’ to the centre of the earth. When you can imagine that there opens a new dimension at the centre of the earth, which maybe we can visualize by taking the volume of the horn-torus as our curved space, and going through the centre point, aka borderpoint of space, we come into the space surrounding the torus surface, that is the zero-dimension of pure light (the 3D horn-torus is peculiar because its one centre point borders its whole surface and in 2-D its surface is a square).
The horn-torus, 4. π².r²  (indeed  the square of the circle =(2. π. r)², with two equal radii), is the pivotal geometry of space on this website and shows consequently its surface equals a square, but is bordered by one point only; this now is what ‘space-curvature’ can do in mathematics, I suppose, but here I miss all the tools to continue; for instance this may also be the ontological key to the holographic aspect of reality, the whole surface of the horn-torus is reflected in the central border-point, in every point of space. No better way to see how all is one.

Pure Light

The ‘pure light’ of the zero-dimension, although ‘outside’ of space, is in every pixall, so every pixall is a ‘carrier’ of light (‘lumini-ferous’ aether), which makes it literally true that everything is ‘made of light’ and this is why light does not travel and ‘speed of light’ is a misnomer. In this conception the ‘ray’ is a string of sparking stationary pixalls, which makes it appear discrete as a photon in observation, but the photon has no self-substance, it is just a rather random sparking local pixall. The notorious double-slit experiment becomes trivial when you realize that the whole interference pattern of the ‘deep-field’ changes when you open or close a slit. The deep-field lines change, so do the the pixalls in their alignments along the field lines, there are myriads of wave-fields going through the slits, so that the pulse travels through a completely different local deep-field in either case.
The whole ‘quantum enigma’ may disappear when you have an aether, a deep-field, and no ‘self-substance’; the ‘virtual particle’ is the ‘dormant pixall’, nothing virtual about it, except for its permanent metamorphosis.
There is a deeper layer to reality than meets the eye and scientific lens, a layer that has to be inferred because by nature it cannot be ‘detected’, it moulds everything, it is the zero-dimension at the heart of every space-pixall.
I do not think quantum-mechanics is as mysterious as it is made out to be, it is a matter of the right description, I predict, and of giving up long cherished stern views about the ‘non-aether’ by embracing the aether again.

The notorious ‘wave-particle duality’ gets also dissolved in our description, because the wave is the form, the configuration, whereas the ‘particle’ is the observed momentary pixall-contents of the, always ‘passing’, configuration through the deep-field.
The wave is the pulse, the form; the space-pixall is the medium, the substance. The ‘relativity’ here is that the deep-field is always at rest, no energy ‘forms’ though are ever at rest, they are always changing, moving. From the point of view of the ‘form’, it is itself at rest and the deep-field flows through it, this is the ‘relativity’-aspect and caused by the substance of the form being fundamentally at rest; that is the paradox of the moving form (we feel at rest while constantly moving through the deep-field, our substance is always at rest, like being at rest in a seat in a moving vehicle).
The form moves through the featureless fixed pixalls of the 4-D space screen, the deep-field, so the form (image) has no substance of itself, no ‘self-substance’, its ‘material’ content is the seemingly fleeting deep-field. Nor does the deep-field have form of itself, it is Substance as an aggregate of space-particles, formless, ‘unborn’. Only together  form and deepfield create an ‘appearance of endurance’.
’Form is emptiness, emptiness is form’. (Heart sutra, Buddha)

So what we try doing in the following is finding a description of the ‘One-ness’ in personal experience and reality, an ‘enlightened ontology’; or how mind and space (-pixall) merge in the zero-dimension, how matter becomes mind, how duality disappears. That is the crux.

Part III

Space and mind

To get to grips with ‘matter’ as manifestation of the mind, we have to carefully describe how it occurs in space. We have to become more aware of this deep relationship of space and matter, and of space and mind (where time falls away), because when you thoroughly catch that insight you must be halfway on the road to enlightenment, to ‘true liberation’. Experiencing the ‘ideality of reality’ is a key to understanding mysticism and concepts like Maya and Emptiness.
A way of understanding space as mind is seeing that space is no ‘volume’, and has no ‘abode’. A volume is finite and is in a place, but space nor mind have a boundary or location.
Our concept of physical space is wrong, because we see it as a ‘vacuum’, ‘void’  containing ‘things’, whereas it is the ultimate ‘plenum’ and the ground of all existence as the manifestation of the One (Mind). Your life, your ‘being (thrown) in the world’, your existential ‘Dasein’ (Being around) is, however you look at it, at basis and in last analysis only: ‘experience’. This is what makes our lives ‘mental’: our body produces ‘experience’ and experience produces ‘memory’ (past) and ‘expectation’ (future), both ‘fill our mind’, it is our ‘stream of consciousness’, ‘our world’, the ‘myth’ we believe in and live by. Our seemingly solid world is no more than the background of our individual and collective experiences; that is the real stuff ‘our world’ is made of. How strange.
We don’t understand the nature of our experience because we do not realize the workings of reality, that is: ‘of our mind’. Again, our mind shapes the experience (you feel buoyant, world is bright, you feel bad, world is bleak), so the mind is the essence of it all, not matter, matter is the surface, the surface of the mind. (see No matter at all)

99% space

What we know of atoms, the entities that give all the ‘qualities’ to ‘matter’, is that they form a neat mathematical system, the periodic table, and that more than 99% of their ‘body’ is space! It’s the physics-teacher’s golden moment when he baffles the class with this truth, matter is 99% space, wow! You would think this discovery was more important than Copernicus helio-centrism, but no. This startling fact namely implies that we ourselves are 99% space. It was already a shock to find out we are some 70% water, but 99% space really beats it all. What ‘matter’ are we still talking about then, you wonder? We are the key to understanding matter, because we are matter ourselves, that is: 99% space. When all that bodily matter is (mostly) space it becomes easier to understand feelings and consciousness, it is just the state of space in the body. The brain is also 99% space, as if we didn’t know. Through the brain we are one with the deep-field and the zero-dimension. We are plugged into the timeless universal storehouse of experience and have instant memories of a distant past, another indication time is not involved. The concept of ‘karma’ is also outside of time, as seems all ‘dreams’ coming from the universal storehouse. (A long dream compressed in a few seconds, no time again)


One of the central points in my approach to ‘cutting edge’ theoretical physics is the re-introduction of the concept of the aether. In many Eastern philosophies, as in ancient Greece, the popular 4 elements : earth, air, water, fire are invariably philosophically accompanied by a fifth:  ‘aether’, also:  ‘space’. In my system this ‘aether’ is usually called ‘deep-field’, as the actual ‘substance’ of reality, although, it must be said,  this goes beyond the classical aether-concepts. Scientific concepts cut reality up in fragments, but this present ‘aether’ is integral with the One, an ‘experience’. Its concept is as in the East: a rather all-encompassing element, space.

This ‘aether/space-as-substance concept’ is an illuminating way of describing the inmost of ‘surface-reality’ (holography) and with it elements of the mystical reality, like Spinoza’s Substance. It is a way of describing the ‘emptiness’ of ‘particulars’, how it is all of ‘one taste’ as the Buddhists say. How it is at the same time ‘Nada Brahma’, as the Hindus assert, the primeval sound, the ground-vibration. (Sound can serve as a door to liberation and -sudden- enlightenment).
How could ‘the Ultimate’ not be these things, these qualities, since these are words expressing the highest awareness and insights of quite diverse human cultures, levels of experience where science plays no part, where ‘enlightened intuition’ rules supreme. So our world-picture should contain these elements.

‘Experience is: the world’

Our immediate environment, that is the way we know the world. It seems a fragment but through the mind it represents the whole. Understanding the character of this experience of the ‘here and now’ as the whole is the doorway to liberation. We can begin to appreciate that our experience is the ‘world’ and that the seeming permanence and firmness of the world ‘out there’ delude us as regards its true character, it has no own substance, it’s mental,  that’s why the Buddha called it ‘empty’, an illusion, a dream.
In the Lankavatara Sutra the right understanding of reality is expressed as follows: ”When an objective world is no longer grasped, there is neither disappearance nor non-being, except something absolute known as Thatata-vastu, Suchness, the realm where the wise have their abode”.
We see here the Suchness of the world brought back to the subjective experience as deepest ground.
Does our body represent a solid world? How does ‘solid’ feel like? In essence our body is only fleeting feelings, sensations, moods, images, nothing solid, it’s mainly space, as we saw.
Herein lies the big mistake, the source of our ignorance, that we separate matter, our body, from our consciousness and from the rest of the world; a double existential mistake. Our body, as it is to us, is only in our consciousness and it is not separate but the centre of our world. We tend to think of the body as a thing, but in last analysis it is the centre of ‘experience’. When unconscious, there is no world at all.
Things only really start to ‘exist’ in our experience, when not perceived the world is in a limbo-state of ‘tasteless’ potentials, that only ‘materialize’ through consciousness into the forms we experience. Nous sommes le Monde, we are the world, we even ‘create’ it. Experience is our ‘being in the world’, one could say that the important Buddhist distinction of ‘sentient beings’, refers to beings that ‘experience’ a world due to their senses.
So, again, the faculties of our mind create the world we experience and because this is so we can improve the world we experience by enhancing our faculties and intuitions. Animals differ from humans in this important respect that they cannot improve their collective and individual conditions, their mental capabilities are exhausted, their evolution of consciousness stopped, where they lack past and future in their consciousness, they cannot reflect and plan other than in an immediate situation. We are definitely ‘more enlightened beings’ than animals are (although in many ways they far surpass us in sensitivity), because of our sense of eternity where past and future melt into the One, where the here and now gets expanded with past and future, in distinction to animals, it gives us in principle infinite possibilities: we are reflective and prescient (if only we were).
It is in enhancing our individual and collective consciousness that our only hope for humankind can rest. The Juggernaut we are on collectively is heading for the abyss because there is no integrated collective control or any agreed perspective for improvement. But the real cause is of course the ingrained competitiveness and self-centred-ness of the industrial age, there are no common perspectives, it is any for his own, a bonfire of greed, until its collapse. Another problem is  that our collective crisis seems not very acute because it seems to hit the periphery, whales, polar bears, ozon-layers, rain-forests (rather  ‘peripheral’ as vital climate and  genetic storehouses), it does not affect our daily lives. But in the mean time humankind has become a devastating plagu“““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““`e to its own natural environment, totally overstretched (finances), a typical condition for the crash of a civilisation. Only a more enlightened awareness of the ‘world-crisis’ can start to solve our collective problems, of which one is of paramount importance: peace, creating a collective aversion to war, the strong conviction that violence is ‘not done’, is ‘no option’, also in entertainment!
We need create conditions for real cooperation, solidarity and respect. What happens is the truth, religions are not so relevant. Today it is not important to which religion or spiritual creed you consider yourself belonging, a spiritual intent can help bridging gaps in communication in every situation. This is what the spiritual path today is about, I think:  awareness of the fragility of the natural environment, care for present and future humankind and opening your heart and communicating, very down to earth; but difficult enough.


Stonehenge, floating the sarsen megaliths

Tuesday, May 7th, 2013

The transport route of the sarsen stones

Devil’s Den, Stukeley

Below you will find the first ever analysis and argumentation for the transport route of the sarsen  megaliths to Stonehenge,  by water over the river Avon instead of overland…..


For over half a century the route of the transport of the sarsen stones (sarsen, specific gravity: 2.5) from the Marlborough Downs to Stonehenge as laid out by Atkinson has not been under discussion or seriously challenged by any archaeologists as far as I know, until a recent find of  sarsen at Marden henge put this topic in the spotlight. This is remarkable because this route would have gone through marshes and over the extremely steep Redhorn Hill, but this is archaeology, a science, which is most of the time in sleep-mode when it comes to important theory it seems and only progresses by chance finds.

Stukeley, self portrait

Fyfield Downs and Clatford Bottom
For three centuries British archaeology is in the invaluable possession of the works of William Stukeley, an antiquarian and close friend of Isaac Newton, about whom he wrote a biography relating the famous ‘apple falling from the tree’- story. To be valued as a friend by Newton you probably had to have a bright mind yourself I suppose, so this is how I treat Stukeley and his work. He believed Druids were the builders of the megalithic monuments and conjectured they built them around 500BC, much earlier then his contemporaries thought. Regarding the Druids it is not altogether fanciful or weird to think that the megalithic architects and scientists were their precursers. Anyway he calculated what he called the Druid’s Cubit as being 52.83cm (20.8 inch), which is less than 5mm longer than the Megalithic Ell (52.36cm) that I propose and use in my work . He probably found this value by dividing the inner diameter of Stonehenge by the number 56 of the Aubrey holes. 56 x .528 = 29. 57m! (Burl’s measure= 29.60m) and got it confirmed in other measures, 208 (16×13) x .528= 109.82m! (Burl’s diameter of the overall perimeter = 110m), 164 x .528 = 86.59m (Burl’s diameter of the Aubrey-hole circle = 86.6m). (164 = 2 x 82, a supposed total number of the Bluestones, 3 sidereal months 3 x 27.3=82!) These are all crucial lengths and numbers at Stonehenge, 300 years ago as they are today. Although every standing stone circle expressed a number, archaeologists are not very interested in their numbers, they have no use for them. (Brodgar? Avebury? numbers not known, only conjectured, as of many other important circles, never excavated for that purpose; where a ‘superficial’ dig suffices).

These numbers and the ratios in the design of their buildings are the only intellectual data Stone Age man has left us, but archaeologists are not interested. Even the highly significant recently found Blue-stone circle near Stonehenge is not further excavated for the purpose of finding a clear number, ‘some 25’ is enough for archaeologists, not ‘some 23’, for instance, as ever there stood in the centre of Stonehenge.
Now it may seem that the apparent accuracy of Stukeley’s Cubit undermines my claims for my own Megalithic Ell, which is 5mm shorter, but I don’t see it that way, because my numbers as there are 210ME= 109.96m (Burl 110m) and 165ME (B: 86.6m) are also meaningful  (165= 5 x 33 year sun-cycle, MacKie, Heath). Besides that I have always stressed my system is foremost a calculator, I don’t claim the precision, for instance Alexander Thom claimed for his Yard, which actually became his undoing, whereas still my measures stand up to any thorough scrutiny and remain more meaningful in mathematical and cosmological context than most others. So Stukeley’s cubit actually strengthens my case because I have found in him an undisputed early authority backing one of my principle lengths, that is, the Megalithic Ell, though with a 5mm difference, like with Thom’s Yard (6mm), differences which, given the inherent uncertainties about the exact measures, if ever there were, and the technical circumstances of the Stone Age itself, are not essential. My measures still make perfect mathematical sense, it’s up to others to further apply Stukeley’s Cubit and even if it turns out to be more precise in other places, which I doubt, it doesn’t touch the peculiar enigma of my calculating system which encompasses the most important measuring units of antiquity in clear whole number ratios.
Once people start to appreciate the practicality, spirituality and beauty of this system, it will be used widely, I’m sure (although it may take a new generation of archaeologists to change their ways).

Aubrey Burl, doyen of the British stone circles
There is obvious logic in Stukeley’s reasoning and measure and I must confess I am not able to get a whole number for the inner diameter length of Stonehenge in Megalithic Ell, but I do get it in Megalithic Yards, Remen and Feet, whereas Alexander Thom did not succeed with his Yard, nor with his Rod.
Aubrey Burl, the doyen of the British stone circles, dismisses Stukeley’s cubit unit measure as erroneous and comes with his own (why not), but had he checked the merit of Stukely’s unit he would have found that indeed his own crucial measurements would have got him the significant whole numbers I give above and moreover his own sarsen circle diameter of 29.6m divided by Stukeley’s ‘cubit’ of .528m would have got him the number 56, being the number of the Aubrey holes, that bear his very name.
I am not a man of ‘statues’ but if one person deserves to be remembered in a bust in every English archaeological institute it is William Stukeley. Here we have someone with, unwittingly, a rare intuition for Stone Age archaeology and architecture, an independent mind and a remarkable talent for drawing. He conjectured and described the Beckhampton Avenue at Avebury, which was done away with as a phantasy by the professionals (left, on the picture below), until he turned out to be right as they found out only recently (by chance): there was a Beckhampton Avenue. (below left ‘tentacle’)

When Stukeley says the cursus near Stonehenge was probably a race-course, to which conclusion I came independently in an ‘aborted novel’, in which it was a ‘dog-race course’, then according to Burl it was something else: it is ‘funerary’ he says! Where have we heard that before? Funerary. They all say it: ‘funerary’; it’s never really substantiated, it’s just a paradigm. Would then the whole megalithic culture be one all consuming laborious dance with the dead? I very much doubt it. (below Stukeley’s ‘Cursus’ as race-course)


The above picture of Stukeley depicts the huge sarsens he found at Clatford near the Kennet river close to the Marlborough Downs. Sarsen A (centre) is described on the picture as some 5 yards long and 4 feet high (thick) (4.60m x 1.20m), this is the right measure of a sarsen for the Stonehenge main circle, as he stated himself. I quote: ‘they are all very large, being about the size and shape of the stones of the outer circle of Stonehenge’ (see extensive quote below). The same stones were even mentioned and pictured by John Aubrey, another giant in early archaeology, nearly half a century earlier. Note the sharp edges and the text ‘rudely hewn’, which makes them definitely different from Avebury’s stones and akin to Stonehenge’s. Aubrey draws a circle whereas Stukeley does not clearly, but mentions a circle as well and draws some stones which seem broken up and moved.

Stonehenge unfinished
Stukeley’s picture may be pivotal in quelling the debate about whether Stonehenge was ever finished or not; it was most probably not! The main circle at Stonehenge counts 17 or 18 sarsen stones that made it all the way, may be a few more, here we find the rest, some 8 -12 stones. That they found indication of more dug stone holes at Stonehenge recently (by chance, because of drought and a too short hose pipe!) does not necessarily mean stones were actually ever placed in them (see the Aubrey holes). Would our Stone Age forbears arduously hew these stones to more regular proportions (it was a hell of a lot of work, the sarsen is extremely hard) at Clatford Bottom to keep such an amount in reserve, if need be, or for setting up a mini-Stonehenge? It seems highly unlikely. Look how regular most are pictured by Stukeley and Aubrey (‘rudely hewn’), compared to Avebury’s irregular sarsens, which seem not hewn at all. These early pictures also tell us that the route of the sarsens was down the Downs, past Devil’s Den, along Clatford Bottom, not up to Avebury as Atkinson argued fifty years ago; mind you: to get blessed by a priest; ‘spiritual benison’, he called it. Megalithic archaeologists seem convinced that they deal with a pristine religion here, priests and all, but it seems though their own self-conjured up religion they believe in, for which some like to act as the high priests themselves one wonders at times.

‘A’ lane
So, no hauling of huge stones up to Avebury for ‘priestly benediction’ but just down the valley on the shortest and straightest way to Stonehenge  from Clatford up the side valley to Oare, Pewsey and the river Avon, that is the better story, as I will show. Like all of nature also humans usually prefer the way of least effort and resistance and that is downwards if possible and over the shortest feasible distance. When one goes down the ‘Sarsen-valley’

fyfield downs sarsen field

one reaches the Kennet river either west of Fyfield or east at Clatford Bottom, the place where John Aubrey and William Stukeley found the huge stones; again, both resounding names in early archaeology, reporting on huge handled sarsens first hand, but completely out of the picture until only recently, as mentioned and still the stones that are known to lie there are ignored. Incomprehensible.
Here are quotes from English Heritage some known for over 300 years:

The earliest reference is a description (late 17th century) by John Aubrey of “eight huge stones in a circle”, all of which “doe lie fall’n down”. In 1723 William Stukeley referred to “twelve stones flat upon the ground in the middle of the road which seems to widen on purpose for it… Eight of them seem to lie in a circle…”.(see picture above,YG) The precise location of the stone circle is unclear; based on Stukeley’s description, place names and other documentary sources it has been suggested to lie south east of Clatford cross roads at ‘Broken Crosses’, or at ‘Broadstones’ where a widening of the road is suggested by the 1792 Inclosure Award.

The Aubrey and Stukeley pictures, with text, clearly indicate where it was: “In a lane between Preshute and Clatford” (Stukeley) or “a lane between Kennet and Marlborough” (Aubrey) This is not necessarily on the Roman Road, as the Romans with their cohorts would not be ‘slalom-ing’ between Stone Age megaliths, you bet, they would move them aside or go around them. The next quote wrongly suggests again it is on the Roman Road itself; this is where all the trouble starts, when you don’t do your close reading, as I will show presently. Here is the next quote:

“In 1723 Stukeley noted that in the lane, or rather Roman Road between Marlborough and Devizes (wrong inference!!!,YG), over against Clatford and on the edge of the meadow just by the side of the Kennet, close to Clatford Bottom “lie twelve stones flat upon the ground in the middle of the road which seems to widen on purpose for it and takes a little turn, too, upon that account. Eight of them seem to lie in a circle (see plan (2)) and the other four may possibly have been the entrance or beginning of an Avenue; they are all very large, being about the size and shape of the stones of the outer circle of Stonehenge. I measured one, 16 1/2ft long, 2yds broad (5.03 x 1.83)…. There are two barrows within sight of it …. a little further westward you turn on the right into the fields which leads (leaving a barrow on the left hand) into the Clatford Bottom”. With close reading we find that it is obvious here that Stukeley is coming from the direction of Marlborough going westward, where you first get Preshute (before modern day Manton, see picture below far right), a very old parish (priest-hut), and then Clatford (far left), so we are here on the south side of the river in a lane, not on the Roman Road on the north side of the river. (Also John Aubrey again speaks of ‘a lane between Kennet and Marlborough’, not the well-known and 2 millennia old and used London-Bath road and he mentions (East-)Kennet (a village), older than West-Kennet (a hamlet), which is south of the river, again). Then Stukeley says ‘a little further westward you turn on the right into the fields’ to get to Clatford Bottom (the bottom part of the Marlborough Downs near Clatford), leaving a barrow on the left hand’, (on the picture) so the site of the stones was before the barrow, which is pictured on the other (north) side of the river with even a windmill in the distance which would have stood on top of the Downs. It even seems Stukely pictures the ‘bend of the river’, since trees get smaller in the middle of the picture and grow larger at the sides, the road then leads to the island in the bend of the river towards Clatford Bottom. So there can be little question about our lane being south of the river, as there is no question about the Roman Road (A4) being definitely north of it.

The white curving line through the length of the picture is the Kennet river, the ‘flexure’ is to the left clearly visible, where there is even a small (manmade?) island in the river and after the bend downstream eastwards is a small creek, with trees today, this would have been the ford probably, somewhat further down there is a real ford today (see Google Earth), so fording the Kennet in that area was and is no problem, that is clear enough. Fording places are usually natural and used since time immemorial by humans. Somewhere in the triangle of land in the bend are big stones close to the river today, just as Stukeley describes. All rather straightforward one would say, but not for archaeologists, they dig somewhere else as we shall see. The trees in Stukeley’s picture suggest this is indeed the bend in the Kennet river, which Stukeley mentions in Burl’s quote: “over against Clatford at a flexure in the river we met with several great stones” (note Stukeley has given different descriptions, some more precise than others) so the picture is in the bend, because ‘against Clatford’, where moreover it is known sarsens are still lying till this day, the Broadstones, completely grown over and forgotten by archaeologists, no interest, but vital clues in ‘the route of the sarsens’, of course.
It is true that Stukeley speaks elsewhere of ‘in the Roman Road’ but one letter change to ‘on the Roman road’ makes it equivalent to saying that ‘West Kennet Longbarrow and Silbury Hill are on the road from Marlborough to Devizes’, this does not mean they are physically on the road, but close by. Since Stukeley does write ‘in’ (at least it is transcribed as such) this is the only weakness in my argument, but he did make mistakes sometimes and there are different versions of the description, besides that ‘in’ may have been a way of saying it in those days, or it was actually written as ‘on’, but wrongly transcribed. Again the overriding argument is that it is hard to believe that  the Romans would be negotiating these stones on their main road to London and Bath, it is rather unthinkable. Moreover there is no reason for another lane next to the Roman Road, since people from old were living on the south side of the river where there is ample space for fields and lanes, not on the north side.

 Parker Pierson dig

As soon as it became clear Marden Henge was involved in the dressing of sarsens, Mike Parker Pearson has launched a new theory on the trajectory for the sarsens: from the Downs over West-Overton, along Knap Hill to Marden Henge. Obviously MPP and a couple of colleagues have thought it might be time to have another look at the Stukeley records, after 300 years, as the website makes clear and they dug twice at the wrong place to see if they could find traces of the stones Stukeley depicted. Mind you, we know big stones are lying in the field in exactly the area Stukeley describes, the Broadstones, very near by. So would it not be a better idea to define the type of these stones and if ‘sarsen’ start digging there to probably find layers and layers of sarsen chips, broken mauls, pottery, you name it, and get radio-carbon dates at sufficient depth and take it from there? It could have been the assembling place of the sarsens, where they got there first dressing to make them as light as possible for further transport. The whole Broadstones area full of chips and broken mauls, imagine!

 From Clatford to Pewsey, the alternative route
My alternative trajectory of the sarsens is simple, straightforward and shortest of all, since it leads from the Downs to the crossroads between Preshute and Clatford and then on southwards through the gently upwards sloping valley, a local road to Pewsey. This natural valley road splits several times to the right, once at the present beginning of Westwood, famous for its blue bells, where a road forks off to a car park a few hundred yards up the road. The track I particularly remember though is probably the next and is a rather straight and broad, definitely man-made, hollow road with flat surface, expressly made for the transport of the sarsens, I claim. Towards the end of it there is a longbarrow in the forest on the right (if I remember well) and coming out of the forest there is a big, rather deep pit, possibly material for the barrow (it is over 15 years ago I was there). When you continue south you will at some point come to the ridge of the chalk escarpment, the physical boundary of the Vale of Pewsey which is down below. At certain places at this point in the ridge there are more than a yard deep grooves several yards apart, which in my view would be testimony of the controlled lowering of the sarsens by grating ropes down the rather steep slope into the Vale from where they would be transported onwards towards the river Avon. (Below Atkinson’s overland route – dotted line-, with mine in red at beginning, sarsen valley, and end, Bluestone-henge to Stonehenge, the rest of the transport by the river Avon in my analysis)

This scenario leaves us at least two routes at the beginning so as to prevent stagnation of transport, once, for whatever reason, a stone got stuck and would block the road for another transport. The other route would be the natural valley road, with several hairpins going down in the Vale to get to the Avon river. They knew how to use water for the transport of the Blue stones, now incontrovertibly proven by Bluestone-henge, so here I argue for the short-cut route from Clatford to the Avon and on to Stonehenge, the route I surmised now 15 years ago.

Transport by water
Once down in the Vale the stones would be dragged past present day Oare and Huish to Pewsey where the river Avon finds a confluence of several of its springs, they may even have canalised it partly. As I claimed elsewhere the sarsens, like the bluestones were transported by water most of the way, which would also have been feasible from Marden henge onwards. Remember this, reader, there is nowhere any suggestion from anyone of using the rivers for the transport of the sarsen stones to this day, because the river is deemed too shallow, not?
Since Marden Henge has forthwith to be taken into account as regards the dressing and transport of the sarsen stones it is necessary to also account for transport over water to that place. So in my account (some) stones could have been transported over the Avon river from Pewsey to Marden henge, which is right at the riverside, specifically for further dressing probably (the lintels!) since they were only ‘rudely hewn’ at Clatford Bottom, and would then from Marden be taken down to Stonehenge by river again. So let me now explain how I see the water transport.

Summer bed, winter bed
A lot can change in nature especially when humans interfere and that is what happened progressively over the last 5,000 years in Western Europe with the advance of agriculture and all that came with it. Europe was mostly one big dense forest some 7000 years ago, today forests are sparse in Western Europe. Enormous amounts of soil have been carried away by the rivers and filled their original beds. Holland, the Netherlands, where I come from is the delta of the Rhine, the largest river of Western Europe. In my country we are aware that a river has a summer and a winter bed, and the mighty Rhine is kept within bounds with long stretches of high winter dams. In Britain they seem to have long since forgotten about winter beds and started building houses on the floodplains; that’s why they get wet feet in places where rivers get swollen in times of heavy rain recently, which is during the winter usually.  In a recent winter the river Avon, the subject of our study here, swelled to ten (!) times its ‘normal’ width in places with a volume of water not on the record yet.

A quick-witted reader now already gets the point: there must have been times that there was enough water and depth in the river to transport the sarsen stones since they knew how to go about it, given the now, since finding Bluestone-henge, established transport of the Bluestones from Wales over the sea up the Avon river. (There simply is no better rationale for the Bluestone-henge being just in the place that it is). This practical knowledge may even have been the very reason they thought hauling the much larger sarsens from the Downs to the Stonehenge area was a feasible project. All they had to do is cut the trees along the summer beds, possibly slightly canalize the beds in places and wait for enough water to do the job, aided by building barrages to raise the water levels locally, all during the winter of course. This was the reason people came to Durrington Walls in winter time and brought their own food in the form of cattle and pigs, not to feast as Stone Age people are perennially deemed to do according to archaeologists, no, just to feed themselves as everyone else does and certainly does when engaging in big exhausting communal efforts and enterprises, like hauling stones over land and water or quarrying moats like  for instance Brodgar and build from the huge quantities of quarried stones (where have they gone? well, to the Ness of Brodgar, of course!) communal halls and kitchens, surrounding them by huge walls against the high and cold winds of Orkney for one thing. It is all practical and useful in my approach, nothing ceremonial per se.
Important always is to wonder what people really need; what you would need in their circumstances, not ceremony and ritual, but shelter and knowledge of the seasons.
So winter was the time to move the (big) stones (no work on the land) and the Grooved Ware people from all over the place descended on Clatford Bottom and on Marden and Durrington Walls henges to participate in the communal effort to build their ‘clockworks for the heavens’. It was this participation with many in something big, which was their spiritual high point and caused their ‘enthusiasm’, it became the hallmark of the agricultural revolution, the communal spirit, as we see it also in the graffiti of the Pyramid workers and as we see it in the ‘Rausch’ or ‘euforia’ of crowds to this day; it is not so much the celebration of the winter-solstice at Stonehenge, as the coming together of a unique and sophisticated culture which was possibly the first to use flat tables, benches and flat bottomed pottery in Western Europe. It was probably the Grooved Ware culture which set itself apart from others in its scientific and innovative approach to daily life as we see it in the insulated houses of Skarabrae built in a refuse heap! with its peculiar dressers, which were probably coolers and storage places for meat and vegetables, built high and deep to keep dogs and goats from stealing the food. All useful, practical and innovative; no showcases, let alone altars.

The engineering
So how did they go about ‘floating the sarsens’, the key question? We have to keep in mind that the sizes of the stones differ considerably. By far the biggest stones are to be found in the horseshoe setting inside the circle of sarsens, the biggest two are estimated at about 50 tons each, but also the circle contains massive stones of up to 30 tons, only the lintels are of relatively ‘reasonable’ size being over 3m long, a metre wide and 75cm thick on average, but still some 6 tons each (3.2x1x.75= 2.4 m^3, times 2.5 specific gravity= 6,000 kg); so for about 33 lintels that is 200 tons lintel weight alone.  I hope the reader agrees that when I can make the case for floating the biggest stones it implies the others were feasible as well, so we go for the 50 ton upright (in weight similar to an army tank).
‘Specific gravity’ is crucial to our calculations so it must be explained first. All specific gravities of materials are ratios gauged against the weight of pure water, which is by definition 1. This means that 1 litre of water weighs 1 kg, sarsen stone has specific weight 2.5, thus 1 litre or better 1 cubic decimetre of sarsen rock weighs 2.5 kg. So our sarsen of 50,000 kg has a volume of 50,000 / 2.5 = 20,000 litre or 20 cubic metre.
All materials with a specific gravity smaller than 1 float in water, like most types of wood, which range from about 0.4 (aspen, willow, poplar) to 0.65 (oak), some tropical types though sink. Let’s make it easy and suppose our wood has s.g. 0.5, this means that a volume of 20 cubic metres of wood weighs 10,000kg  or 10 tons and can carry about another 10 ton of material before it definitely sinks, the principle of the raft, well known in those days. The Stonehenge builders had already found out with the Bluestones that when you submerge the stones in water they become much lighter because of buoyancy, you don’t have to be an Archimedes for that, it’s just experience, so you need less wood to make them float. This is the first trick they knew from past experience: submerge your stone and it becomes (40%) lighter, that is 20 tons in our case, that’s not nothing. So effectively for the 50 ton stone submerged we need 30 ton buoyancy which means 60 cubic metres of wood of 0.5 s.g. The crucial question is then how many tree trunks from what length deliver that volume of 60 m^3. We think of straight poplar trees (s.g. o.45) and then that we need a trunk section of an average of 1m diameter, this gives us a volume of  0.5×0.5×3.14=0.785m^3 of wood in 1 metre trunk, this times 13 metres = appr. 10m^3 wood, so we need about 6 of those average 1m diameter trunks of 13 metres long to get our 50 ton 9 m long sarsen stone just afloat, that’s all.

The pictures show two different ways of doing it of which the right one seems the best because it uses the depth of the summer bed. It all depends on how deep and wide the Avon was in its summer bed, but it is obvious this is all feasible with between 1.5 to 2 metres of water depth and, as said, this could be achieved by building barrages, to begin with south of Upavon downstream of the major confluence of the Avon, where the initially two arms of the river, one from Pewsey, the other from Marden, join. This barrage would make sarsen transport from Pewsey to Marden an option and in the meantime they could start building barrages down river, with the last one downstream of the natural ford in the bend of the river at the end of the Stonehenge Avenue, near todays Amesbury.
The picture below shows the summer and winter bed of the Avon in dark ands light blue. As far as I remember the river drops only 20 metres over 20 km or thereabouts, so 1 metre per km, which is very, very little and makes for a gentle stream, easy to manipulate and even to go against, firmly embedded as it is between river banks. (Remember they transported the Blue stones upstream from the sea)

The lower picture shows the final route of the sarsen transport from the new-found ‘Bluestone circle’ and the ford in the river, not over the Avenue but along the easiest and most gradually climbing route through the valley nearby and then to Stonehenge, which is the black trajectory. This route presents itself in the middle black relief in the upper picture. QED. Mystery of transport solved!

Again climate change

When I take this argument further it could very well be that the sarsens that Aubrey and Stukeley found at Clatford Bottom are indeed the proof sarsen-Stonehenge never got finished and that the reason for this could very well be another minor climate change, that is, that the rivers for a considerable period of time did not carry or never again carried enough water to move the stones and that this became the loss of momentum in the building process which eventually halted the whole enterprise. The amount of water could also be decisive in the movement of the biggest stones ahead of smaller ones, so that they took every opportunity to move the big ones, which could also imply that most of the biggest stones were already at hand and ready before they started moving any stones whatsoever. (You see this at Banks chamber in Orkney where the form of the available big roof-stones decided the form of the individual side-cells, they had their most important material at hand before they started building)

This first assembling and dressing could have taken place at Clatford Bottom. This all ended when there was just not enough water anymore to move anything. Then when people would come year after year from far in vain, they would eventually stop coming.
For half a century everything went well it seems from evidence at Durrington Walls and then the weather-gods turned against the Grooved Ware people again as this may have happened in Orkney where the land started to sink into the sea after the millennium long Neolithic Ice Age, with rising sea levels heralding the end of the flourishing Maeshowe Grooved Ware culture.
Maybe others outside the Grooved Ware culture began to perceive it as an Evil culture with its cosmological knowledge interfering with Heaven, causing climate change and therefor sought to expel its people from the islands and eradicate all traces of it, as the Ness of Brodgar may be testimony of, buried as it got, as if it never existed and that even, with a warmer climate, their now obsolete refuge places with remains of some of their dead were completely sealed so their ‘evil’ spirits would no longer roam the land.
Skarabrae, Ness of Brodgar, Durrington Walls and Stonehenge, all seem to have come to a sudden end. Will we ever know why?

The Shaman and the Standing Stone

Tuesday, April 24th, 2012

The Shamans


In the three years I have been running this website the truly innovative ideas I have launched here as regards archaeology are several. The concept of a climate change between 4000-3000 BC, coined the Neolithic Ice Age (NIA), is one such idea which then is seen as the primary reason for the profuse building of artificial caves as refuge places on the windy sea coasts of Atlantic Europe, this being a new approach to the well-known megalithic chambers, deemed burial places by all archaeologists.
Then I have argued that subsequently some of these chambers were explicitly built as maternal clinics, like Newgrange and Maeshowe or single delivery rooms like Knowth, Dowth and Cairn L, Loughcrew and that only after the Ice Age was truly over, say 2900BC, the chambers, which had lost their primary functions, became used as burial chambers (like a church can contain burials from long after its erection, however it was never built for that purpose).
So I place the megalithic chambers initially in the context of bare survival, as refuge under forbidding weather circumstances, a fortiori survival of women and new-borns, which would subsequently evolve into centres of cosmology (observers staying over at night), medicine and mathematics (Loughcrew, Maeshowe, NessBrodgar), in short, centres of learning. The alarming demographic under-representation of women (1:3 at thirty) must have had a profound impact on society with a substantial amount of ‘idle’ men with a lot of time on their hands. This is the group which must have formed the core of the work forces which brought these giant megalithic works about.
In continental Europe the end of the Neolithic Ice Age also meant the end of the megalithic culture (Trichterbecher,TRB) with its communal outlook and burial practices around 2900 BC. The individualistic, fully agrarian, axe wielding and alcohol drinking Beaker culture became dominant, the ego had triumphed and the erstwhile spiritual ‘Rausch’ got degraded into a ‘recreational Rausch’ and began conquering Neolithic Europe bringing to an end a spiritual era which would never return again.
Last but not least I claim the standing stones, be they single, aligned or in circles, were found to be lightning conductors and, placed in circles, became instrumental in triggering downpours from a thunderstorm, that is, it was a technological innovation in the long standing tradition of shamanic ‘rain-making’. The shaman is the ‘master of fire’ and impervious to it, some do even aspire to be struck by lightning, so the shaman is the perfect conductor to orchestrate a stone circle with drums, you bet, create a thunderstorm and ‘let it rain’.
On top of that I have, I think convincingly, shown that many megalithic designs are intricately geometrical and symbolize a mathematical truth which is found in the ratio of the double rainbow, 14 over 11. (Maeshowe, Brodgar, Stonehenge)
Although I had mentioned ‘rainmaking’ as relating to shamanism earlier, it is only since I have been reading more extensively about shamans that I realized that also ‘lightning’ as symbol of fire, the ‘rainbow’ as the ladder by which the shaman climbs to heaven and not to forget the ‘circle’ itself, as ring of people, are key symbols of shamanism, so I came to the conclusion that unawares I had been describing the shamanic horizon underlying the design of the megalithic buildings and that therefor shamanism is likely to be the key to the spirit of the Stone Age and its final scientific apotheosis in the Atlantic Megalithicum of the British Isles.
I once attended a ‘shamanisation’ by an Hungarian shaman, the late Joska Soos, in Oibibio, Amsterdam, and I found recently in an ‘autobiography’ of him and in his paintings an explicit confirmation of the link between mathematics and shamanism (see below). Since Soos explicitly denies that shamanism is a religion – which I sensed about the Stone Age all the way – he nevertheless sees shamanism as ‘the source of all religion’. Let us take that authoritative insight as the theme by which we can try to weave a ‘spirit of the Stone Age’ out of the above mentioned cultural threads.


30,000 year old West-European cave painting of a shaman

There can be no doubt that the shaman has come down to us from deep in the Stone Age, that the shaman, in an important sense, is the spirit of the Stone Age, all over the world. Although the shaman was of the highest authority in the community, they usually were a ‘primus inter pares’, a ‘first among equals’ in daily life and up until this day, in those rare communities in Siberia, where the ‘original’ shaman still exists, this is the case. The spirit of equality and compassion is very strong in the shamanic society and symbolizes the egalitarian character of the Stone Age, since both men and women were shaman.
The shaman is in a sense the Ur-typ, the arche-type, of the enlightened personality, the truly ‘inspired’ teacher, the accomplished human being, and whatever their many baffling skills, it is the healing of body and mind, the restoring of harmony to the soul, be it individual or communal, which is their essential function, their ‘raison d’etre’ (reason of being). The shaman is there to help and cure others, that is what they are chosen and educated for, he or she is originally a ‘chosen-one’. Their spiritual training of seclusion, celibacy and meditative practices, lies at the root of yoga and meditation, their spiritual flight is at the heart of our dreams, their ‘Rausch’ (trance) has produced visions, medicines, arts and sciences, they are the masters and magicians of the mind, the sufferers of every human illness, the seers of future and truth; in their extra-ordinary practices they need the help from the spirits of animals, the spirits of trees and of rocks and sometimes they engage with the dead, indeed. This extra-ordinary personality was the spiritual refuge for Stone Age society; it’s a world where the Mind rules supreme.

A heavy way

Only few are called upon to become shaman, it is a heavy way and only few possess the personal integrity to deal with the power they can wield over others; if a shaman makes a (moral) mistake it can mean their own death. The essence of the shaman, and this is their deep humanity, is that her or his life itself is a sacrifice to the community, because to become a healer they have to go through great ordeals of body and mind themselves, so as to be thoroughly purified and free from ego and selfish motives to serve the ‘world’; they must be prepared to face death for the well-being of others without harming others. Where do you find that today? Only in the highest teachings on selflessness in our religions.
We see that this highest ideal of self-sacrifice for the ‘benefit of the whole’ comes straight from the shamanic ethics, that is, from the Stone Age.

No doubt there have been evil shamans as there are evil thinkers, but the essence of the general occurrence of both these types of teachers is that they use the gifts of their extraordinary or superior insights for the good of all, not for the greater glory, power and advantage of themselves and their kin. The profound compassion embodied in the shaman is the most solid proof of the sincere humanity of Stone Age society, thus all the prejudice in talk of ‘barbarian practices’ is obviously blown out of all proportion when set against the essentially ego-less altruism of the shaman, as a moral role model of his age. It is in the ‘outstanding otherness’ of the shaman, that we can understand the basic equality of the Stone Age society, its natural caring for the fellow human, its purity of intentions, its purgations by psychotropic substances, processes in which there are no spectators only participants, where the family or the community as a whole become the healing agent guided by the shaman, be it man or woman, as the medium, as the messenger of the Beyond. This can only function when all barriers between people fall away and this can only happen where egos dissolve and equality arises as the source of healing harmony in humanity.


When we do not grasp the profoundly egalitarian spirit of the Stone Age, the personal sacrifice of the shaman and the highly communal emotional cohesion, which during deteriorating weather conditions was further fostered in the long spells of hardship when they were sometimes packed together in the refuge spaces, collectively surviving forbidding circumstances, with a lot of patience and forbearance, (communal chambers became huge in Orkney, several of them 20m inside), if we do not see the spiritual purity and humanity of their society, then all our imaginations about life in those ‘halcyon’ days start from a biased ‘primitive’ perspective, a misunderstanding, if not a caricature of ancient human beings, the very people we descend from.
The equality of the Stone Age, as evidenced in the communal burials of people of all ages, is best gauged in the equality between male and female shaman, this is the root of the egalitarian society and why shamanism cannot be called a religion. Religions are institutions based on collections of moral stories and prescriptions, ‘received and passed on’ by enlightened male teachers, usually put down in scriptures, guarded and ruled by men, inside male bastions also known as monasteries. Women have traditionally no role of substance in religion, this is so all over the world; a priestess occurs only in a shamanic setting.
It is this hierarchical social inequality between male and female which justifies all inequality in society. ‘Inequality’ is a state of mind, it is the ego, which is urged to stand out and triumph (hero), but on the other hand also ‘equality’ is a state of mind, it is without ego, it is the union with the One as the source of compassion and ‘enlightenment’, which is a female aspect. Shamanism has all the characteristics of the female, religion has none.
So where did women lose equality along the way, one may wonder?

Agriculture, a female revolution

In my analysis the whole Neolithic agricultural revolution is female in character, because it springs from the female side of the hunter-gatherer society, that is, from the gatherers, the women. To this day it are usually women who have and tend herb-gardens and we know that all early medicine is based on herbs and plants. Our Ur-typ of this female preoccupancy with herbal concoctions is, of course, the witch. She even flies like the shaman on her broom. Hunting and gathering is initially a state of ‘pursuing and taking’ from nature (male), whereas nurturing herb-gardens and nourishing animals (ducks, geese, goats) is a state of ‘giving, caring’ (female), so in effect it was a deep cultural revolution which took place and at first this gave women the upper-hand, they became the masters of the house and its direct environment, while the men ‘worked’ for them and their offspring in a secure and comfortable setting. Instead of ‘killers’ men became ‘carers’, instead of free-roaming, he had to manage a schedule, he started to manage time; many men psychologically never really took to the change, they have always remained hunters.
Hunting is not ‘working’, of course, otherwise it wouldn’t be a sport of the elite. Hunting and fishing is the male in his erstwhile free and pristine independent natural state, which he lost since the Stone Age and is still desperately trying to revive in the few ‘free’ hours that are left to him.

It has often been wondered why, for heaven’s sake, the hunter-gatherer lifestyle which was so much less labour intensive and cumbersome than agriculture was ever given up for the sedentary agricultural lifestyle. The answer in my view is clear, it was: the women. The women brought this about, they wanted a husband-man (husbandry) with a farm and a permanent shelter (house) for their children, no longer the exhausting life with a hunter without a home, than a feeble hut and sometimes even nowhere to go. So it was the preference of women which made young men go out into the woods with their axe and clear the land, build a firm wooden house, wait for local women to appear and start a family. And so it spread and came about (There seems to be genetic evidence for this scenario)

Master of the house

So agriculture was a female revolution and brought unwittingly a take-over of domestic power, the woman became the master of the house(hold) and had under most circumstances her man nearby; it became a success as we see today. In the Stone Age women languished in an alarming minority of 1 to 3 males at about 30 years of age, on average the end of their short lived lives. They lost their infants and own lives at an abominable rate under exhausting circumstances and with insufficient comfort to recover. Today though nearly everywhere women get older than men and are in the majority. Taken that it is a bare 10.000 years on the hundreds of thousands preceding, that is a resounding success for women.
Losing his dominance in the family life and the control of his transient ‘hut’, seeing women taking over on the domestic front, the male ‘conspired’ successfully to regain his dominance outside the house in the communty and did it by building himself the bigger house, where he and his ‘brethren’ were the boss again, he built himself the ultimate house, he built himself a ‘temple’, and he convinced woman that this was not for himself and his dominance, but for the ‘gods’ (his gods) and their dominance.
This gender inequality is the root of all social inequality and the bastion of male superiority in society. This is the paramount reason why religion is different from Shamanism, why the priest is just a shadow of the shaman, why women are banished from spiritual authority, why mysticism and morals are no longer at the heart but at the periphery of society. Religion is a male message, an urge to convert, a conviction of being right, of embodying the ultimate truth etc., shamanism has none of it, because it is from the heart not from the head, because it is rooted in compassion with fellow humans, whereas religion usually does not practice the compassion it preaches.

What I want to stress is that shamanism can be seen as the genealogy and evolution of the human mind, but also of its compassion, breaking free in susceptible individuals, where it manifests itself as, indeed, the ‘soul’ of the human condition, be it male or female. Undergoing and managing the hidden powers of the mind, that is the extraordinary importance of shamanism and the cause of its resilience and resurgeance in our age. It is an endangered wisdom which should not go lost.


‘Information’, ‘getting information’, are terms often used with respect to the ‘spiritual flight’ of the shaman which gives it an unexpectedly modern flavour. We read the shaman moves into ‘the other world’ to retrieve ‘information’, on how to heal a sickness, help a community, or do fortune-telling, or any other ‘incredible’ thing these shamans do in the eyes of an ‘ordinary mortal’. By encountering the spirit of a medicinal plant, for instance, the shaman, during the ‘Rausch’ (trance), gets ‘information’ by its spirit on how to use the healing qualities of the plant, this way they gain knowledge by, what seem to us, ‘supernatural’ means.
But then, if scientists, artists, composers find creative solutions given them in their dreams for free without the ‘supernatural’, would then not the shaman, the tireless worker and expert on dreams, be the most well-placed to retrieve even more knowledge at will, just by being ‘extremely natural’ about it all? This is more than likely precisely the way the most precious of knowledge has originally been gained by mankind, through mind-altering plants and fungi, producing extraordinary insights and revelations on plants, animals, the cosmos and themselves. A prodigy is born, not made, a prodigy is a shaman, a shaman is a chosen-one, a genius, answering a fateful call, without much choice. If it were common there would be more genius in humankind.

Shaman and the standing stone (part 2)

Apart from the fact that the Atlantic Megalithicum culture put so much energy in big stones, megaliths, they appear to have been actually very fond of stones in general as is again and again shown in the ‘uselessly’ small neatly polished stone axes found at the Ness of Brodgar and several mace-heads as well, sometimes small stones seem to have no practical function at all and are just extremely smooth and nice in the hand, the ‘nature and feel’ of the stone is the essence of its specific ‘healing influence’, you’d say. A lot of people today still believe in the healing influence of stones, a very old belief, no doubt.

The megalith

Putting the stone upright was honouring it and making it enter the world of man, who go upright. Everyone who ‘encounters’ a big standing stone knows it has an enormous ‘presence’, which is its ‘uprightness’, it has been singled out, been chosen to stand in the world of man, but it could not get there on its own, that is its mystery. Where do you come from, what are you doing here? In Holland the megalithic boulders come from the glaciers of Scandinavia, a thousand miles away carried by the glaciers of ice-ages. The more foreign a stone is to its environment (no living rock in the whole of Holland), the more magic and enigma it is bound to exude. Where do you come from? It could be, and this holds especially for the single standing stones, that it was raised where a shaman had been hit by lightning, or lightning had hit the ground, marking an auspicious place in the subtle power grid covering the upper surface of the earth. It could be. Nobody believes standing stones were placed at random and nearly everyone who knows sufficient places and gives it a thought will conclude that it usually enhances the power of an already special place by singling it out, giving it focus; a spot to experience the whole environment. I am pretty convinced that the triad lightning-shaman-standing stone is a valid one, that the standing stone through its ‘sun-spirit’ can be seen as a bridge between Us, the Underworld and the Beyond. The standing stone is associated with the energy of the Sun as well as of the Earth, but also with the spirit of male and female reflected in its shape.

Cult of the dead?

In Japan, where shamanic Shinto animism has reigned from time immemorial, spirits reside in the house preferably in the upper corners of a room. So the buttresses with their standing stones in Maeshowe, when pictured flat-topped and level at 10 Megalithic Feet height (2.96m), would provide an abode for the ‘cardinal spirits’ at the cardinal points of the central space of the building, North, East, South, West……, which figure prominently in shamanic invocations.
In the shamanic animistic worldview every block of stone in a building has a spirit (animus), how much more when these are upright stones like the buttresses in Maeshowe and all the other standing stones, aligned in line or in circle at given points in the landscape. The spirit of the stone though is not the spirit of the dead, it is the spirit of the stone. But since spirits are essentially equal (because all is One -which also explains the natural enlightenment and equality of the shamanic society) and since spirits are interchangeable, no doubt the spirit of the dead can enter the stone and find a temporary abode there. When you touch the stone and hold it for a while it may tell you its wisdom, like trees do. So that stone circles were places where shamans could convene the spirits of the dead, is plausible, but is it a cult of the dead? No, it is shamanism, and in shamanism everything is ‘alive’ and ‘spirited’, so how could it be a cult of the dead? The shaman communicates, as we say, with the spirits of the dead and many other spirits, that are very much ‘alive’ and no one with any knowledge of shamanism will judge that it is a cult of the dead, although the dead can play an important role in it, like they can in our own lives. We see the same ‘entering the sphere of the dead and of spirits’ in Tibetan Buddhism with a different emphasis, but in many of its cultural manifestations close to the shamanic tradition; but no one can seriously call Tibetan Buddhism a cult of the dead.

Listening to trees

We know it is not given to just anyone to listen to a tree (let alone talk to it), but this does not mean that we have some kind of superior rational understanding, when we think it is nonsense. That attitude of ‘rationalist science’ and the prevailing general ‘common sense’ in its wake, that attitude of ‘knowing’ something about the profound mystery of life is just preposterous, if not really stupid, and certainly one of the biggest spiritual deficiencies of modern day ‘scientific’ Western culture. It stems from the all-pervading materialistic scientific interpretation of reality, of which of course the Big Bang is the ultimate symbol. That ‘superior’ rational interpretation though always immediately breaks down as soon as it cannot explain phenomena outside its own reference frame, the mystical, the magical. Scientists ignore these facts or simply don’t believe them or, worse, ridicule them, as if they know something about the mystery of life. (I hope this site will provide a start for a vocabulary and logical framework which is able to translate also the ‘inexplicable’ in a consistent ‘logical’ description, because in my view all and everything that can be talked about can descriptively be brought back to a common ground.)
We know the druids revered the oak and its mistletoe and no doubt listened to the holy trees in their sacred groves, but we know also from Caesar that they were of the opinion that writing ruins the memory of man, so all knowledge should be memorized, not written down. Unfortunately there is great wisdom in those words, on the one hand their knowledge is lost, because not recorded, on the other it seems indeed the power of the human mind has deteriorated and that that is the reason why educated Stone Age people have the edge over most of us and our scientists today. Like druidic pupils also the shamans went through a long schooling and of course this was all memorization and direct experience, no reading and writing.


After having Socrates denounce ‘writing’ in his dialogue with Phaedrus, Plato wrote: ‘The first prophesies were the words of an oak” and he goes on to state about ‘the people of old’ that they thought it rewarding ”to listen to an oak or a rock, so long as it was telling the truth.” Socrates even lectured Phaedrus for not heeding the words of an oak, as if it were not the content of the words which mattered, but who they were related by.
You would never have thought this would come from Socrates or Plato, would you, and where do these oaks and rocks come from you may wonder, maybe Wales or Brittany? Or is Plato talking of the people of old in general as: all the Shamans, the innumerable enlightened-ones preceding the Buddha, the Immortals absorbed in the eternal Tao?

The state of knowledge that I attribute to the Stone Age is based on the assumption that they discovered the cycles of Sun and Moon, had to make calculations and records to do this, mastered prediction of eclipses, found the ratios between square and circle and derived the double rainbow proportion.
Patience is the key to this knowledge and to all the countless lost crafts of the past, it was after all a different state of mind, where time did not count and perfection was the norm. Already the audacity to dare start charting the night sky, given the tools they had, is awe-inspiring by itself. (Did you ever look up at the stars and imagined you had decided to understand and make sense of what you are seeing?) Night after night making observations, or waiting for the sky to clear in the North, generation upon generation memorizing the assembled knowledge.
Discovering after hundreds of years the intricate cycle of the moon must have been a momentous event, for which there was every reason to cast such knowledge into stone so as to never ever let it go lost. To this spirit we owe the great scientific megalithic works, also those of Giza, where the spirit of the great healer and architect Imhotep would hover over the work force.

European shamans

In the late eighties I attended a ‘shamanisation’ by the Hungarian Joska Soos in Oibibio in Amsterdam. He used Tibetan singing bowls in his performances. Soos tells us in an auto-biography how he was ‘born with the caul’, auspicious, and coming from a shamanic family background. He was educated by the shaman Tamas Bacsi who indeed listened to the rustling birch for predictions and exposed himself to lightning (to no avail). Lightning is the symbol of the power of the shaman: they control fire and are impervious to it.
In the context of this website Soos has some interesting experiences to share in that he tells how he as a kid had a profound vision on the nature of reality which he describes as follows in ‘I don’t heal, I restore the harmony’ p.39: ‘Then I did experience the mathematical structure of the micro- and macro cosmos and it were mathematical structures and fabrics of power-lines like you’ve never seen. This was not Euclidean mathematics, this was something completely different and they moved, they were alive. The one changed into the other and the differing mathematical figures were space and time at once. I felt space, time and light, the white light”.

I feel very much at home with such a description because it so closely describes the mathematics and ontology I propose in these pages. It was ‘alive’, it ‘moved’, it was ‘complementary’ as one moves into the other and there was the ‘white light’; so then we are ‘at home’.

In Soos’s adagium on healing we recognize the shamanic humbleness as: ‘it is not me who is doing the healing, I just put things back in the right place, the healing happens from the restored harmony of the soul itself”.

When we see Soos’s pictures it is obvious this deep early ‘revelation’ stayed with him all his life and evolved into light-beings and singing bowls.

The use of circles, half circles, straight and parallel lines remains part of Soos’s way of expressing the ‘Beyond’ and what he ‘sees’.

What I want to argue is that just as the shamans were capable of retrieving ‘information’ from a plant or tree they would have been able to ‘supernaturally’ retrieve information about mathematical rules and physical laws by producing circles and making geometric connections between the standing stones in the circle. This way they could not only physically enter the geometry of the circle, but in their ‘Rausch’ connect to the cosmic (mathematical) spirit of the design, which was revealed in ‘eternal’ ratios of numbers, held in patterns.
According to Soos the polar star is object of a cult in shamanism, because as central star, circled by all the others, it is ‘the great door to the Divine’. But also the Great Bear is singled out and revered. Do we call this religion? Or better shamanic cosmology? (I have argued elsewhere that the engravings in Newgrange and Orkney recumbent stones depict the Polar star and the Great Bear, see Skara Brae and Science)
Even more striking in my view are the descriptions of space by this old shaman Tamas Bacsi. The universe is full of tiny holes, he said, the cow, your hand, full of tiny holes. “You look at the sky and then you will see the little coloured holes which you can go into and enter a different space, a different existence”.
It certainly combines well with my concepts of space-pixall, zero-point, zero-dimension and pure light. As Soos relates his actual experience of the holes 30 years later, he describes something which I experienced myself several times long ago, that space is an aggregate of coloured sparks, that it is indeed granular and full of light. This experience is the basis of my certainty that my concepts are sound, I’ve ‘seen’ it, so to speak. Also Tibetan ontology has ‘sparks’ and ‘space-particles’ as fundamental concepts.

It may be clear that I have been inspired again by ‘the people of old’ where it comes to their piercing insights and intimate feeling for ‘reality’. I always come away wiser when I let them be wiser than me, that is also the best way to let them come close. The silence of the Stone Age has still a lot to tell us.