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Imagine you are walking along the beach and come across an empty seashell. Intrigued by the spiral form of the mollusc, you cut it lengthways and see a shell of a very similar shape to the spiral logarithmic mathematical function (see photo 1). Imagine too that days later you are watching the weather forecast on television and learn that there has been a tropical hurricane. The weatherman shows satellite images of this atmospheric phenomenon and we see that the clouds sketch an immense series of logarithmic spirals converging in the "eye" of the hurricane (see photo 2). Finally, let's say that an astronomy expert friend shows us the image of deep space through his powerful telescope. One of the things we can see, our friend tells us, is galaxy M51, or, the Whirlpool Galaxy. Looking at it through the lens of the telescope, we marvel that it forms a pair of immense spirals spanning millions upon millions of kilometres, and that this too looks like two logarithmic spirals converging in the centre of the galaxy (see photo 3).
The resemblance of forms (spiral logarithms) among these natural entities of vastly different scales suggests that there could be a relation among them. Or at least that there could be some law that compels them to organise themselves morphologically so that they resemble each other. Is the resemblance of forms between entities of vastly different scales a constant in nature? Or an isolated event? We have just discovered the resemblance, or correspondence, among certain shapes in nature, by our experience with the seashell, the hurricane and the galaxy. This phenomenon has been noted and studied by Pythagoras in his Golden Verses in the Third Century BC. The work is a collection of aphorisms compiled by his followers, which make up Pythagorean philosophy. Verse no. 28 reads as follows: φυσιν περι παντος ομοιην In Fabre d'Olivet's translation into French4, "La Nature, semblable en toute chose, est la m�me en tout lieu". "Nature, similar in all things, is the same in all its parts". It is therefore clear that the Pythagoreans were already aware of what we have just discovered.
This idea, according to the same Olivet5, can be seen in the myth of Thoth, called Hermes Trismegistus by the Greeks, in a text titled The Emerald Table (or Tabula Smaragdina in Latin). The text exists today thanks to its translation into Arabic by Geber6, an eighth-century Iraqi mystic. Many centuries later, it was translated into Latin on several occasions, as in I. Petreium's On Alchemy7. Another translation into Latin was published by the German Heinrich Kunrath8, and there was even a version by the eminent eighteenth-century English physicist Isaac Newton9. The Latin version of The Emerald Table which appears in On Alchemy by I. Petreium (1541): “Verum sine mendacio, certum, et verisimum. Quod est inferius, est sicut quod est superius. Et quod est superius, est sicut quod est inferius, ad perpetranda miracula rei unius. Et sicut omnes res fuerunt ab uno, meditatione unius. Sic omnes res natæ fuerunt ab hac una re, adaptatione. Pater eius est Sol, mater eius Luna. Portavit illud ventus in ventre suo. Nutrix eius terra est. Pater omnis telesmi totius mundi est hic. Vis eius integra est, si versa fuerit in terram. Separabis terram ab igne, subtile a spisso, suaviter cum magno ingenio. Ascendit a terra in cœlum, interumque descendit in terra, et recipit vim superiorum et inferiorum. Sic habebis gloriam totius mundi. Ideo fugiet a te omnis obscuritas. Hic est totius fortitudinis fortitudo fortis, quia vincet omnem rem subtilem, omnemque solidam penetrabit. Sic mundus creatus est. Hinc erunt adaptationes mirabiles, quarum modus hic est. Itaque vocatus sum Hermes Trismegistus, habens res partes philosophiæ totius mundi. Completus est, quod dixi de operatione Solis.” The text has such historical importance that authors such as the historian T. Burckhardt10 have commented that "the alchemists of the Arabic and Latin tongues consider the table to be the true source of the law of their art". And inasmuch as we can interpret the Table, we can confirm in the paragraph below that: "Quod est inferius, est sicut quod est superius. Et quod est superius, est sicut quod est inferius [...]" Or, translated, "that which is below resembles that which is above, and conversely, that which is above resembles that which is below." In other words, the text establishes a correspondence between what is "below" and what is "above", confirming that both are similar (though not equal). The ambiguity around the use of the words "below" and "above" allows a wide scope for interpretation. Thus we have the exchange of multiple entities, not only those in nature like the seashell or the hurricane, but also others amplified to an infinite degree by this idea of similarity. Not only is there a link between the morphology of different entities in nature, but this correspondence also exists within completely different categories, whether abstract or concrete and tangible. This can extend to similarities in the life cycles of entities as far removed from each other as civilisations, men and stars; to similarities in the process of creation of a novel, a scientific theory, or the Universe itself. There can thus be a resemblance between the life of a man ("below") and the life of a star ("above") or between the creative process of a novel ("below") and the creative process of the Universe itself ("above"). To further illustrate this idea, think about the similarity between the life of a man, that of a civilisation, and that of a star like our Sun. The similarity these three share is that they come into being through birth and depart through death. The "below" here is man and the "above" is the stars. In between the two, we can place nations or civilisations.
On the Man level are stills from the "Nantes Triptych"11 video installation by contemporary artist Bill Viola. The first image shows a woman about to bear a child (Birth), the second shows a man suspended in limbo (Maturity), and finally the artist?s own mother on her deathbed (Death). On the Ancient Egypt level, we can see a representation of the life cycle of ancient imperial Egypt in three works of art. The first is a sculpture by Kephren (4th Dynasty, 2556-2530 BC), one of the first great pharaohs of Ancient Egypt (Birth). The second is the effigy of the famous Tutankhamen (18th Dynasty, 1336-1327 BC), representing the intermediary period of Ancient Egypt (Maturity). Finally, we have an image resembling a sculpture of a Romanised pharaoh, the emperor Nero, dressed like the ancient pharaohs, of whom we have no knowledge (Death). On the Star level we see the life cycle of a star. In the first photo we see proto-planetary discs forming around various stars that resemble our Sun, and which have just come into existence on the Orion Belt12. (Birth). Next is an image of our Sun13, currently about half way through its life cycle (Maturity). Finally, there is a photo of a star, also similar to our Sun14, exploding at the end of its existence (Death).
Relationship to the insula smaragdina poem All these different ideas can be expressed more succinctly through the insula smaragdina visual poem. As we saw in the Description section, the words Quod est superius, est sicut... were written in such a way that, strung together they formed a fractal called Koch?s Island. This implied that the form of the poem had the quality of self-similarity, regardless of the scale on which it was observed. Moreover, as I have just explained in the Meaning section, Quod est superius, est sicut... in itself can express the idea of similarity between what is "above", and what is "below". Consequently, the same principle is expressed both by the form of the poem, and the text of which it is composed. Thus through modern mathematics, the insula smaragdina demonstrates one of the deepest and most ancient axioms of the overwhelming harmony of the Universe.
1. More information on the seashell is available on Wikipedia http://en.wikipedia.org/wiki/Nautilus. 2. Photo from the American Metrological Agency NOAA, by John Weells (Science Photo Library). Scanned from the "Science and Health" supplement of La Vanguardia newspaper, 15 March 1997, p.12. 3. Photo of the Whirlpool Galaxy, Messier 51, NGC 5194, taken by the NASA Hubble space telescope. 4. D'OLIVET, FABRE, The Golden Verses of Pythagoras. Kessinger Publishing Company, Montana, USA, pp.114 to 121. 5. D'OLIVET FABRE, op cit., pp.251 to 252. 6. Abu Musa Jabir ibn Hayyan (Geber). Second Book of the Elements of Foundation. 7. On Alchemy, published by I. Petreium, Nuremberg 1541, p.363. 8. KHUNRATH, HEINRICH. Amfiteatre de la Sapiència Eterna, Tarragona, 1st Catalan edition, 1991, translated by Desideri Forner. 9. NEWTON, ISAAC, Tabula Smaragdina Hermetis Trismegisti Philosophorum Patris, published by DOBBS, B.J.T. in The Janus Face of Genius, Cambridge University Press, Cambridge, 1991, p.274 to 275. 10. BURCKHARDT, TITUS. Alchemy. World Meanings and Images. Paidós Orientalia, Barcelona, 1994, p.18. 11. The Art of Bill Viola, edited by Chris Townsend, Thames & Hudson, London, 2004, p.186, The "Nantes Triptych" installation was made in 1992. 12. Proplyd or PROto-PLanetarY Disk situated on the Orion Belt. Photographed from the NASA Hubble space telescope on 29 December 1993, using the "Wide Field and Planetary Camera". 13. Image of our Sun in false colour (x-rays) from the Yohkoh Japanese solar satellite. 14. Hubble image of the NGC 7027 proto-planetary formation situated at 3,000 light years from earth in the Swan constellation, resulting in the explosion of a star similar to our Sun at the end of its life cycle. 15. GONZÁLEZ URBANEJA, P.M. Pythagoras, from the collection "Mathematics and its Characters", no. 9 Ed. NIVOLA, 1st edition, May 2001, Madrid. 16. MANDELBROT, BENOIT. FRACTAL GEOMETRY of Nature. Metatemas, no. 49. 1st edition, October 1997, Barcelona. 17. CASADO, JAVIER. ART in the UNIVERSE. The most beautiful images of our spatial environment. Ed. NIVOLA, 1st edition, November 2003, Madrid.
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