To my mother

La science, la nouvelle noblesse! Le progrès. Le monde marche!
Pourquoi ne tournerait-il pas? C’est la vision des nombres.

Nous allons à l’Esprit, c’est très certain, c’est oracle, ce que je dis.
Je comprends, et ne sachant m’expliquer sans paroles païennes, je voudrais me taire.
A. Rimbaud

1) Among the more obscure enigmas that the past of humanity has left to anthropologists and historians we can surely include the reason why the pure mathematics and the mathematized astronomy – in very far and very different times from ours – could have fascinated so much the human beings to the point to induce the more powerful, rich and educated castes to dedicate sometimes the whole life to this kind of studies. Nowadays, the reason of the fascination of mathematics – and of the whole of the empiric science – coincide more or less with their actual or possible applications, that’s to say with the enormous and absolutely irreplaceable practical role that they have into the mechanisms of development and survival of our society, that if deprived of empirical sciences and mathematics would disappear within some weeks (or even earlier). But, as we all know, even in the near past of humanity things didn’t go on like that. Still in Galileo’s times high mathematics and mathematized astronomy remained more or less completely unemployed – at least in relation to the practical and really vital aims of society. The reason of that was that any sort of productive techniques were connected to the handicraft ability and not to a kind of abstract knowledge managed by hi-tech machines, as things go on nowadays. But, as strange as it could seem, historiographers has the tendency to pass over with questions of this kind, and is very, very unusual to find the problem of the interest to pure mathematics and mathematized astronomy put down like that: “Why in Galileo’s times – just like in Babylonian and Pythagorean times – there were people who spent their lives trying to explicate what was the relation between diameter and circle, or what were the orbits of planets like Mars and Venus etc., if from this researches there were nothing useful to get?”. A possible motivation of this “missing problem” could be that no much time went by from the first experiments of Galileo and the epoch in which high mathematics and the connected empirical sciences begun to hold that fundamental importance in the western economical and productive apparatus of which anyone is nowadays more or less conscious. This current importance of mathematics and mathematized astronomy give us the possibility to rationalize the efforts of Galileo and its contemporary scientists and mathematicians imagining that the period elapsed between the first steps of the empirical-mathematic science and its first really important applications was the time of an evolutionary phase, necessary to the disclosing of the potentiality of empiric science, in the moment it would get to the point of its maturity. So, the period in which our mathematics and astronomy were almost completely useless it’s unconsciously compared to that in which a child is busy to learn to read, without he could understand why he has to do this effort. This effort has a very deep and actual practical meaning, even if he is incapable to understand it, and this is the reason why no one in the world ask why schools are considered useful, or why children have to learn to read. In a similar way, historiographers have the tendency to pass over the fact that about two or three centuries ago scientific research was an aim in itself, of the same kind or even more abstract than the aesthetical pleasure.

2) But it is completely obvious that an argumentation of this kind can’t be made to give sense to Babylonian astronomy and Classic Greek mathematics. Those civilizations – that is a thing we are absolutely confident – never developed a kind of technology in which mathematics or the mathematized astronomy resulted really useful: so we can’t justify the presence of these sciences in these cultures thinking about them as a preparation to something else. Furthermore, we are acquainted with the fact that in Classic Greek times the practical applications of theory were considered a stain or even a shame in the life of a true philosopher. In spite of that, with amazement we discover that pure mathematics and mathematized astronomy were often considered in ancient times as the most important kind of study, regarding no less than the divine essence of the world. With a certain consternation we discover that a genius like Plato considered geometry a basic study for politics – a thing that to any modern politician seems as absurd as ridiculous, as to the common people result absurd and ridiculous the Pythagorean disgust of the beans, and all this kind of strange beliefs that we can find in all the ancient world. Anyway, since this kind of thinking has passed from the Classic Greek culture to the Western-Christian – and precisely by means of the Plato’s philosophy – for many centuries some mathematical entities have been put in relation with God and the divine in the western world too, as they were in the eastern. Just like Pythagoras – that we can reasonably believe a former student of Ancient Egyptian priests – Saint Augustine had not doubts on a certain kind of issues: for instance, God created the world in 6 days because 6 is the perfect number. But we can’t deny that in the western world this style of thought seems exhausted from many time. Nowadays scientists do not believe in any kind of sacredness of numbers, and only astrologists go on to believe strange things like that. But astrologists are people that usually the respectable intellectuals consider impostors, tricksters that take advantage from the credulity of so weak of mind people that let themselves influence by the rests of old and absurd superstitions (old and absurd superstitions that, as strange as it could seem, once was firmly believed as truths by very serious thinkers, even by the man who we can define the Founding Father of the empirical science, Galileo Galilei).

3) Maybe, the last big western scientist that considered sacred certain mathematical entities – and so aprioristically worthy and respectable a kind of study – the pure mathematics – that does not seem to offer any sort of comfort to the human life, if not a particularly tortuous kind of distraction – was Kepler. This big scientist, let’s say this genius – before to make some discoveries that made of him another of the Founding Father of modern astronomy and so of modern science – made a thing that nowadays we are hardly capable to attribute to a man with some good sense. From the observation of something that to us is just one of the millions of chances that we can find in the universe – the strange fact that the relation between the circumscribing and the inscribed circle of an equilateral triangle was very similar to those of the orbits of Jupiter and Saturn – Kepler was capable to deduce no less than the mathematical relations of the entire solar system. In his idea the dimensions of the orbits of the six planets – that at that time was believed the whole of the solar system – were aprioristically derived from the geometrical relations that we can find – among other things – between figures characterized by the golden number. No doubt that nowadays an intellectual enterprise of this sort appear no less idle than deprived of any kind of fundament. But, on the other side, there’s no doubt too that to Kepler it appeared completely sensed, and surely he didn’t believe a waste the energies he spent on it. Tested with facts, this geometrical model of the solar system turned out almost immediately as a total failure, first because it was evidenced that the actual dimensions of the orbits of the planets had not any sort of characteristic proportion – let alone a proportion someway similar to a progression of regular solids. Furthermore, early it was discovered that the solar system is formed by more than six planets, a fact that to our eyes make this geometrical-astronomical construction still more childish and unrealistic. Nowadays, at the best, it could appear to scientists no more than that the result of a fantasy of an artistic or architectonic kind, rather than the fruit of an authentic scientific effort.

4) The reason why this theory – that was divulged by means of a book with the very meaningful title of “Mysterium Cosmographicum” – appeared to his inventor worthy of consideration was probably – between other things – that at that time they had a scarce empirical knowledge of the solar system. So they could still think – in Plato’s way – that “God is an architect”, and so that He created the universe with measures that had a sacred meaning, beyond the mere and banal quantity that they fix (certain numbers, at Kepler’s time, could still be seen as an eternal attribute of the divine mind, and so they were considered bearer of an obscure theological meaning, a sort of revelation of a secret side of the divinity and of its creative power). This idea of a “geometrical God” in a platonic style – that still was conserved more or less intact in the scientific western world after two thousands of years from Plato’s death – was however doomed to a swift ending. In a few time, and precisely on the basis of the problems posed by Kepler himself, Newton introduced maybe for the first time in a complete and conscious way that empirical-quantitative method that is so familiar to scientists and intellectuals of modernity. Following this method, the western thinkers was almost compelled to renounce to the idea that in the God Mind we can find privileged numbers or special geometrical figures, as this method seems to repeat, each time that we use it correctly, that we have to renounce to mathematical entities endowed of an intrinsic theoretical importance, aside the facts that by them we can predict and describe. That means that the modern cosmological theories seem to have not qualities that result more important than the absolutely fundamental one which consist in “to correspond with the facts”, whatever numbers they hold in. For instance, the constant “c²” in the Einstein equation “E = m c²” – which represents the square of the light speed – seems to us a number that, given its enormousness between other things, could not on principle to correspond to any human need of proportion. Another very famous constant, the constant of Plank, has the very contrary problem, because it is so little to result humanly unfathomable too, not to say that no one of these constants seem represented by ciphers that could be the pretext of any kind of easy or even difficult numerologic consecration. But, practically, already at Newton times, the idea that in those that we can call the “characteristic numbers” of the solar system – or of the whole of the universe – is hidden some kind of revelation about mysterious sides of the divine world has begun to fade away, as the gravitational constant too does not seem to anyone a number that could have any kind of interest, except that formula into which it results so useful to put in relation the gravitational attraction that is generated by “m1”, “m2”, etc.

5) In a certain sense, we could even say that this very general thinking attitude of modernity toward numbers has some kind of very, very far relationship with which of Classic Greek toward the golden number. We say that because aside Plato and platonic solids (and maybe of the Pythagorean pentagram) it seems that the fascination of this number – which in the mathematical formulas is represented by the Greek letter “ϕ” – came out only from its intrinsic mathematical qualities. In no way it was connected by anyone with some secret or invisible side of the universe or of the goddess. We risk this hypothesis because the Euclid text seems in no way to turn out with the idea that ϕ could have any particular importance except for pure geometry – as other numbers had before and after Euclid work (for instance, to the Israelites the 7 was synonymous of perfection, as in St. Augustine times the 6 was: but the number of examples we could do is so enormous that it’s convenient to stop at the first). Actually, in Euclid times or in the following centuries and in the former millennia, it does not seem that a religious attention was reserved to golden number, and surely we have not acquaintance with any belief that the world was someway created on the basis of it. If we discovered something like that, it would appear to us a very, very strange thing, not to say a completely inexplicable enigma, as ϕ does not seem to have something to do with any natural or cosmic cycle, nor with any natural phenomena of some importance of which in our far past some people had some kind of notion. Otherwise, it would be really, really extraordinary to find it adored only on the basis of its intrinsic mathematical properties, as it is usual that in ancient time numbers were hold as sacred only when they had something to do with some kind of astronomic cycle, and so with some heaven object that was considered a goddess (the 7 for instance has clearly something to do with the lunar week, and so with the moon, the 4 with the number of the solstices and equinoxes and so with the cycle of the seasons). In contrary, as far as we can know, the discovery of the golden number had nothing to do with observation of natural phenomena, and any kind of philological and historical evidences suggests that instead it was discovered only by researches of pure geometry, precisely thinking about a possible subdivision of a line, as that we can see below.


If we take a line AC and we divide it by the golden ratio we will have that the relation between the major segment AB and the minor segment BC is the same of which that exists between the whole of the line AC and the major segment AB. From this ratio, that remains identical whatever the length of the chosen line could be (as  remains identical whatever the diameter of the circle could be) results an irrational number, equal to 1,61803398874989…, a cipher that beside to be on the basis of very important regular geometric figures, as the pentagon and the pentagram, has some characteristics that seem very peculiar. For instance, if we raise it to the second power, we find rather surprisingly that ϕ² = 1,618033988588432…² = 2,618033988588432…, that’s to say a number that conserve all the decimals of its square root; and no less surprising is the fact that the same happens with the reciprocal, as 1/ϕ is equal to 0,6180339888115682… Given these characteristics, absolutely unique in the range of the numbers that we know, we have the possibility to build equations with a someway strange look, that result anew absolutely unique, for instance: 1 / ϕ = 1 – ϕ and 1 + ϕ = ϕ².

6) As we said, from what we know of, they were precisely these purely mathematical characteristics to arouse the wonder and admiration of the Greek philosophers and mathematicians, since at that time nobody was aware of the fact that – actually – the golden number characterizes many and very different natural phenomena. If we shot an even incomplete list, we find that it is at the basis of the genealogical tree of the drones, of some phenomena of optic reflection, of the sunflower inflorescence, of the disposition of branches and lives in a big number of plants, that of the rose petals and a big number of other flowers, and then the seeds of the apple, the flakes of the pineapple, lives of any sort of succulent plants, the spiral of the shells of Nautilus king, the tail of the sea horses and of the ram horns, the fossil foraminifers, the nose-dive of the peregrine falcon, the whirlpools, the hurricanes, the tornados, the form of the spiral galaxies, the bidimensional projection of the space-time (that, as we have seen in the preceding article, probably characterize any Ancient Egyptian sacred space1) and, according to some scientist, the DNA spiral too. But we have to underline that all these are very recent discoveries, and so it seems that we can’t hypothesize in a sensed way that Classic Greek mathematicians or those of the following centuries could give importance to the golden number for other reasons beside its abstract qualities. And that seems very reasonable. If they didn’t see the golden number to have effect as a rule and measure in any place of the creation, very unlikely they could connect it to the creative power of the divine if not – as we said – to a merely abstract and mathematical power, that at first sight has nothing to do with universe and nature. That is the reason why, in the moment we discovered that probably the whole of the sacred artistic and architectonic Ancient Egyptian production is characterized by a very complicate application to the space of the golden number2, on the first moment we thought to have discovered only that the golden number and the high geometry connected to its application was discovered various millennia before the time we used to believe. What anyway remained unchanged was the idea that the discoverers should attribute importance to it only on the basis of its intrinsic mathematical qualities, as at that point of the research we had not any reason nor any clue to believe that in an epoch as Nabta Playa or even older, Ancient Egyptian knew that ϕ has that kind of importance in all those natural phenomena that we have seen above. But in a very few time we were induced to change this way of thinking on the basis of two astronomic discoveries, that were possible with a careful analysis of some Ancient Egyptian reliefs; discoveries that seem to reveal the decisive role of golden number in two heavenly cycles of which this people demonstrate a perfect knowledge, and that had an absolutely decisive importance in their religious beliefs.


7) Maybe, the best way to introduce this argumentation is to recall that the whole of Ancient Egyptian architecture, and particularly the Giza Plateau, has a deep astronomical meaning (an “astronomical meaning” that, it’s opportune to recall this, at that time was synonymous of a “theological meaning”). Starting from this fact we can reasonably arrive to demonstrate our first thesis, that’s to say that not only Ancient Egyptians had discovered the phenomena of the precession of the equinoxes, but above all that they were aware that this phenomena is characterized by the golden number. This discovery – in our opinion – was one of the theological-astronomical motivations why the golden number was therefore attached to the sacred art and architecture (as we already said, there is another motivation too, that we will expose in a little while). From what we know of Ancient Egyptian religion we can easy deduce the theological weight of this motivation: if in the golden number they saw the hidden rule of the more fundamental of the cosmic cycles (and so well hidden that up until now no western astronomer had noticed it) it is completely clear that inserting it in the deep geometrical structure of the sacred images and buildings Ancient Egyptian intended magically incorporate in them the vital and creative power of the divine mind that, following the measure represented by the golden number, gave life to the world. So, it’s easy to hypothesize that the golden number was considered by Ancient Egyptians as a sort of very powerful amulet, that inscribed in sacred spaces of any kind would have hold in check the force of the Chaos, always in ambush and always ready to destroy the universe destroying its harmony.

8) The starting point of our argumentation – that Ancient Egyptians were aware of the phenomena of precession – at this point results well founded by a relevant number of archaeoastronomical data, and the doubts that still exist come mostly from a kind of academic culture so obstinate in its dogmas to result unworthy of any sort of consideration (the official Egyptological culture – if we can still define “culture” an interminable repetition of a theory censoring all the facts that dare to contrast with it – is in this sense similar to that cardinal Bellarmino who refused to look into the Galileo’s telescope saying that what he would have seen would have been only a diabolic illusion). Particularly, the Thomas Brophy work have demonstrated in a way that appear very hardly refutable that the Nabta Playa megalithic circle – datable at least around the 5000 BC – were actually a sort of precessional meridian, in which by stone alignments was marked the higher point that the Orion Belt could reach in the southern part of the horizon during the precessional oscillation at the moment of the summer solstice heliacal rising (a fact that from a point of observation as Nabta Playa happened just around the 5000 BC). Furthermore, still in the Brophy interpretation, the circle marks another point too – a point very near to the lowest possible at the same moment of the year – that the Orion Shoulder reached around the 16500 BC . Before this work of Brophy about the stone circle of Nabta Playa, a work that seems to have an absolutely decisive importance, there was which of Bauval and others, about the IV Dynasty. This work demonstrated that the layout of the Pyramids of Giza and Dahshur it’s nothing but the reproduction of the position of some stars and constellations in a year around 10500 BC3, that’s to say the moment in which Orion was at the lowest point at the horizon at the moment of the vernal equinox heliacal rising. The obvious consequence of this discoveries is that Ancient Egyptian should have a perfect knowledge of the phenomena of precession, as it would be completely impossible to build a site like Giza or a stone circle like Nabta Playa without the knowledge that the configuration of the sky changes slowly but inexorably in a cyclic way.

3 Actually, the lowest point would be reached in the 18000 BC, and it’s our opinion that the date marked by the stone alignments should be precisely that, and so that about this particular Brophy was wrong: but we postpone the analysis of this particular to a following work.

9) But, to these objective proofs of an archaeoastronomical kind, we can surely add some other considerations, that don’t seem less relevant, founded on something that we could define “an historical-scientific common sense”. To people who are not expert about the observation of the sky with the naked eye it could appear a strange thing, but actually to discover the phenomena of precession it is not necessary a scientific-mathematic apparatus so refined as we could believe. The continuative observation of the starry sky and the registration of the position of stars at a characteristic moment of the year – for instance at the vernal equinox heliacal rising – is absolutely sufficient to get acquainted with the fact that the sun, with the passing of the centuries, rises on different constellations. In fact, we could say that it’s completely impossible that a people which has as essential part of its religion the observation and the registration of the heavenly phenomena could not notice that after some centuries the position of the stars changes slowly but unceasingly. But Ancient Egyptians was actually a people like that and actually had this kind of religion. So it’s practically unthinkable that in various millennia of observation of a big number of stars and constellations they didn’t notice that their position in the sky changed following a certain rhythm, and certain mathematical and geometrical rules. That’s because you need some knowledge of mathematic and geometry to establish and to mark the reciprocal position of the heavenly entities and to establish calendars and relations between calendars (and so between heavenly cycles like, for instance, the solar and the lunar ones). So, given their familiarity with the precession, they must have discovered it’s must hidden rule that, as we are going to see, is connected with the golden ratio. In this case, the golden ratio concern the circle, and results from the fraction 360 : 1,618 = 222, 4969, with a reciprocal angle equal to 360 – 222, 4969 = 137,5031. Now, if we think about the polar axis as a sort of cosmic pointer and we imagine to take as a point of reference the plane of the equator and keep it virtually immobile – let’s suppose at a summer solstice – we discover that to the 90° of inclination of the polar axis in relation to the equator we must add the about 47° of which this cosmic pointer oscillates in about 13000 years (that’s to say one half of the precessional cycle). This way we change from a section of 90° + 270° = 360° of that summer solstice we have taken as a point of reference to the 90° + 47° = 137° + 223° = 360°, when that same point of the orbit of the earth around the sun has become a winter solstice (it’s clear that we can reverse this reasoning, taking as point of reference the winter solstice). So, in this period, the earth oscillates between a section of 90°/270° to another very near to the golden ratio (the difference it’s only of one half of degree). And, in a certain sense, we can say that a similar oscillation take place each six months, as the same point of the earth is inclined in relation to the ecliptic of ± 47° at the two opposite solstices. We can have a visual idea of the phenomena in the below image.



Now, if we return to another article we have already published not much time ago in this same site, THE ANCIENT EGYPTIAN RELIEFS AND “THE SNEFRU CODE”: FIRST HISTORICAL AND SYMBOLICAL CONSIDERATIONS AND NEW GEOMETRICAL DISCOVERIES, we can recall to have already seen two images (one on the text of the article, the other in the gallery of the same article in the section named “precession”) that seem to be an unequivocal clue that not only the Ancient Egyptians perfectly knew the phenomena of precession, but also that they were aware that the apparent oscillation of the star goes on between a section of the round angle very similar to the golden ratio – 137°/223° – and another of 90°/270° (this section is numerologically interesting too, as the ratio between these angles is equal to 3, or, reversed, to 0,33333…: we recall that during the precessional cycle the earth change for 3 times its polar star, a thing that probably has originated very important religious beliefs: for instance, the Parcae were 3, in the Tao-Thè-King is said that “the 1 generate the 2, the 2 generate the 3, and the 3 generate all things”, that the Christian God is an Holy Trinity, etc.: watching at the Ramses image below, we can notice that the flowers that the Pharaoh hold in its right hand are 3 (the 7 bellflowers that he holds in his left hand, and that he offers to Sekhmet could be an allusion to the days of the lunar week)). It could be useful to watch again at these images.


If we think about the vertical plain of these two reliefs as a symbol of the position of the equatorial plan of earth at the summer (or winter) solstice and instead to the inclination of the Ramses skirt and to which of the hawk into the Snefru relief as two symbols of the inclination of the same plain at the winter (or summer) solstice after 13000 years, we have this way reconstruct in a symbolical-religious way that purely astronomic-geometric image that we have seen above.

10) Naturally, noting a similar fact, everyone is free to think whatever he wants, including that – for the hundredth time – this scientific information too is contained in very ancient sacred images just for chance. But, in contrary, observing these images we could consider them as a new, very clear clue of the fact that Ancient Egyptians not only knew the phenomena of precession, but above all that they were aware that the apparent oscillation of stars follows a rhythm characterized by the golden ratio of the round angle, even if in an approximated and hidden way (we have reason to think that the deviations of the cosmic cycles from entire and so perfect numbers should be considered by ancient people as the product of the interference of the Chaos forces with an original and perfect order: we recall that the solar year is approximately 365 days, that the precessional year is approximately 26000 solar years, etc., but that they tried to report all time calculations to entire numbers, as Ancient Egyptians did: 360, 5, 12, 10, etc.). These considerations are reinforced by the fact that the golden number is in connection with another heavenly cycle which importance in the astronomical Ancient Egyptian religion is up until now remained in shade, and that instead seems to show itself very clearly on the basis of a deep analysis of the reliefs: the cycle of retrogradation of the nodes of the moon. The first clue that Ancient Egyptians knew this archaeoastronomically very important heavenly cycle comes from the Pharaoh crown, that in a couple of its characteristic versions appear with very strange forms and no less strange dimensions, which up until now remain completely unexplained. If we take these two crowns of the most typical kinds and imagine to rotate them and put them in an horizontal position, immediately we can notice that they get a different look, a look that to many astronomers will seem immediately very familiar. We can say that because the form of these crowns seems to overlap in a perfect way to the curve of a modern Cartesian diagram of that same lunar cycle.


This extraordinary resemblance drive us to recall that the two animals with which was formed the symbol that we can find on the Pharaoh crown, the vulture and the cobra, were connected with two feminine goddesses of which no one has never explained the exact symbolic role and function. But, given the context, we could considerate them as a solar sign the first and as a lunar one the second. If we compare the sun to the vulture, we find that the association could spontaneously turn out from the fact that in whatever period of the year, once it is risen up at the horizon, the sun has always the same form and the same dimensions, that’s to say, metaphorically speaking, that the sun “fly” always goes on with “wings” completely spread, just the vulture does, as it glides with immobile wings or flaps them in an almost imperceptible way. Instead, the cycle or “fly” of the moon could be compared with the lobes of the cobra, that take different forms and appear of variable dimensions according with the situation: and the moon has exactly a cycle of phases in which its dimensions appear very different, so that the full moon could be compared to the cobra when it spreads up the lobes, as the half moon and the various crescents could be compared to the intermediate spans that the lobes could reach in other moments, according to the circumstances. Furthermore, given the complex asymmetry that exist between the lunar cycle and the solar one, we can have periods in which during the night time, when the moon is much more visible than in the day (and so its presence is much more impending to the observer), it seems almost “slithering” on the horizon, when the sun always seems “climbing” to the sky at the dawn and “alighting” on the horizon at sunset, despite of that its maximum highness changes during the year.

11) It is clear that the presence of a lunar symbol on the pharaonic headdress drive us to hypothesize that a calendar connected with a lunar cycle could have had a role of very big importance at the level of an esoteric culture – that in the end was the one which had a real importance in the Ancient Egypt – since we don’t have any knowledge of a folk culture with the slightest independency from that of the cast of the priests, that seemed to correspond more or less with which of the nobles. If we give a look anew to that image of the pharaonic headdress we have the way to realize how deep is their astronomical meaning.



Actually, the High Egypt crown seems exactly a diagram of retrogradation of the nodes of the moon, made like that to change the head and the body of the Pharaoh in a sort of its ideal and symbolic prosecution, and so to change its body in the divine, heavenly entity that he hermetically represent (instead, that the Pharaoh were a personification of the sun and/or of Orion seems a kind of faith openly professed by the whole of the Ancient Egyptian people, not only by a caste of initiates). Instead, the Low Egypt crown seems formed in the rear side by the line of the horizon, as the front side seems a characteristic segment of a very familiar mathematical symbol, the braces, that we see almost entirely drawn from the point in which the crown goes to fit in with the ear. Instead, the strange “bar” that ends in a sort of doodle seems to mark a characteristic point of the cycle, that’s to say a fourth, or, as we could say, the half of one half. To these remarks we can add that another very important symbol of the Ancient Egypt culture, the Horus Eye, shows too very impressive similarities with the diagram of the retrogradation of the nodes of the moon . It is true that our Cartesian-style diagram is perfectly symmetric and the Horus Eye is not, but we have to consider that that symbol exist in the right and left version, and that that perfect symmetry that we can’t find in a single eye could be built overlapping the two eyes, for instance, at the level of the “bar” traced under the lower eyelid.


12) But we can appreciate completely the importance that this horizon diagram had to the Ancient Egyptians only analyzing some of the most famous sacred images, where it seems that the figures are systematically constructed, so to speak, assembling this diagram or parts of this diagram following rules that at this moment are completely unknown: even the minimum details, that in this graphic analysis we couldn’t emphasize, as eye and eyebrow, seem at a rough guess derived in this way.


In altri casi la proiezione del diagramma non è possibile realizzarla in modo così preciso ed olistico, anche se questo genere di curva la si può ritrovare in pratica dappertutto, e sembra caratterizzare gran parte della produzione figurativa Antico Egizia4. Vi è inoltre un ulteriore e molto importante indizio di un’importanza ermetica del ciclo di retrogradazione dei nodi della luna all’interno del pensiero astronomico-religioso Antico Egizio, che forse è quello più importante di tutti. Uno degli elementiche vengono considerati più enigmatici nel mito che racconta dell’unificazione dell’Alto e del Basso Egitto, è il fatto che questo evento avvenne al termine di una battaglia fra Horus, il figlio di Osiride, e Seth, il suo fratello assassino: la durata di questa battaglia – 18 anni solari – non è mai stata compresa, fra l’altro anche perché, almeno prima facie, non sembra corrispondere a nessun ciclo umano o naturale di un qualche significato. Al termine di questa battaglia Horus, pur risultando infine vittorioso, perse un occhio, un occhio che alla fine venne dato in dono a Osiride. Ma, dopo aver visto le precedenti immagini, e considerando che i 18 anni possono essere un arrotondamento avvenuto per motivi magico-numerologici (allo stesso modo in cui si sono arrotondati a 360 i giorni “normali” dell’anno solare in relazione ai 365,25 effettivi, cosa che ha consentito l’istituzione di 12 mesi di 30 giorni, ognuno costituito da 3 settimane di 10 giorni) ecco che viene del tutto naturale e spontaneo immaginare che questo numero possa riferirsi al ciclo di retrogradazione dei nodi della luna, che dura appunto 18,61 anni solari: la “battaglia” fra Horus e Seth non sarebbe dunque una battaglia fra due aspiranti sovrani effettivamente vissuti e poi, come si dice, “mitizzati” (non foss’altro perché non si vede come sia possibile che una singola battaglia duri il tempo che – storicamente parlando – può riferirsi solo a quello di una guerra, per giunta assai più lunga della media). Al contrario, questa battaglia dovrebbe essere considerata senz’altro una battaglia cosmica fra entità celesti, una battaglia che dura per un intero ciclo di retrogradazione dei nodi della luna e al termine della quale Horus, molto probabilmente il Sole, diventa l’entità divino-umana “ufficialmente” e dunque pubblicamente rappresentata dal Faraone, entità che fa da trait d’union fra il Basso Egitto e l’Alto Egitto, vale a dire fra l’Egitto Celeste, identificato con il cielo diurno e notturno e l’Egitto terrestre.


4 Senza entrare in ulteriori dettagli, che affronteremo in una fase successiva, dall’analisi geometrica che abbiamo condotto si può ricavare la pratica certezza che tutte le curve tipiche dell’arte figurativa Antico Egizia, ove non si tratti del diagramma lunare che abbiamo appena visto, vi si può riconoscere senz’altro la curva della traiettoria del sole durante il solstizio d’estate. Anche quando non si possono sovrapporre in modo perfettamente “realistico” dei diagrammi d’orizzonte, si ha la sensazione che in qualche modo anche questo genere di variazioni possano essere ricondotte a delle motivazioni religioso-astronomiche: forse la corona faraonica rappresentava, a seconda delle esigenze o delle credenze teologiche in voga in un certo momento storico, o dei momenti del ciclo cosmico, i diagrammi di differenti corpi celesti e non necessariamente quello lunare che abbiamo visto sopra.

13) L’occhio che Horus perde al termine di questa battaglia e che lascia in dono a Osiride potrebbe essere proprio la Luna che, come tutti sappiamo, è un astro che compare all’orizzonte tanto di notte che di giorno, e che dunque può essere associata tanto al Sole che a una costellazione divinizzata come Osiride-Orione. Il significato di questa parte del mito sembra essere proprio questo: che Horus-Sole figlio Osiride-Orione, ereditando il trono del padre stellare, gli cede in cambio la Luna-Iside, rimanendo con un solo occhio, che, come è stato più volte osservato, probabilmente non è altro che il disco solare (in questo senso, Horus che perde un occhio dovrebbe essere considerato, in senso ampio, come il tempo diurno che perde un astro, la Luna, che in effetti, siccome di giorno è enormemente meno visibile del sole, può essere considerata in relazione a questo “occhio” tanto pieno di luce da non poter essere guardato come una sorta di orbita vuota). Ma una volta che ammettiamo che il ciclo di retrogradazione dei nodi della luna abbia questo genere di importanza tanto nella sapienza che nella religione esoterica Antico Egizie, ecco che scopriamo che questo ciclo ha – proprio come la precessione degli equinozi – una relazione che appare davvero profonda con la sezione aurea, una relazione che possiamo constatare in modo inequivocabile nelle immagini che seguono, dato che il suo diagramma si può costruire a partire da una doppia spirale logaritmica, e che con essa si possono individuare i suoi punti fondamentali e caratterizzanti


n other cases it’s impossible to obtain the projection of the diagram in a so precise and holistic way, but it seems that we can find this kind of curve practically everywhere, so that it seems to characterize a big part of the figurative Ancient Egyptian production4. Furthermore, we have another and very important clue of an hermetic presence of the cycle of the retrogradation of the nodes of the moon in the astronomic-religious Ancient Egyptian thought, that is maybe the most important. One of the elements which are considered more enigmatic in the myth that tells of the unification of the High and Low Egypt, is the fact that this event happened at the end of a battle between Horus, the Osiris son, with Seth, his murderous brother: the length of this battle – 18 solar years – was never really understood, because does not seem to correspond to any human or natural cycle with some kind of deep meaning. At the end of this battle Horus, yet victorious, lost an eye, an eye that in the end was given to Osiris as a gift. But having seen the previous images, and considering that a length of 18 years could correspond to a rounding down of the 18,61 solar years of the retrogradation moon cycle, made by magical-numerological reasons (in the same way in which Ancient Egyptians rounded down to 360 the 365,25 days of the solar year, so they could divided these 360 days in exactly 12 months of 30 days each, and these 12 month in 3 weeks of 10 days each). This way, it is natural and spontaneous to imagine that the battle between Horus and Seth was not a real battle, occurred between two real, historical aspiring kings, that in the end turned into a legend, if nothing else because it seems almost impossible that a single battle could last the time that – historically speaking – can refer only to an entire war, furthermore far longer than the time of an average war. In contrary, this war should be considered a cosmic battle between heavenly entities, a battle that lasts the time of an entire cycle of retrogradation of the nodes of the moon. Probably, this battle ends with a conjunction of the moon with Orion, as the myth says that Horus (most likely the sun, that in ancient times become the human-divine entity officially represented by the Pharaoh) gives its eye to Osiris. With this battle and with this sacrifice, Horus – that’s to say the sun – become the trait d’union between the High Egypt, the Egypt of the starry sky, and the terrestrial Egypt.

4 From the geometrical analysis that we made of some other Ancient Egyptian reliefs, we could derive the practical certainty that the whole typical curves of the figurative art are characterized or by this kind of curve or by those of the two solstices. Also in the cases that we have not the possibility of a perfect overlapping we can have the sensation that this kind of variation have been caused to motivation of an astronomic-religious kind. Maybe, the Pharaoh crown too, with the changing of the theological tendencies or of the moments of a cosmic cycle, could represent the diagram of different heavenly entity, and not necessarily that of the retrogradation of the nodes of the moon.

13) So, the eye that Horus lose at the end of this battle and that he let as a gift to his father Osiris should be the moon itself, that is a celestial body that appear in the sky both in the night and in the day, and so could be attached to the sun or to a divinized constellation as Orion-Osiris. The meaning of this part of the myth seems to be exactly this: that the Sun-Horus, son of Orion-Osiris, inheriting the father’s throne, gives him in exchange – as a gift of reparation – his mother Moon-Isis (considering this situation in a psychoanalytical way, we could talk of a restitution of the mother to the father). This way Horus remains with only one eye that, as it was many times noticed, probably is nothing but the solar disc (so, in a large sense, Horus should be considered not only as the sun, but as the daytime: only with this logical-mythological step we can understand the affirmation that “Horus loses an eye”: from another point of view, we can notice that the moon during the day is actually very less visible than the sun and, in comparison with this so glowing “eye”, could be considered as a sort of empty orbit). But if we admit that the cycle of retrogradation of the nodes of the moon could have this kind of importance as in the wisdom and in the esoteric Ancient Egyptian religion, than we discover that this cycle – just as the precession – has a relation that appear really important with the golden ratio. We can find out it in an unequivocal way in the following images, where we can clearly see that we can both construct the modern Cartesian diagram of this cycle and to single out its fundamental or characteristic points by a twin logarithmic spiral.


At this point, it results clear the astronomical-theological reason why many thousands of years ago a model of golden ratio space was built5 , aside its possible connections with the relativity theory6. And if nowadays this reason results to us more or less extraneous, if not completely incomprehensible, still at the times in which Alexander the Great occupied the Persian Empire was instead commonly accepted as so obvious and fundamental that neither it was assumed in a conscious way and discussed, at least in the south-oriental part of the Mediterranean sea (but it is possible that to the Macedonian invaders this way of thinking could already appear rather amazing, even not in the same measure that to us; in fact, Aristotle wrote that “once was believed that the Goddesses were planets”: so in the Greece of his times this belief was no more actual, or it lasted only as an echo of an ancient faith but without any real and vital worth). Actually, still in the 300 BC, it was spread in all the Middle East the belief that the “under world”, that’s to say the human world, should be a copy of the “superior world”, that’s to say of the starry sky. And it was precisely to be the terrestrial reflection of the cities and the monuments that they believed to contemplate in the night sky that terrestrial cities and monuments were built. For instance, the so famous and celebrated hanging gardens that was possible to admire in Babylon were not a monumental but meaningless decoration, made to entertain nobles and common people with an empty distraction: in contrary, they had been made to be a copy of the celestial Eden. Not very far from there, the terrestrial Jerusalem had been built to be the reflection of the Heavenly Jerusalem, not a whatever place where to live in.

5 We have found the first traces of this space in Paleolithic remains that results about 40.000 years old. But we will see and discuss this material in a new work, that we will publish on this same site.

14) It is not clear what the origin of this form of religious and philosophical thought could be, but it is clear that it should have been shared by the Ancient Egyptians, because otherwise they had not built a terrestrial image of the starry sky as it results from the layout of the IV Dynasty Pyramids, or from the alignments of a megalithic circle as Nabta Playa. But as the Ancient Egyptians had discovered the intimate connection between the precessional cycle and that of the retrogradation of the nodes of the moon with the golden ratio, it is historically logical to hypothesize that their astronomers-priests got to the persuasion that the secret of the divine creative and regulative power itself should lay into this mathematical-geometrical proportion (particularly, the precessional cycle is a sort of background of all the other astronomic cycles, as it is on the basis of precession that skies and maps of the skies are made up and destroyed). That was the reason why the Ancient Egyptian sacred buildings – that as images of the sky was built on earth – not only should be located in analogy with the position of stars and constellation in a certain point of the precessional cycle. They got beyond this, and arrive to think that the geometrical essence of the space in which they were drawn should be determined – point by point – by the divine measure that determined the fundamental rhythms of the precession and of the retrogradation of the nodes of the moon. It was from this mathematical-religious belief that came out the effort to arrive to that inhumanly complicated and refined way to define the space and the form of the space that we have seen in the previous article: as in this religious view the sacred rhythm of the sky are believed the origin of the generation and regeneration of the life on earth, the believers produced images that could contains in their intimate structure the fundamental and so divine measure, that’s to say the golden number. We have seen in the previous article “THE GOLDEN RATIO SPACE IN THE ANCIENT EGYPTIAN FIGURATIVE ART AND ARCHITECTURE: AN HYPOTHESIS OF SOLUTION STARTING FROM THE STRUCTURE OF THE SUNFLOWER INFLORENCE” that the diagram that fits better to describe the coordinates of the points of the Ancient Egyptian reliefs and the profile section of the Big Pyramid seems to be that of the projection of the space-time on two dimensions. Now we can see that this kind of diagram seems to be at the basis of the plan of gigantic architectonic spaces as Giza and Dahshur, and of the megalithic circle of Nabta Playa.


From these images we can note how that particular “geometrical rhythm” – from which the whole and any particular of each work of Ancient Egyptian sacred art and architecture seems mysteriously to spring out – seems corresponds to that of the two dimension diagram of the space-time elaborated by Vincenzo Zappalà. And this diagram, in turn, is fundamentally constructed by circles (and so by ) and by semi-circles that follows one another according to the Fibonacci’ sequence (and so by  and ϕ). So it is not a case that we can find  and ϕ in all the fundamental proportions of the Big Pyramid or in those of the Ramses statues of Luxor, as it was demonstrated by the Christopher Dunn works.

15) On the basis of what we have seen in the previous pages, we can very reasonably hypothesize that one of the deep reasons of this very complicated construction of the sacred space must be sought in the structure of the fundamental cosmic cycles that was observed and followed by the priests-astronomies as signs of the changing of the heavenly world, and so of divine Eras. Even if we don’t want to believe that Ancient Egyptians knew the relativity theory, all the argumentation we have carried out up until now should be sufficient to convince ourselves that they must be aware at least of the fundamental part that the golden ratio has in the precession of the equinoxes and in the retrogradation of the nodes of the moon. And if a religion has as a fundamental belief that earth and each terrestrial entity are generated as image of the Sky and Heavenly Entities, the consequence is that those images of Sky and Heavenly Entities (represented sometimes by means of hieroglyphics, sometimes by means of human beings, or animals, or a mix between humans and animals, or by entire architectonic sites as Giza and Dahshur) could become literally capable to take the Sky on earth, if the spaces, the proportions and the forms was designed and built according to the hermetic measure of the Sky and its rhythms. This way the sacred images had took on earth also a piece of the creative and so protective power of the divine, playing the role of a very powerful “”. To build pyramids and sacred architectonic spaces of any sort, to engrave reliefs or inscriptions according to a geometry founded on the golden ratio was so an active way with which man collaborated with the goddesses in the fight against the ubiquitous and threatening forces of the Chaos. To the Ancient Egyptians, it was not considered a sufficient contribution to the conservation of the universe that human being acted to a spiritual, moral and social level according to the Order and the Justice (Maat). In this sense, we find an analogy with the Aztec culture. The Aztecs too believed that they must collaborate with the Gods to delay the end of the world, only, they believed that this duty consisted in offering human blood to the Sun – because the energy he dispensed to give life to earth was restored. Instead, Ancient Egyptians thought they had to do the same “salvation work” building monuments intimately permeated by the secret law of the Skies.