Posts Tagged ‘shamanism’

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 and can be appreciated even without any mathematical knowledge)

Natural Number Geometry 

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, applied to the geometry of circles and squares, with evidence and clarification given in many chapters. This ontology is based on the assumption that mathematics underlies the manifestation of Nature, a modern version of Pythagoras’ ‘All is Number’, as resonances in vibration.
I’ve called this theorem 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 at eastern platform level = 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 in ratios 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, but ‘megalihic’ cultures may have been partly contemporaneous.
Both the Atlantic and the Mediterranean ‘Megalithicum’ ‘eternalized’ their scientific/cosmological knowledge in megalithic 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).
It is an early manifestation of the awareness of rhythm and cycles in Nature and the Cosmos, of scientific interest.

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 Euclidean 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, 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 perimeter in blue circle. I conceived of this symbol of my mathematical theorem before I found it in the pyramids and in Maeshowe, both were spectacular findings and experiences, “I had understood and did understand their message, I was working on it”. Rare moments in a lifetime, but this mathematical work has given me several profound ‘recognitions’, as the ‘little theorem of Fermat’ in relation to repunits and primes, which was a huge confirmation that my math was all right.                                                                                                                                       

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’

Its True Significance

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, here. (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 (+0.002). Nowhere does the theorem differ in the hundredths, always only in the thousandths, so it is very close to conventional math.
The eventual order is achieved in the Lost Theorem by using a ‘rational’ approximation of π as a squareroot-2 number= (20/9)V2 or 3.14 2696… (ratio circle circumference and its Euclidean diameter, π = 3.14 159265…) and by a rational value for π²-squared ( 9.87654…), respectively by the rational Pi-numbers: 22/7 (3.14 2857 1428…) and its complement 2800/891 (=3.14 25 36…) and the new Pi-squared as 9.87654321 (= 22/7 x 2800/891 = 800/81), all expressed in natural and rational numbers, the undubitable basis of any early mathematics.

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 big blue circle diameter is 10, the red circle diameter is 9, as ratios. The big blue circle perimeter 40, the yellow square perimeter is 36, which is again ratio 10 : 9.
(The 10/9 ratio is the basis of the 1.11111…., 2.2222….., 9.9999…, 11.11111…. nine-number system which rules the algorithm and this, in turn, is a bridge to the repunits and their root primes, number waves and more)

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 the nearby Ring of Brodgar ( its standing stone circle and rock-hewn moat with 2 sloping edges give 3 ratios), there though 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’ now as 140/99 and 99/70. (see Rationalizing the Ir-rational).

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, or 0.524, less precise) 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), this measure has, as diameter-size, an ingenious mathematical relation to a slightly revised Megalithic Yard (A.Thom).
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 (= 2 x 0.4114 = 0.8228m, 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 (from heaven) 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 at (eastern) platform level, 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 Megalithic Yard = 4×99).

Natural number Logic of Menkaure design
Mathematical height (from Eastern platform level) is 9×14 = 126 RC, the side at that level is 2×99 = 198RC, the diagonal there is 280 RC, which is the height of the Great Pyramid. We see that 126 is the smallest grain to directly relate to measures of GP and to express the mathematical theorem, which would be even more rudimentary expressed as 63 and 99 and 140 for the double cubit of 104,72 cm. Here we see the geometry of sqrt2 in the ratio 140/99, or half-diagonal for 70/99.
(John Legon found a possible even more basic expression of the Theorem in (900 palms for the height 67.32m, 1000 for half diagonal = 748m, full diagonal 2000 palms = 1496m and finally a side of 1414 palms (1000V2) as 202 RC of 105,8m. This the ground level of the base of the Third Pyramid, 9, 10 and 10V2, at the ‘mathematical level’ it is a side of 1386 palms (198RC), of 1960 palms diagonal and a height of 882 palms, at sun-platform-east level some 2.5 RC above ground level)

Moreover it shows the intrinsic numerical 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 hold 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 (-math)  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 F.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 – possibly a misprint-  Cole’s average = 230.36m-;   230.33m  is definitely wrong).

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, the discovery of Pi as 22/7, the change to 7 palms in the Royal Cubit, it is not about cynical 20th century, so called critical modern scientific scepticism. The data I use are solid, the differences or inaccuracies are trivial compared to the overwhelming mathematical logic that is revealed.
It’s about possible synchronism 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 ‘real 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; I am not a mathematician, only logician. 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, their orientations, in their obvious involvement in cosmology and calendar making (solstices) 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 calendrical and cosmological cycles. Avebury is about recording the moon cycle, with 3 circles of which one has the diameter of Brodgar, again 103.67m, Stonehenge is obviously dedicated to the sun. (See further pictorial analyses of the Orkney 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 as well, 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 perpetual drumbeat together with a huge pendulum in Maeshowe (see picture), were instrumental in creating day-time, clock-time, at the equinoxes.


Pendulum Maeshowe + starry sky

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 marvels of the Northern lights, to hear the echo of their voices at the centre of Stenness, as in caves, 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 at platform level, 198RC.
Lehner’s unreliable 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’s, so I maintain that the Third is the prototype top of the First, measured exactly at airshaft exit levels, and there must be a correlation between the different sides of the Great and Third Pyramid

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 at east-platform level. This is the reason for the angles of the shafts, they had to mark out on the outside the ‘theoretical mathematical principle’ that is 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. (always note this play with numbers: 9, 10, 11, 14)
The number 11 (22/7) is the number of the Great Pyramid (440), with 9:10 hidden, 37 (3×37=111) 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 at platform level =√(198²)x2 = 280.
The differences are very small here but can be decisive in our argument when it comes to an overall picture; the accumulation of arguments within the same framework. It is only because they had so great reverence for the building technique and genius of their ancestors (Imhotep, 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 on the site. 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 in harmony.

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), which I attended, 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 it 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 magical 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 at all 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 are still available to us all, in principle, I think.
Tibetan Buddhism is possibly the only highly 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 sharp intuitive mystical insights it is based upon.

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 geometry’ 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; when grasped  it is amazingly ‘solid and aesthetic’.


Part I


At first sight the complementary natural number theorem I have ‘uncovered’ (it has always been there of course), looks like a ‘Kindergarten mathematics’ and you certainly could make a nice puzzle for very young children on the basis of it and making them acquainted with numbers and geometry in one go, it is very pictorial as well, but when you get to the amazing intricacies of it, it can become dizzying, however simple the ingredients remain, because only the numbers 1-14 are involved.
I am not a mathematician and I can only do very simple things, so most people with education can follow this exposition and enjoy math maybe for the first time. It may also be the reason why I have done what no mathematician would obviously ever do, that is changing the rules of mathematics by holding the integrity of the natural numbers as ‘sacred’ and follow where they lead. The new rules are: only natural or rational numbers, complementarity, transparency, approximation, resonance in ratios. So irrational numbers, like sqrt2, or transcendental numbers, like Pi, are off limits. Now one could argue that this is just a nice set of rules to make exquisite designs with squares and circles involving the golden rule, so: an aesthetic device, but I claim not only that it was discovered already thousands of years ago, but also that it concerns the basic mathematical rules that govern Nature as such. That is, in all its simplicity: a profound scientific device.

Given the fact that it chimes very well with the new drive for a fundamental paradigm-shift in Science, which places Mind at the centre of reality, I hope my work can contribute to strengthen this movement (Laszlo, Church e.a.) towards a spiritual foundation of science and philosophy.

The mathematical model

Decimal nine-number system (1.111…; 2.222…./ 9.999…(=10); 11.111…)
Initially this peculiar geometry evolved from a ‘decimal nine-number system’ (for lack of a better expression), that was inspired by cuneiform clay tablets from ancient Sumer (Iraq) I saw in a newspaper (’98) and worked out.  This 9-number system, Sumerian or not, is a rational number system and still at the heart of all the calculations I do here, 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 rational numbers in the space of 10 natural numbers) 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 the dynamics of Reality and it shouldn’t; this shows nicely in the universal ‘fine structure constant’/α (alpha), alias, ‘coupling constant’/e (0.085424 55….) 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) = 0.085424 21…. [because it is about repunits  (numbers consisting of only ‘ones’, 111….111) and the ratio  10 : 9, this squared = 1.234567..]]

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 0.085424 55... (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 (and without secretly putting it in anywhere).
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 0.085424 55… as the square root of 27/3700 = 0.085424 21…, 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 theoretical 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’, that ‘repunits’ are deemed to be in professional circles,  this is the ‘real stuff’, however hidden and enigmatic it still may appear now. It is the powers of 10 in relation to 9, with only one unit difference; formula repunit, only one spark changes the structure of numbers. In the powers of 10 the extra-primes 2 and 5 are stored, plus or minus 1 produce a sum of totally different primes, either 111….111 or 100….001 and multiplicated they form a new repunit and new primes.

I have given here two examples of the stunning approximations I can make with this number system and with immediate numerical transparency. But my exploration of the number system brought a large variety of natural constants to light.

Rainbow Proportion
As my personal contribution to clarifying 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 aether waves at all levels, 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 and that the 2-D circle and square natural number ratios can stand for sphere and torus (ball and ring) geometries in space and for the necessary interaction of their wave-systems in natural number resonances. 

The magnetic field is the effect of the horn-toroidal shape of spatial fieldlines. The electric field has spherical incoming and outgoing fieldlines, so presenting the interrelation between sphere and horn-torus mathematically as fundamental for space geometry seems more than justified.

Resonance is always a whole number phenomenon, keep that in mind, this is also the ultimate scientific ‘rationale’ for this peculiar mathematical model I developed 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 super-strings. Nevertheless it is simpler and more straightforward than super-strings, I think, (no ontological questions: what is a string made of?) and it 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 when right, I would say, as it shows that something fundamental is missing in gravity theory regarding resonance; and numbers.

Since I have never stopped thinking about the implications of these fundamental number 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 (TOE), 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 the Mind and I hope it is consistent and universal and that what I do with my very limited knowledge is setting the first feeble Kindergarten 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’ (which is ambiguous in itself, but you know what I mean, by context). 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 and capacities. Our, limited, senses ‘create’ the world as we ‘experience’ it. This insight is in the West usually attributed to the philosopher Kant, heralding the European ‘Enlightenment’ epoch, but it was an analysis the Buddha (and others) 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’ and, of course, the ‘BigBang’….., that even spiritual people feel forced to believe in; such is the force of a scientific paradigm.


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 one. 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 fabric, 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 it is 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, -granule) 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’ (Formentera ’72), subsequently triggered by the fundamental mathematics I found when in Ireland (Beara ’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; maybe the pixall annihilates when sparking.

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 vibrating 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 recurring 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 else, 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 that 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.
At least it is reported that the mysterious centre of the earth appears to have clockwise and anticlockwise motion, which is perfectly feasible in a horn-torus geometry and as such the doorway to the zero-dimension.
No better geometrical 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 randomly hit and 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. At quantum level what we see as matter, the sides of the slit, is merged with the surrounding space, it is blurred, besides that the photon, has no eigen-substance, is a travelling excitement of  space-pixalls. 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 inner 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 and stern intransigent views about the ‘non-aether’, by embracing the aether again, however modified and ‘aetherial’. That there is something rather than nothing after all.

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, the substance, 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. It is here now and gone the next moment, a flux
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 about atoms, the entities that give all the ‘qualities’ to ‘matter’, is that they form a neat mathematical system: the ‘Periodic Table’, one of the basic proofs of the mathematical basis of reality, and that more than 99% of the atom’s ‘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 of pure light.
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 seem all ‘dreams’ coming from the ‘universal storehouse’. (A long dream compressed in a few seconds, no time again)
The most enlightened state of humankind is the bliss of the pure light (‘clear light’ Buddhists say), becoming one with the deep-field, Akasha.


One of the central points in my approach to ‘cutting edge’ theoretical physics is the re-introduction of the concept of the aether. It is good to see that over the years more and more scientists come round to an aether concept again. Referred to is, for instance, the Hindu Akasha-field, the Chinese Tao/Dao. So we see also here a re-introduction of very ancient human knowledge, but knowledge based on enlightened insight.
In many Eastern philosophies, as in ancient Greece, the popular 4 elements : earth, air, water, fire are invariably philosophically accompanied by a fifth:  ‘aether’  or:  ‘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 , an aggregate of space-pixalls.

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 ‘scientific’ 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 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 or body 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 (although they do remember by resonance), they cannot reflect and plan other than in an immediate situation or by collective instinct. 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 still no really integrated collective control or any solidly agreed perspective for improvement, a lot of words so far. But the real cause is of course the ingrained competitiveness and self-centred-ness of the industrial age, the socially corrossive capitalist order and economy, 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), oceans in cms per century; it does not affect our daily lives, until recently. But in the mean time humankind has become a devastating plague to its own natural environment, by totally overstretched production and waste in every area (also finances), an ominous condition for the crash of a civilisation.

In the ancient past one of the incentives to stone-building was to build for many, many generations, that should be again the perspective of collective endeavers.


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.