Sam C Ziegler 2008-2009

Abstract.The following thesis examines the origin of the Möbius strip, by testing the hypothesis that this (Möbius strip) and related single-surfaced topologies were originally discovered by A. F Möbius and J. B Listing. This is achieved by conducting a study of verifiable historical documents.

This study is limited to characteristic devices which correspond both visually and semantically to the related topologies. The devices studied here are the following: Möbius Strip, Lemniscates, Pelican, Ouroboros, and Ankh.

By making a study of historical examples of these symbols the purpose of this essay is to seek knowledge of the Möbius Strip existing prior to 1858. The range of this study is European history, the four centuries spanning 2000BCE - 2000 CE.

Evidence indicates with a degree of certainty that the shape of the Möbius strip is the historical origin of the Chrysopoeia of the early Alchemists.

The evidence gathered here gives insight to the studies of iconography, epistemology, the philosophy of science, natural philosophy.

Key words: Möbius Strip, Lemniscates, Pelican, Ankh, Ouroboros, Chrysopoeia, History, Topology, Alchemy, Klein bottle

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Table of Contents

Abstract 1

Table of Figures 3

Introduction 5

The Möbius Strip 6

The Lemniscus 9

The Pelican 11

The Ouroboros 13

The Ankh 20

The Cycle 27

Conclusion 30

References 36

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Sometime, we see a cloud that’s dragonish.

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Introduction.Humankind is learning, of this we can have no doubt. Many simple truths have been discovered only recently, many we can remember from millennia ago. The world of conceptual knowledge is a colorful realm, of opinion and imagination, art, emotion, and shifting logical truths, a fractured, brilliant and growing landscape. This essay stands as an enquiry into the knowledge of the Möbius strip.

The Möbius strip, heretofore attributed to two sound minds of the intellectual revolution, is most often defined as a manifestation of the mathematical science, topology.

The first aim of this essay is to test the hypothesis that single-surface topologies were first discovered by August Möbius and Johann Listing around the time of the mid 19th century. This can be achieved through the discovery of evidence of the appearance of single-surface in historical record prior to this date.

The hypothesis that humankind has never, ever, known of the Möbius topology before Möbius and Listing is strictly unable to be proven through scientific reason. However, as it is supported by the evidence at hand, and completely unopposed by any other proven fact, the suggestion that it has only existed since the mid 1800s is taken to be the simplest explanation. As a hypothesis,

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this can be tested, and although it cannot be proven to be true, it could be shown to be false. Naturally, it is given as a possibility that the first origin came in the 1800’s. But it is also possible that mankind has somehow erased all previous records, has not discovered past record of it, or has never produced any record of it. Thus it can never be scientifically proven that our ancestors did not at one time know of the Möbius strip. This essay cannot be said to offer conclusive scientific knowledge on all points, but it is worthwhile simply if it articulates the fine balance of truth, fallacy and possibility.

‘Moebus Strip II’

The Mobius Strip.So then, let us begin by sifting through all evidence of the history of the Human race. Or perhaps we can save this for later. It is clear that we can best conduct this enquiry only if we first have a full understanding of the forms of the Möbius topology.

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We know it to be true that in Europe in the 19th Century there was discovered an idea which may have existed since the most ancient of times. As if an ideal fossil, the geometry of the Möbius strip today remains unchanged by all time, a potential link to times past. Assuming that the laws of mathematics and geometry are unchanging, as it is today, it is as it would have been centuries ago. With this knowledge, we can limit our search of history, and begin our enquiry seeking only information which contains a resemblance to the geometrical forms which bear one of the potential forms of the single-surface topology.

Records state that the first known discovery of the Möbius strip was made in the year 1858, independently and almost simultaneously by Johann Benedict Listing (1808-1882), and August Ferdinand Möbius (1790- 1868), both men established academics in what is now Germany1. The official discovery is credited to be Möbius’ 1965 paper, ‘On the Determination of the Volume of a Polyhedron’2.

The Möbius strip has been represented in art in many ways, however the boundary was undoubtedly set by the original Dutch master of the lithograph, Maurits C Escher, whose exceptional skill in weaving the impossible with the possible alike is still rare to be seen. The images above and below are Escher’s woodcuts. As ‘Moebus Strip II’ shows, we can imagine that an ant walking along the surface of a Möbius strip can journey across the whole length, and return again at her beginning, whereupon she can repeat the journey again, ad infinitum.

Johann Benedict Listing

1 Pickover, Cliff A, The Möbius Strip, Thunder’s Mouth Press, New York, 2006, p.282 ibid

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August Ferdinand Möbius

Naturally, the Möbius strip has since been interpreted by many as a metaphorical figure of the infinite or the eternal. Many will already recognize the Möbius topology, perhaps even unaware of it. As a symbol it bears the atlas task of persuading humans to remember the fact what we dispose of does not go away. As to the ultimate state of humanity, this symbol may prove to be of a far greater good than the symbols of the human brands, which, like factories of waste, only consume and digest the ancient form of the natural world, to feed our craving for intangible self-gratification.

‘Moebus strip I’ Recycling Emblem

By Plato’s standard, the ideal form of the Möbius strip has existed for at least as long as time itself3. Yet one of the most curious of the Möbius strip parlor tricks is the enigma of its novelty. For, while humankind has been measuring and imitating nature geometrically for ages in areas of science, philosophy and artistic expression, from these millennial foundations, we are told that we humans have been aware of the simple shape of the Möbius strip for the piddling span of 150 years. This is fair, there is little or no historical evidence that any humans were aware of the existence of the Möbius topology prior to this era of modern western science4.

Like the point, or the line, only imperfect reflections of the ideal Möbius strip form exist in nature. In fact, perfect or imperfect, it never seems to occur naturally in the human sized world of existence. As by this earlier stated argument, this non-being cannot be proven to be true. Assuming that the Möbius topology is partly artificial, it would seem to be a factual fiction come to 3 Plato, ‘Allegory of the Cave’, The Republic, 4 Pickover, The Möbius Strip, 2006, p.29

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being though human creation, an imagining brought into the stage of physical being through artificial technology, much like a can of soup. Like many forms, it is intelligible to the language of mathematics5.

The standard method to create a physical Möbius strip is to take a flat long rectangular strip of flexible material, such as paper, fabric, fibers or metal, perform a half a twist over its length, and join together the ends so as to create a ring with a half-twist. Möbius and Listing are specially honored in that they also made mathematical descriptions of the form.

The Möbius strip itself seems in all first appearances to be completely unnatural. It is not an unnatural figure; however it may adopt this quality from the fact that it is an artificial creation. This might be because nature rarely conducts herself so two-dimensionally. Few naturally flat forms exist to exhibit a single surface, most have two, like a leaf or a soap bubble.

How to make a Möbius Strip

A two dimensional form which can only exist in three dimensional space, the Möbius strip bears the property of a continuous single surface on the body of its form. What makes this curious is that unlike a sphere or torus, it is almost without width in section. When taken as a whole though, because it must curve in a twist, it requires a ‘manifold’ of three dimensions width, length, breadth, to form. While it does not truly ‘exist’ in

5 Möbius, August Ferdinand, Gessammelte Werke. Herausgegeben auf Veranlassung der Königlich sächsischen Gesellschaft der Wissenschaften, Edited by Richard Baltzer, Felix Klein, and Wilhelm Scheibner, 4 vols, Leipzig

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this reality other than within the imaginary plane of mathematics, the Möbius strip has nevertheless become an inspirational curiosity for artists, scientists, philosophers, and dreamers alike.

The Lemniscus.It is a common speculation that the Möbius strip was the original idea behind the adoption of the lemniscates symbol, i.e. ∞, as the standard mathematical notation for the figure of infinity6. There are a few good reasons for this. First, the surface of the Möbius strip can be philosophically associated with idea of the unending procession of the infinite. Secondly, the creator of the symbol, the mathematician John Wallis, referred to this shape the specific property of boundlessness. Thirdly, the title of the symbol, lemniscates, means in Latin ‘ribbon’. Ribbon is a material is unlike string or rope in that it has a flat surface with definite width but little depth, which is the necessary structure of the Möbius strip. Finally, due to the curvature made by tension held by the form in paper, a side on perspective of the Möbius strip of moderate width, resting on a flat surface, will form precisely this figure-eight shape, unless forced otherwise. A regular loop will not behave like this.

John Wallis first made use of this symbol, referring it to the infinite in 1655 in the short work De Sectionibus Conicus, and again the following year in his treatise Arithmetica Infinitorum7.

Möbius strip- top view: ring

6 Clegg, Brian, A Brief History of Infinity: The Quest to Think the Unthinkable, Constable and Robinson, London, 2003, P.757 Clegg, A Brief History of Infinity, 2003, pp70-71

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Möbius strip- side view: lemniscus John Wallis 1616-1703

It is important to note that Wallis did not give any particular title to the figure; in fact the title ‘lemniscates’ is attributed to Jakob Bernoulli, from his later description of the lemniscus curve. Lemniscus sometimes translates as ‘pendant ribbon’, which is possibly a reference to a strip of fabric tied to a pole, as would be found on the pennant flag of a medieval European House.

It is an often disputed question as to Wallis’s motive in assigning this symbolic notation8. It has been suggested that Wallis was adapting the roman numeral ‘thousand’, which was often the word used to simply mean vast amount. Sometimes the Etruscan notation denoting 1,000 was reduced to CIƆ in shorthand9. This symbol when written in ink can flow to form the symbol ∞. This figure corresponds in a way to the idea of a large sum, and to the graphic icon used by Wallis. Clearly it is on similar footing to the Möbius strip, and provides a highly plausible explanation to the origin of this symbol.

One more thing to note, perhaps of importance, is Wallis’s literally cryptic reputation. He held the office of Chief Cryptographer to the English Parliamentarian Army, at war with the Royalists10. Cryptography is the art and science of making and deciphering secret codes. It is a fact that one of the oldest and simplest methods of making a cipher involves wrapping a thin strip of paper around a cylinder of known size, and writing letters in a straight line the length of the coil, then unwrapping the paper strip and filling in the blank. Regardless of the method of cryptography, a small rectangular strip of paper is a common way of transporting a coded message, as it is unobtrusive when attached to a pigeon’s leg, and highly concealable.

We can easily speculate that John Wallis spent a good piece of his life, however obliquely, in the manufacture of long rectangular strips of paper. It is not hard to suggest that John Wallis, scientist, mathematician, cryptographer, an unsung polymath genius of the enlightenment, knew the simple fact that a strip of paper joined at the ends with a half twist makes single

8 Clegg, A Brief History of Infinity, 2003, p.749 Clegg, A Brief History of Infinity, 2003, p. 7410 Clegg, A Brief History of Infinity, 2003, p.73

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continuous, infinite seeming, surface. If we suggest this hypothesis, the fact that Wallis used a symbol with an identical form as the reference to the quality of boundlessness would support it. As it is this reference which prompted us to suppose that he had knowledge of it, this is makes for a circular argument.

The Pelican.In 1882, advancing on the work of Möbius, Felix Klein was accredited with the invention of the surface form known as the Klein bottle11. Unlike the Möbius strip, the Klein bottle is an enclosed surface, and can bear binary symmetry. An open composite of the both the Klein bottle and the Möbius strip results in a semi-enclosed surface, which can bear binary symmetry.

Pelican Klein Bottle

This illustration of the Pelican was given to me by my father. It is from Lancaster, Pennsylvania, and dates around 1800. The similarity between the structure of the Klein bottle and the ancient fable of the pelican indicates correlation in form. In mythological accounts the pelican bird was thought to pierce its own breast with its beak, to its heart, so as to draw blood, which has the power to resurrect its slain offspring12. This image was

11 Pickover, The Möbius Strip, 2006, pp.81-8212 Borges, Jorge Louis, ‘The Pelican’, in The Book of Imaginary Beings, trans. A Hurley, Penguin Books, 2005, pp 151-152

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especially prevalent through Europe in the middle ages, as a symbol of resurrection of the innocent through self sacrifice.

The philosopher Melanie Purcell, a colleague of the noted artist Sheila Morgan, has made light of this similarity of this figure to an alembic flask, the basic equipment of an experimental alchemist13. An alembic flask is made from blown glass, and thus is very similar to the manufacture of the Klein bottle.

It is difficult to say whether there was any causal connection between the Klein bottle topology and the Pelican alembic flask, or between the Pelican alembic flask and the Pelican mythos. The origin of this myth is not certain. It is unlikely that the myth originated from the observation of pelican birds actually stabbing themselves. Although pelicans have sometimes been observed doing strange things, this is not currently one of them.

It is unlikely that anyone can cover the idea more comprehensively than Purcell, and I will not attempt to do so here. It is sufficient to say that the cycle of regeneration from self, combined with the correspondence in form of the bird in this pose of self-intersection, shows an outstanding metaphorical and physical correlation to the Klein bottle.

13 Purcell M, Towards A New E.R.A.: Epistemological Resolution Analysis By Through and From Klein-Bottle Wholeness and Transdisciplinary Education, 2006

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Ouroboros- ‘Hen to Pan’

The Ouroboros. The Ouroboros is a myth of giant primal serpentine dragon which consumes its own tail for food. It was a familiar symbol in Ancient Egypt; there are examples of it inscribed on Tutankhamen’s tomb, designated as the figure Mehen14. The symbol has survived over the centuries by remaining in use, and is an important symbolic metaphor within the alchemical tradition, known to refer to many principles, processes, and ideas. It is without doubt one of the most significant themes of alchemical work. While it has adopted many different symbolic lights and shades, it is primarily used as a symbol of Time, or of an eternal self-sustainable whole. This sense of an eternal cyclical progression is very similar to that of the Möbius strip. Despite the connection can safely assume that the origin of this symbol was the imaginary implications of a creature flexible enough to eat its tail, so as to thus exist in perfect self-sufficiency.

This above image is widely circulated as an example of the Ouroboros, in fact it is sourced from the Leyden Papyrus, a fragmentary collection of alchemical documents dating to

14Ellison, Taylor R, An Enigmatic Book of the Netherworld From a Shrine of Tutankhamen, http://www.touregypt.net/featurestories/enigmatic.htm

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around 250 A.D15. It is suggested that this papyrus is among the earliest known evidence of Alchemical knowledge16. The words in the center of this illustration translate as Hen To Pan, ‘The One- The All’, or ‘One is all’17. Here again is the picture in its whole context. The page of illustrations is grandly entitled The Chrysopoeia of Cleopatra, which can alternatively be translated as The Gold Making of Cleopatra. The illustration shown is a very accurate hand copy of the original illustrations. The original document resides in the Greek Alchemy Collection of the Bibliotheque Nationale, Paris18.

If you did not see it before, hidden in the pattern of spots, there is a diagonal line from the inner edge of the ring to the outer. There is also a slightly curved line shorter line. The points of these internal lines meeting, with the inner edge, form a curving right-angled triangle. The right angled triangle was a sacred symbol of the Pythagorean movement. However it is doubtful that this is a mathematical geometric figure judging by the organic curvature of the lines. Given intricacy of the diagrams, it appears a deliberate inclusion. The simplest explanation is that this detail in the illustration was devised in order demonstrate that the image shown had the physical form of the Möbius strip.

A. B.

Figure A is a simple line drawing of a Möbius strip. As we will have to consider with the key topology- Ankh correspondence,

15Taylor, Frank Sherwood, The Alchemists: Founders of Modern Chemistry, Kessinger, 1992, p.5116 ibid17 Taylor, 5818 Marshall, Peter, The Philosopher’s Stone: A Quest for the Secrets of Alchemy, Pan Books, 2002, p.199

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the Möbius strip it is difficult to depict well in a two dimensional medium. If we did not already know what this picture was intended to represent, it might be difficult to see the three dimensional perspective of a Möbius strip in this simple line illustration. Figure B is also a simple line drawing of a Möbius strip. Topologically the figure is homeomorphic to the one above, but this loop is fabricated by a different method. While the topological features of this figure might still be a challenge to discern, it may seem more comprehensible than the first. The surface forms of these two figures are not identical. Figure A shows a round band with one small half twist. Figure B shows a flat band with one bend 180 degree bend like a rainbow, and another 180 degree in a third dimension, like the side of a pipe, which meets the other. Topologically, they are homeomorphic, both Möbius strips. As you can see, the outline of Figure B bears an effective resemblance to the Alexandrian illustration.

Pan Dragon19

The curious copper ring above has a similar shape to the picture in the Alexandrian document, were the picture of a Möbius strip. In contemplation this sculpture appears humble, as if it might have started out as a clothes pin. The design operates with a simple elegance in this respect.

Upon study, it appears that the anomalous diagonal line in the ancient illustration serves to indicate that the outer edge of the body continues as the inner edge of the tail. In this period, the artistic technique used to show depth through shading is not well developed, but the illustration is clear. The semantic nature

19Pan Dragon, Author

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of the Ouroboros, combined with the text reference and the form of the illustration, it appears that the Ouroboros in the Leyden Papyrus is in fact a 2nd century illustration of a Möbius strip. This could be taken as solid evidence of the Möbius strip existing sixteen centuries prior to Möbius. But it is now it is as it was then, alone; the one, the all.

‘The Chrysopoeia of Cleopatra’20

These drawings represent Alchemical information, a blend of ancient chemistry, metaphysics and mysticism. Luckily, enough practitioners of Alchemy survived to help it develop into the first sciences. The title of the page describes the figures as ‘Chrysopoeia of Cleopatra’. Chrysopoeia is a Greek word. The prefix chryso refers gold or golden, the word in full translates to something like ‘gold maker’20 Taylor, 52

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We cannot immediately guess that the title of this page referred specifically to the Ouroboros diagram as the Chrysopoeia, it may have simply implied be that the Ouroboros was part of the equipment or process. The picture in the bottom right is identified as a still, and the small drawings above the Ouroboros as alembic flasks; equipment used for what we now know as chemistry. Above the Ouroboros are two concentric circles surrounding the Alchemical notations for gold, silver, mercury. The text within the circles reads as follows: One is the serpent which has its poison according to two compositions. One is All and through it is All and by it is All and if you have not All, All is Nothing.21 This does appear to be a statement in a chemical sense, but rather a metaphysical, or perhaps even physical description. This paragraph of text would appear to refer to the Ouroboros symbol below it, the round dragon marked Hen To Pan, ‘One is All’, or ‘All is One’.

This figure is shaded half in black, and half in white with specks. There is an equal balance of light and dark. Some can interpret it to mean the concept of good and evil, or health and sickness. Some clearly have interpreted it to mean the transition of leaden metal to golden metal. It is likely a reference to the composition of the ‘poison’ of the serpent. It seems to function well as philosophical statement, provoking thought on a range of ideas which are relevant to the nature of binary opposition within a whole.

Chrysopoeia, which was later termed ‘Philosopher’s Stone’ is an object of legend. In myth, it is imbued with magical properties, most prominently the power to transform of base metal into gold, and the power to bestow an individual Life. Naturally a substance of these properties has not yet been discovered, nor is it likely to be. Often these days, people view the Philosopher’s Stone no longer as the prime elemental substance, but rather as an ultimate principle or ideal.

The creation of this substance is purported to be the highest goal of the practice of Alchemy. Perhaps even due to this document, the Ouroboros has been one of the most highly

21 Taylor, 58

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regarded of symbols to represent of the Philosopher’s Stone22.The philosopher Cleopatra was one of the few alchemists reputed to know the secret to it23. The Leyden Papyrus contains some of the earliest known documents of the western alchemical tradition24. The Leyden Papyrus suggests that the Möbius strip was at one time regarded as a central part of ‘Chrysopoeia’, the original philosopher’s stone.

The Möbius strip form is not the only explanation for the extra lines in the Chrysopoeia diagram. Purcell has already noted the anomaly. She writes: “When scrutinised closely, the Ouroboros becomes far more than a serpent biting its tail. As with the Codex Marcianus, and other’s that depict the dual black and white aspects, the Ouroboros clearly is devouring itself but it is devouring what has been turned inside out… These correspondences are often related to the fact that snakes shed their skin.25” This is a valid interpretation of what the illustrator could have intended the lines to portray, the shedding of the snake’s skin.

By way of another explanation, there is a motif of particular significance, relating to serpent themed bracelets and bangles produced by Roman and Hellenic jewelers in the late BCE early CE centuries. Sometimes, the head and tail of gold work snake bracelets had intricate artistic decorations and chasings. Instead of carrying the carved section the length of the bracelet, it tapers shortly after the head and the tail into a sensible polish, which then meets in the middle. This taper into the polished midsection takes the form a sharp V. This is a frequent motif in surviving snake bracelets. I have examined armbands of this kind retrieved from a site in ancient Pompeii2627. The design motif hints to a possible reason for the inclusion of the lines of the diagram, specifically that anomalous triangle may have

22 Marshall, Peter, The Philosopher’s Stone: A Quest for the Secrets of Alchemy, 2002, p.20023 Linden, p.44 24 Taylor, p.5125 Purcell, Melanie, What are the relationships between Infinity and Zero? The Implications of A Cyclic Universe and the Diagonally Woven Single Joined Thread Klein Bottle , http://www.towardsanewera.net/infinity_and_zero.htm, 1999, Accessed July 2009

26 Melbourne Museum Exhibition, ‘A Day in Pompeii’, 200927 For further examples see Trustees of the British Museum, Jewelry Through 7000 Years, British Museum Publications, 1976, Fig. 164, p. 114 and Coarelli, Filippo, Greek and Roman Jewelry, trans D Strong, Hamlyn Publishing, 1970, Fig. 59, p. 131

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simply been a fashion statement of the illustrator. The timeframe, the snake figure, and the triangular form all correlate. However, this explanation is made improbable by the fact that the position and appearance of the triangular shape in the Ouroboros illustration is almost completely un-like the position and appearance of the bracelet tapers.

The theory that the Chrysopoeia Ouroboros is an illustration of a Möbius strip could be supported by the description. The first line reads ‘One is the serpent which has its poison according to two compositions’. In the context of the theory, we can recall that every Möbius surface must have one of two possible compositions. This is to say that it must have either a clockwise or anti-clockwise half-twist, not both. This fact causes most Möbius surfaces to be asymmetrical along any straight line, although there are exceptions to this rule. It is difficult be certain of any interpretation made with design, especially of isolated passages of ancient text. In this case the interpretation is made of a very general statement.

The Chrysopoeia illustrations were not discovered with, but are linked to another manuscript, The Dialogue of Cleopatra and the Philosophers, through the author’s name, Cleopatra. This philosopher likely should not to be confused with the famous queen of the same name28. However these works are both dated between fifty and three hundred years after Queen Cleopatra VII (69-30 BC)29. Naturally it is possible both that the surviving manuscripts were copies of earlier documents. To be suspect, there would be a motive for many authors or schools of philosophy would wish to attribute their work to the name. Of course, it is just as likely that the manuscripts were composed by a philosopher who shared her name with the queen. The Dialogue, a work of poetic, natural philosophy, was discovered in the Grecian manuscript, The Book of Komarios, which dates to around the same time as the Chrysopoeia diagrams30. In this, Cleopatra’s philosophy is related by the philosophers as a description of ‘how the highest descends to the lowest, and how the lowest rises to the highest, and how that which is in the

28 Taylor, 5729 ibid30Linden, Stanton J, The Alchemy Reader: From Hermes Trimegistus to Isaac Newton, Cambridge University Press, 2003, p.44

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midst approaches the highest and is united to it’31 This is probably relates not to the Möbius strip, but to the interdependence of the substances which sustain life.

The alchemist Zosimos of Panopolis was living in Egypt while The Chrysopoeia of Cleopatra was in circulation, sometime around 300CE32. What follows is a passage from Zosimos’ On Virtue, Lesson One: Linden suggests that this section refers to a Chrysopoeia.

“And that I may not write many things to you, my friend, build a temple of one stone, like ceruse in appearance, like alabaster, like marble of Prosonnesus, having neither beginning nor end in its construction. Let it have within it a spring of pure water glittering like the sun. Notice on which side is the entry of the temple and, taking your sword in hand, so seek for the entry. For narrow is the place at which the temple opens. A serpent lies before the entry guarding the temple; seize him and sacrifice him. Skin him and, taking his flesh and bones, separate his parts; then reuniting the members with the bones at the entry of the temple, make of them a stepping stone, mount thereon, and enter. You will find there what you seek.”33

This passage is significant for a few reasons. It confirms the fact that the early Egyptian alchemists knew of the Chrysopoeia as a physical thing, rather than a process. Ceruse is white lead, or possibly lead carbonate, the mineral ore of this substance. It is a very soft mineral. Alabaster and Marble of Prosennesus were both highly valued materials in the production of stone carving in ancient Europe. Alabaster made for a particularly beautiful medium, for its color and its translucent quality. The fact that these materials were mentioned by Zosimos suggests that what he knew of as the Chrysopoeia was in fact a hand-carved stone, rather than a naturally occurring substance.

Zosimos describes the temple as ‘having neither beginning nor end in it’s construction. This idea relates particularly well to the Möbius strip form.

31 Linden, p.4532 Linden, p.5033 Linden, p.52

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Finally, the ‘temple’ is described as having an opening on only one side. The edge of a ring shaped Möbius strip resting flat on flat surface must always, by necessity of form, lift up from the surface in part of the section. When water is poured into the middle of the ring, it will flow out from this lifted part only. It could be this feature of the topology that Zosimos is referring to as the side with the entry.

A carving of a Möbius strip, perhaps carved from alabaster or quartz crystal, might certainly appear mystical to a person from ancient and medieval times, perhaps sufficiently so for them to attribute to it mystical, spiritual, or philosophical properties. The evidence of the diagram in the Chrysopoeia of Cleopatra clearly opens the possibility that the mythical Philosopher’s Stone may have had a real historical basis in the form of the Möbius strip. This theory is supported by the description of Zosimos.

In contemplation of the form of a Möbius strip like the one in the Chrysopoeia illustration, two shapes become immediate to the viewer. One of these is of course the Snake. The other, the presence of which is less apparent from the photograph perspective, is the form of an Egg, nestled within the serpent. The mythologies of ancient times spoke often of an egg and a snake, as the source of the creation of this universe. It is an excellent if unconfirmed explanation of how this myth originated.

Investigating the origin of the idea of the Ouroboros, we know that many ancient people believed in giant beings which in some way or another appear as a giant flying serpent, which we term as dragon. This idea might have come from the discovery of massive reptilian bones in the rock surface of mountains, cliffs, or quarry sites. Rare and magical seeming things, such as mummies, tigers, narwhal horns, albinos, mercury, and so forth, are the prime ingredients for ‘magical’ medicinal concoctions, potions which are spruiked to give whatever beneficial properties a wealthy buyer might ask. It is plausible that dinosaur fossils, or ‘Dragon Bones’, were once such a commodity. If this was the case, it would seem then that they were utterly consumed, probably in the production of ‘magic’ elixirs. This is an empirically baseless theory; no positive evidence could really remain to be discovered; however it is at least true that at least some

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primitive, ancient, and modern people are known to practice belief in potion-based shamanism.

The idea that a sculpture in the shape of the Möbius strip can bestow restorative properties is probably sufficient to account for the non-existence of any sculptures dating prior to the 1800s. But the fact that almost all Alchemical ‘recipes’ which call for a philosopher’s stone state that it must be completely incinerated to be effective would make the extinction of the form seem absolutely inevitable.

Ankh, 1200BC34

(centre figure)

34 The Book of the Dead: The Papyrus of Ani, trans E. A Wallis Budge, Dover Publications, NY, 1967

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The Ankh.Egypt contains some of the best preserved examples of sophisticated ancient art forms. In wall carvings, sculpture, painting and script, the ankh pictograph is shown to be a common and revered symbol in Ancient Egyptian religious cultural practice. This meaning of this symbol is ‘Life’35. It is surprisingly often under-referenced in academic and historical works.

During its adoption into the Latin speaking world, an interesting and eventful journey, it was described as Crux Ansata, ‘cross with handle’36. The Ankh is depicted in ancient art as a droplet loop upon a angular shaft extending from its lower terminus, with a horizontal bar in at the intersection of the loop point and the shaft. It was used freely as a hieroglyph- a pictographic word symbol; and also as an abstract symbolic figure. Paintings, stone carved statuettes and bas reliefs, papyrus, and evidence of this kind often show the Ankh as being carried, loop in hand, by Royalty and Gods. It was a common design motif for jewelry, amulets, and other worked crafts. We do not know with any real certainty when the invention and assignment of this symbol took place, or where, by whom, or from what source.

Through experimentation and thoughtful consideration, it is evident that the ring method described above is not the only way to create a single sided two dimensional surface. While there is a number of variations on forms which show a single flat surface, some methods, such as the one detailed above, are simpler than others, and so these stand out. Another of these methods is as simple as the first, if not simpler. Beginning again with a rectangle of paper or semi flexible material, split the length to the midway into two even sections so as to make a Y shape. Join the two small ends to make a loop; it should finish with a shape roughly similar to a key.

35 Cox, Simon and Davies, Susan, ‘Ankh’, in An A to Z of Ancient Egypt, Mainstream Publishing, 2007, pp.29-3136 ibid

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How to make a Möbius strip: Method 2

This Möbius key method creates a different topological configuration from the original. While ring strip has a single continuous surface and a single continuous edge, this form shows a single continuous surface with a terminating edge. This surface is not to be confused with an Anorankh, which is an Ankh wearing an anorak37.

A loop on a shaft is a very general description, but human history, defined by the signifiers we construct, records some examples of this thing. By the specification of the key topology, we must be seeking a droplet shaped loop, pointing into the shaft. It is for this reason we consider the Ankh.

We know it to be certain that it was possible to construct the key topology before even the first civilization of Ancient Egyptian. Construction requires no more sophisticated technology than flattened plant fiber, like a reed stem, and some simple means of binding, such as string or glue. On the river bank of the Nile grows a sturdy reed plant, papyrus, which could easily have served this purpose.

Unlike the Möbius strip method, the key construction spreads the ‘twist’ over the simple bend of the loop, and so unlike the Möbius strip, it does not place a twisting force on the material of the construct. What this means is that the key topology is simpler to produce from semi flexible materials than the Möbius strip, which can break from the sheer force in the torsion. What

37 http://www.lspace.org/fandom/afp/timelines/holy-anorankh.html Accessed July 09.

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we must put to question here is the possibility that the key topology was the origin of the Ankh symbol. But four thousand years down the line, the question is not whether it was possible to suppose, but whether it is at all plausible.

Key topology with cross-bar

It is clear that there is no available historical evidence which specifically relates the Ankh symbol to the Möbius topology. The fact is that it is clear that all surviving historic examples of the Ankh which I have witnessed, the paintings, carvings, amulets, sculptures, and written illustrations, show specifically that the physical Ankh was not a single-surface topology.

In all historical examples the loop section of the Ankh, when visible, is shown to be spread solid and flat over the curve like, a rainbow. Sculpted three dimensional versions of the Ankh are also shown the same way, flat and solid, making a straight line from the side view perspective. The cross section is usually square or rectangular, not thin and flat. Therefore it would seem that in this way all real, tangible evidence actually stands against this theory of origin.

The fact that the key topology differs significantly from the Ankh in another way does not discredit the theory so much as it might credit it. The most obvious variation between the two is the absence of the horizontal cross line, which appears in the Ankh structure but is not an immediate part of the key topology.

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It follows that if the hypothetical pioneer of the key topology wanted to draw or write the shape, but was unable to portray depth, it would have seemed to be impossible to display the specific qualities of a single recurrent surface. The next best means to indicate the significant topological features of the loop would have been by the suggestion of a straight bar passing through it. By the natural design of the topology, any such bar would easily come to rest in the cradle formation of the middle split in the form. If this was the case, it would be confirmed by the position of the crossbar on the Ankh symbol. This disparity is explained if we make the secondary hypothesis that the key topology was discovered at a time when the technical ability to represent depth perspective in two dimensional modes of art, such as drawing or painting, was not yet sufficient to accurately portray the figure. Evidence shows that this skill developed slowly and incrementally over many great spans of history for all races on all continents, and that even arguably the most advanced painters and carvers of ancient times, the Egyptians, had difficulty showing perspective on a two dimensional medium. For instance, the conventional Egyptian method of in portraying the human body was to show head and legs in side profile, while the torso and shoulders were shown facing forward, so as to imitate depth by splitting the difference.

The musical instrument, the sistrum, was an important artifact in Ancient Egyptian culture, and was often depicted in carved and painted scenes of daily and ceremonial life. The topology of the upper third of this instrument is almost identical to the upper third section of the key topology. It is usually drawn from a front facing perspective. If there can be found any instance of sistra depicted in ancient Egyptian two dimensional art with the depth perspective, this would prove that Egyptian artists were capable of drawing the key topology from a depth perspective.

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Late period Sistrum Nefertari with Sistrum

We must of course remember that the ancient Egyptians were not only capable of drawing and painting, but also highly skilled in the art of wood, stone, metal and clay sculpture. If an artisan of the Ancient Egyptian civilizations had the plans and the will to create a three dimensional sculpture of the key topology, it would have been a small challenge to find some durable method of doing so.

Illustrations of the Ankh show that the pseudonym, Crux Ansata, is a fair title. This name means in Latin ‘cross with a handle’, and probably refers specifically to the fact that paintings of royal or deity figures often bear the Ankh, carrying it in hand by the loop section, using the loop form as a natural handle. All this really tells us is that the loop was thought of as open, like a hoop; rather than closed, like a looking glass. The Ankh is often also depicted held by the stem section, usually when the figure is in a seated position, or when using the Ankh to perform a ceremony. If we let our imagination drift, this artistic preference to hide the loop might give some rise to the speculation that these figures were intentionally concealing a significant section of the Ankh. Whether or not this sounds far-fetched, it brings our mind to the

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fact that most religious institutions can be often secretive and esoteric, withholding special knowledge from the public and lower hierarchies.

Censorship was and often still is a common practice within theocracies, perhaps more so than in secular societies. For the Ancient Egyptians, there was often little or no distinction made between priests and the governing powers. The reasoning here is that if the key topology was the source of the Ankh symbol, even if remembrance of this origin was retained or re-discovered, it is not at all inconceivable that the true meaning was deliberately withheld or suppressed from publication by an authoritative body, by motive of religious or epistemological dominance, and this could have occurred at any time in the long theologically turbulent history of this geographical region.

Horus with Crux Ansata Gifting of Life

Luxor Temple Hieroglyphs Funerary Goods

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If we still persist under the hypothesis that the key topology was the source of the Ankh symbol, perhaps the only explanation left is the Ancient Egyptians portrayed it without single sided characteristics because they themselves did copy the topology, without knowing of it.

As incredible as this does sound, it must logically be true that all preserved ancient illustrations and Ankh artifacts must have been produced some time after the invention of the symbolic figure, and therefore cannot define with certainty the original nature of the Ankh. All existing productions of the Ankh are in fact reproductions, imitations, of the first original source. As the circumstances of this origin are concealed, we cannot say how long the intervening period between the invention of the symbol and association with meaning, and the production of the existing representations was; nor can we say what occurred during this period. It may be that the origin dates to a far earlier people, and it is possible that the appearance of the Ankh has changed in the past, in the many generations. But if all we have to reach for are straws, then we may need something more to stay afloat.

There have been many explanations already put forward as to why the Ancient Egyptians used this specific pictogram to represent the word and concept of Life.

One early modern explanation is that the meaning is actually phonetic. The explanation stated that a picture of a ‘sandal-strap’ represents the homophone Ankh to the word for sandal strap. Homophonic substitution is a common method of representing meaning in the hieroglyphic method of recording words. But even if the correlation between the visual sign and the phonetic sign in relation to the sandal does imply a link in origin, it does not suggest the direction of correspondence. This is to say, while it is possible that originally the link was made between the words references as claimed, it is also possible that the link was made between the precedent icon of Life and the later image of the sandal, leading to the sound of the symbol being adopted the phonetic reference of sandal strap.

A fair theory is one that the Ankh is a picture of a knot or loop, as of a belt or sash, which was held in sacred regard, perhaps

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the garment of an ordained priestess. This is a popular belief of those who regard Isis.

There has been forwarded a theory which explains the Ankh as a pictorial representation of the thoracic vertebrae of a bulls spine. This theory is supported by the Egyptian symbiosis with cattle. If so, the Egyptians would not be the only early civilization to sanctify the image of this species. For a cattle farmer, the natural place to look for the source of life is just above a bull’s balls. This is further confirmed by the trend of belief in ancient Egypt that the spine was the source of virility38.

Given that the simplest is usually the probable explanation, then it is highly likely the pictogram is a composite picture of man and woman merging in the act of reproduction, an act of love, the source of Life, a simple naturally inspired symbol. But perhaps even more visibly, it represents the stem, leaves and blossom of a plant.Upside down, the symbol resembles the image of a drop of water on the verge of falling.

It a fact unchallenged by any known evidence that water is a component substance all Living beings. It is known that the Egyptians had an appropriate respect for the role of water in the substance of Life. The pulmonary artery of human life in the Ancient Egyptian civilization was the flow of a vast River System. The Nile could be described with fair accuracy as the source of life for the civilization of Egypt. The Nile River was the reason why some of the first agriculturalists managed to sustain a sufficient harvest to support an ancient urban civilization, just as our use of water today dictates the construction of our cast technological civilization. More importantly, the flow of water in this Earth’s rocky atmosphere supports the great thriving wilderness of all Life known to exist in this universe.

Perhaps for an Egyptian or pre-Egyptian priest or farmer, the key topology would appear to chart the flood of the Nile waters, in and out the upper surface of the stem, to then flow and recede over the lower side, perhaps symbolic of the reflection of the

38 Schwabe, C. W, Adams, J, and Hodge, C T, Egyptian Beliefs about the Bull's Spine: An Anatomical Origin for Ankh, Anthropological Linguistics, Vol. 24, No. 4, 1982, pp. 445-479.

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underworld, the unseen world of where it all goes, only to then return above again in still another cycle.

This idea of the loop procession extends to other steady cycles, solar, lunar, planetary and seasonal. The key topology also can be perceived as a standing human figure, a seed, a tree, and an open eye.

There are many more theories, but as many as we list, the theories of the origin of Ankh symbolism are not in the slightest way mutually incompatible. It is possible in this case even that multiple points of origin were in fact necessary. A single origin may not have been sufficient, and that only because of synchronized expression of this symbol in important aspects of daily life that it was adopted for such an esteemed use. These individual explanations are only made true on the grounds of a connection, strong or tenuous, to the idea of Life in ancient Egyptian society, but together.

To fit with the evidence, if the key topology was related to the origin, then the only remaining possibility is that the figure was discovered and partially forgotten in the period between its original introduction as a written symbol, and the later period responsible for development of significant technology of art. A key topology drawn in the mud will look like an Ankh, without perspective. It is certainly possible, but not easily plausable, as it requires huge assumptions in place of evidence.

So what would warrant us to persist in suggesting that this original source was the key topology, given that the historical evidence goes against it? The answer to this is in the meaning which was given to the Ankh, ‘Life’.

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Enso, ‘’39

Life in essence extends beyond the contingencies of thought, because it is an embodiment, a physical thing. If it was not a real thing, it would not exist: meaning in turn that thought could not exist. For us to seek the meaning of life, however, is a sure assurance of Life’s existence. Our purpose here is to understand the nature of Life in relation to the idea of a cycle, which the Möbius strip can appear to represent.

The circle is prominent in almost all human religions, of life through time. The concept of a continuous progressive cycle lends itself well to the idea of Life. This idea of continuity of being formed in a cycle is common in the expression of Eastern philosophy. Given that the substance of the world is change itself, the only way for form to retain a steady state of being is to change its future state into a something similar to what its past state was, which is to say it must repeat itself in a regular fashion.

Life, the substance, is something which retains a semblance of constituent semi-stable being in form, while maintaining a state of flowing liquidity. A simple loop pattern is one conceptual method to suppose a four dimensional Cycle of movement. For a being to sustain longevity within a consistency in form, under the condition that the form of the substance is bound to change, we can see that it could achieve this by producing a four 39 http://www.absoluteastronomy.com/topics/Enso in July 2009

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dimensional loop. This Universe, ours, us, it is four dimensional; which is to say that it is a composite of the three dimensions of Space in Time. The cycle, wheel, is one of the most predominant images of time. This may be because fluid composite bodies, such as our own, to retain our form, our self, during constant change, must in some way exchange the composition of it’s substance, with minimal loss in form. We can imagine, provided we don’t look, that this goes very much like a cycle.

Finally, the cycle alone is an excellent metaphor of the being of Life. To put it another way, for form to remain in being, to exist relatively unchanged over time, form in the end must meet form in beginning. What sustains a body must be similar later to what it once was, in order to achieve constancy in the bath of eternal change.

For the seasons to sustain a constant temperature, pressure, humidity, and molecular composition of the surface of this planet, there must be in effect a constant arrangement of being in form. We humans are changing the form of this arrangement. We may well fear that if we change too much, too fast, too differently, our actions, our creation, will cause the cycle of constant being to alter, and perhaps cease entirely. We know that if instead of the slow, ordered change, our bodies somehow contact rapid, powerful, differences in form; the cycles which sustain our lives fail. Rapid forceful change is physically and metaphysically injurious, destructive, and possibly deadly to the cycles which sustain our living form. Slow change over time can be beneficial; as demonstrated by evolution to greater harmony with the world and its beings.

Conclusion.Artistically, the form of a simple loop of any dimensions can be made in many different ways, to suit many different forms of the mind, designed, even subliminally, to forms chosen of the semi-infinite possibilities of imagination and free will. It could be that this is true, and when combined that the physical structure the loop may always generally resemble the ideas of Life, the Infinite and Eternal, the correspondence in forms can be explained with a real causal link which does no relate to historical circumstance. Therefore, the validity of the historical link is balanced against the possibility that evidence of the theories are

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made, ad hoc fiction, and the evidence is intentionally designed to fit the facts. If this is the case, much of this essay is simply pseudoscientific fallacy. But if it is a fiction, it may be even more profound, made so by the fact that it was written by both humanity and the universe.

There are very few simple Möbius configurations known even in this age. Diverse numbers of viable Möbius-like forms are not known. This is because they must fit within certain allowances and disallowances of natural existence. The laws of the universe, symmetry, topology, severely limit the surfaces available, as with all forms. The form of each topology is in part created, limited by mind, and in part allowed, limited by Nature.

The final accounting recalls the trilemma that some, all, or none of the aforementioned topological sculptures could have provided creative inspiration for the historical origin of the ancient artworks. What makes the first two possibilities (that some or all of the forms helped cause the symbolic trends) so incredible is that there are no historical records. It may be less incredible that forms of single surface forms echo in synchronously, causa sui, with symbols of Life, the Infinite, and the Eternal.

Discoveries can occur at many times, though often it happens simultaneously, such as in the case of Möbius and Listing. Perhaps we alive today are the growing edge of mankind’s history, knowledge building by accretion to a point when knew information can be observed. The Möbius strip does not seem to fit this theory, as it is a simple geometrical form, like the sphere, square, triangle and line. Seemingly, these were known almost from the first times of humanity. The Möbius strip differs greatly from these figures in that it is semi-artificial. This world does not seem to contain the Möbius strip without the assistance of human invention.

With the current evidence, it is not possible to state it true or false that any of the logographs mentioned correspond in origin to the Möbius topology, though positive evidence indicates that it is possible, and the (almost) infinitely vast and (almost) infinitely miniscule absence of evidence in these cases weighs still against this possibility. It is only a feeling, but it seems at

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least to me that the absence of such a simple form in the vast span of human creation, is amazing enough to want to indicate something in its absence, some unknown truth, either about history, or about humanity, if even we know not what. However absence is more unfathomable than the pure unknown. The unknown at least meets knowledge in the source of both, the natural material world.

This essay has shown that there is definite theoretical evidence in all cases, Lemniscates, Pelican, Ankh and Ouroboros, have the form of a visual and symbolic impression which to varying degrees silhouettes the figure of the Möbius topology:

It is entirely certain that John Wallis made a profession in part out many small strips of paper, and it is he who applied the silhouette ∞ to infinity. However, as far as I can tell, this intensely intelligent, pious cryptographer and mathematician, a founder of the Royal Society, did not make any explicit note on the Möbius strip.

Despite the wide knowledge of the Pelican myth, culminating indeed even its consecration by the Christian church, again not a single artifact referring precisely to Klein bottle topology has been recorded in official documents prior to Felix Klein. However we know that a Klein ‘bottle’ structure can be made from blown glass, like the alembic flask of the medieval scientist. The correlation between the pelican and the Klein form remains.

The above three examples offer, in differing degrees, possible clues. However, in all cases these inferences must speculate on possibilities which are not in sight. None are confirmed by specific examples of knowledge of single surface topology of the kinds we understand. To be fair, for the period which we are discussing, few people actually knew how to write. But when we speak of truth, imagination counts as poor evidence for anything outside of our minds.

The Chrysopoeia Ouroboros, accompanied by a reference to singularity and wholeness- ‘One - All’, is highly likely to be an actual illustration of the Möbius topology. As the manuscript at the source to this image dates to around 200 AD, it makes for

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clear first hand evidence that the Möbius strip was known prior to 1858, a difference of at least 1650 years.

We know the idea of a four dimensional loop, a cycle, applies to the nature of Life. The preceding metaphysical discourse may not demonstrate that this example of life as a loop was believed or known of by the Ancient Egyptians. However in this regard we do have some knowledge of their religious worship of the sun, moon, and seasons. If you follow it with your eye, the key topology, like the Möbius strip, suggests the conceptual picture of a perpetual flowing of a cycle, running over and under the loop, and up and down the stem. This imagined pattern of flow suggestively resembles the annual flood of the Nile, from a primitive perspective, among other images of Life. Only, for the long history of archeological interest in the remains of the great ancient civilization of the Nile, it must be said that not a single artifact, of stone, metal, wood, paint or ink, specifically resembling a Möbius topology has been discovered, even in tombs which are opened only recently. This suggests that the peculiar shape of the key topology could represent an a priori figure which sits with these symbols of Life only by the hand of fate.

We can no longer say for sure that the Möbius strip was invented in 1858. If the Möbius strip did exist as an artifact in ancient times, we must still provide an explanation as to why little or no evidence of it remains. So far, we are not entirely without possible explanation for this non-occurrence.

Considering the case as it may have been in ancient or medieval times, the Möbius strip would have appeared to synchronize with some of the major symbols of world. We can clearly see that given the correct context, the Möbius strip may have been sufficiently remarkable to people that it could become a sacred object, an icon of a communal idea. However if this was ever the situation, we know from experience that a combined theocratic and military occupation of whatever persuasion is rarely peaceful and enlightened; rather it is often a vicious and unforgiving battle for power over human actions, beliefs, wealth, and social destination. We have grown much in religious tolerance and compassion since ancient days, yet religious persecution still occurs widely around the globe. Complete

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erasure of the icons of an opposing religious movement is often a goal during times of conquest, expansion, and conversion.

With biblical regard for the Ouroboros, the dragon, the serpent, as it is, it is possible that a Christian would regard the Möbius strip to be possibly tainted by evil spirits. This would be due to the stigma of being associated with the Devil Dragon, Satan. This spirit figure of ill repute was regarded as the source of evil or the ordained opponent of God in Christian mythology. This figure is associated with the icon of the serpent. There are many points of reference to connect this superstition, and the effect of superstition and suspicion in a harsh time can be dangerous. For a period, the standard consequence of the even the suggestion of association with the Devil was cremation of the living body. This makes it very unlikely that a person of this time and region would publicize or tolerate the publication of the idol-like serpentine figure, especially not by any practitioner of a secret and mystical occult philosophical metallurgic chemistry.

Our last possible reason as to why the Möbius strip could have fallen from existence is that images of it could have been regarded as lapis philosophorum: a legendary magical substance prized by the scientists and philosophers of the long and ancient school of Alchemy. The idea of the Chrysopoeia is said to have been the central, defining concept of Alchemy, and the Ouroboros in the Leyden papyrus is the earliest known illustration of the Chrysopoeia. The evidence from the Cleopatra illustrations and from the writing of Zosimos strongly suggests that the Chrysopoeia of early Egyptian alchemy took the form we know as the Möbius strip.

It should not be immediately assumed that the Möbius strip form was known of by anybody other than the by the author of the Chrysopoeia of Cleopatra, and the Zosimos of Panopolis.

However, if we make the conclusion that later alchemists, informed of by the Dialogue of Cleopatra, really did possess in secret the knowledge of the stone, then we assume that it was in the hands of human beings, who always act in their own perceived interests. If it was the belief that ancient relics, perhaps gold or rock crystal, in the form of a Möbius strip were of a substance capable of creating gold or prolonging their life,

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they would have been highly sought after, and the competition to get them to somehow turn lead to gold, fierce. It is not entirely unthinkable that they could have destroyed them all in fire, which actually is the clear instruction of many surviving Alchemical documents. Successive generations of the experimental application of heat, force, and corrosive substances will have a detrimental effect on anything as fragile as stone, and the number of forms on our planet is finite.

Memory and thought fly as ephemeral as shadowed sparks across the tapestry of the ages. If we in ignorance, greed, apathy, and consumption, happen to shift the cycles which bring the conditions of Life on this planet; the air, the water, the sun, our brother and sister animals, and the plants which eat light; all things which are a true meaning of Life, then we will fail. Destroying the forms which sustain this, be forcing them open to dissipation, in avarice, is the price of our creation of new things. Our power is our responsibility. Intelligence, hope, love, and courage have proved time and time again in the preservation of lives. Our purpose in Life, if we can be said to have any, is guard in preservation of the clean water, air, temperature, soil, and of the myriad beautiful forms of Life which cultivate, groom, and sustain it all over time.

The life of earth has evolved for millennia without the existence of our artificial creations. We alter the fabric of the universe to make new forms out of our environment in a massive, massive way. We beings cannot know what will become of creations as they fallow their own paths through the time ahead of us. We learn so slowly, while incredible change rushes from us at an increasing pace. But we must somehow remember unimaginable time span of the survival of life on earth it required for us reach to this point now.

If we change the conditions of earth today away from the conditions of this ancient multi-millenial history, in the shuddering span of two or three hundred years, it could act as a grievous blow to the constitution of the cycles which life has evolved against for a long, long time. We must now decide whether we will choose actions to heal the conditions of Life on Earth which we and those who came before us have unwittingly

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laid to waste with our poison, or whether we will allow the fangs of humanity to wholly consume the garden of Life.

I hope that we, with compassion extend our power of creation and destruction to heal our greater self, Life, Earth. By breaking our addiction with death, and allowing our ingenuity, and true self-harmony, not just with our own minds but all with all things, we may yet return our potential for creation and destruction to move with the harmony it once had with Life. The death of our amazing species is both a tiny sadness and the foregone conclusion, against the irrevocable loss of this beauty of the free existence of Life. Always our own death is given to us by the universe. If we cannot cease to harm the cycles of Life which we have broken, then our species, for all its sapiens brilliance, our self expression, our morality, our love, we should pray that we as we are forever cease to be, a limb torn from the tree of evolution, bound into the eternal extinction which we ourselves may have already condemned to so many ancient noble beings.

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Coarelli, Filippo, Greek and Roman Jewelry, trans D Strong, Hamlyn Publishing, 1970,

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Möbius, August Ferdinand, Gessammelte Werke. Herausgegeben auf Veranlassung der Königlich sächsischen Gesellschaft der Wissenschaften, edited Richard Baltzer, Felix Klein, and Wilhelm Scheibner, 4 vols, Leipzig

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Pickover, Cliff A, The Möbius Strip: Dr. August Mobius’s Marvelous Band in Mathematics, Games, Literature, Technology, and Cosmology, Thunder’s Mouth Press, New York, 2006

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Purcell, Melanie, Towards A New E.R.A.: Epistemological Resolution Analysis By Through and From Klein-Bottle Wholeness and Transdisciplinary Education, http://www.towardsanewera.net/, 2006 Accessed July 2009

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Klein bottle courtesy of Acme Klein Bottles

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