ANCIENT OMPHALOS AT DELPHI: GEOMETRICALLY ASPACE-INVERTING ANAMORPHOSCOPE*

P. W. KUCHEL

School of Molecular Bioscience and Centre for Mathematical Biology,University of Sydney, NSW 2006, Australia

Whether the artisan who made the Omphalos at Delphi over 2500 years ago recognized theoptical transform properties of its shape or not, its geometrical features are nevertheless thoseof a space-inverting anamorphoscopic mirror. Specifically, it is similar in shape to a circle-inverting anamorphoscope. We speculate that select members of the ancient religious sect atthe Temple of Apollo at Delphi realized the symbolism of inverting all of space outside theOmphalos into the image field inside its base, thus making it the virtual centre of the World.

KEYWORDS: ANCIENT MATHEMATICS, ANCIENT MIRRORS, ANCIENT MYSTICISM,OPTICAL TRANSFORMS, OMPHALOI, PYTHOGORAS

INTRODUCTION

Delphi

Delphi was the most famous religious site in ancient Greece, and it reached the zenith of itscultural influence in the fifth century bc. The Temple of Apollo was where the Pythia issued theiroracles to the poor, and to the rich and famous. The pharmacological/geological basis of thetrance-like state that the Pythia entered when ‘communing with Apollo’ to generate her oracleshas been the topic of recent analytical archaeology (de Boer et al. 2001; Spiller et al. 2002; Haleet al. 2003); and an account of the scientific quest for the answer to the question of the effectorsubstances that induced her trance-like state is the topic of a recent, engaging book (Broad 2006).

Associated with the Temple of Apollo, and the Pythia, were many rituals and objects used intheir various practices. While there are few written insights into the proceedings of the mysticalcult that ran the affairs of the temple, one object that is well described in ancient texts is theOmphalos. Actually, there were various Omphaloi produced at different times (see below); butthe one that is the topic of this paper is the biggest of those that have survived, and most likelybecause of its plainness, it has attracted little attention or interest from the archaeologist and thetourist alike.

Omphalos shape

The main ancient Omphalos at Delphi is now positioned outside, next to the Athenian Treasury,at this very popular archaeological site. It is an unadorned, smooth, bullet-shaped (conoidal)marble object, with a circular base of diameter 77 cm and a height of 72 cm (Fig. 1). Accordingto ancient legend, it marked the centre of the World (Flaceliere 1965). Close inspection of theobject (Fig. 1) reveals that the top has been cracked off, and yet curvilinear extrapolation of the

*Received 25 October 2009; accepted 13 January 2010

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© University of Oxford, 2010

edges of the digital image reveals a conoidal top with an included angle of ~90°, and thus a totaloriginal height that would have been approximately the same as the diameter of the base (Fig. 1).

According to Levi (1971) in his translation of the second century ad, Pausanias guide to Greece,a similar conoid ‘ . . . is a dismal object, and there are some suitably gloomy photographs of it inFouilles de Delphes (La Terasse du temple, 1927, pp. 73–7)’. Notwithstanding the dismissal of aconoidal Omphalos by this commentator as uninteresting, our aim here is to present an interpre-tation of the shape that gives it much greater meaning and hence ‘beauty’: it is contended that thisbeauty was based on ancient mathematical reasoning that, in turn, was used to surround the objectin mystery for those not versed in mathematics, or those mystified by mathematics.

Spatial and circle inversion

If an object like the Omphalos is made with a smooth reflecting surface, then it serves as anoptical transforming device known as a space-inverting anamorphoscope (Kuchel 1979, 2004).When such a device is placed on a flat plane and viewed from directly above its apex, it gives rise

Figure 1 A side view of the conoidal Omphalos at Delphi. The photograph, which was taken on 29 August 2009, showsthe Omphalos on a modern concrete plinth. The ballpoint pen at the base is 14 cm long, indicating a base diameter of77 cm and a height of 72 cm for the Omphalos. The solid lines drawn on each edge of the image of the conoid wereextrapolated empirically to fill in the apex that is missing, giving a predicted apex that has an included angle that is ~90°.The inset shows an oblique view from above the centre of the Omphalos, and more details of the concrete plinth.

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to a perceived image that lies within the confines of the circular base (see Figs 2 and 3). A curiousoptical property is that points of graphical objects on the plane near the outside edge of the conoidlie near the edge of the base in the image field and are transformed in size and/or shape; and thefurther away from the base of the conoid, the closer is the corresponding image to the centre ofthe image field (Kuchel 1979, 2004; Hunt et al. 2000; Cargo et al. 2002; Sharp et al. 2011).

In theory, in the limit of an object being an infinite distance away, the image appears at thecentre of the base. In other words, the whole of the external ‘world’ is inverted to the inside of theimage field by this optical transformation. In effect, the base of the conoid becomes the virtualcentre of the (planar) universe, no matter where it is placed on the plane.

ANALYSIS

Straight-sided cones have been used in anamorphic (Greek, ana, again; morph, form) art for atleast four centuries (Rijksmuseum 1975). On the other hand, circle-inverting anamorphoscopesare a newer invention and are a special case of space-inverting anamorphoscopes (Kuchel 1979,2004; Hunt et al. 2000; Cargo et al. 2002; Sharp et al. 2011). What is being proposed here is thatknowledge (and practice) of this general form of art is much older than previously thought. The

Infinity

Mediumdistance

Near

A

B

CC’B’A’

Figure 2 A depiction of a circle-inverting anamorphoscope. A circle in the object field is transformed to a circle in thevirtual, image field (inside the base of the conoid); all points at infinity (light rays parallel to the plane on which theconoid stands) are inverted to the centre of the base of the conoid.

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(a)

(b)

Figure 3 A scale model of the Omphalos, and an illustration of some of its properties as a space-inverting device. (a)An aluminium conoid of base diameter 70 mm and height 70 mm, thus making it a ~1/10 scale model. (b) The aluminiumconoid viewed from directly above its apex, surrounded by red, green and blue discs in the base-plane. (c) An oblique viewof the aluminium conoid. (d) A view directly above a previously described brass conoid (Kuchel 1979) that rests on aplane containing an inverted image of the Union Jack. [This figure is available in colour in the online version of thisarticle.]

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(d)

(c)

Figure 3 Continued.

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circle-inverting anamorphoscope has the special property that a circle in the object field istransformed to a circle in the image field by the conoid (Kuchel 1979, 2004; Hunt et al. 2000;Cargo et al. 2002; Sharp et al. 2011). A key feature of all total-plane inverters, whether circleinverters or more general non-preservers of circles (apart from the shape of their generatingcurve), is the solid angle of 90° at the apex: this implies that the tangent plane at the apex reflectslight from infinity on the basal plane directly up to the viewer, whose eye is centred above theapex (Figs 2 and 3).

FILM MIRRORS

It is implied in the classical literature that Omphaloi at Delphi, and elsewhere, might have beenanointed with water or oil (Flaceliere 1965). Since the Delphian Omphalos is made from hardmarble, it could have retained a high polish with high specular reflectivity; and under theappropriate conditions of lighting the anointing could have made it into an effective mirror.Hence if it were viewed from directly above the apex, the effect of spatial inversion could havebeen apparent.

The idea of using thin films of oil or water to form a mirror has recently been proposed as theraison d’être of low-sided terracotta ‘frying pans’ from the Aegean region in the Early BronzeAge (Papathanassoglou and Georgouli 2007). Thus there appear to have been precedents for theuse of liquid films on surfaces to make mirrors; so perhaps this applied to the Delphian Omphalosas well. However, the persistence of a film of liquid on a sloping marble surface might have beena problem, making it less likely to have been water (unless it was frozen, which could have beenpossible in winter), but rather something with more of the consistency of a wax.

It is also conceivable that the Omphalos was covered in gold, silver or bronze leaf or sheet, tomake it a reflector. In the case of bronze, this is well known to have been used in ancient mirrors.The use of such metals is a testable hypothesis, as fragments of the metals and various possibleadhesives, such as resins, might be detected in the cracks and crevices of the marble. Also, ancientorganic matter such as oil could be detected by using modern analytical methods of chemistry.

OTHER OMPHALOI

At Delphi there is another much more elaborately carved Omphalos that is on display in themuseum on the site. It is labelled as being from the Roman period, dated after the Roman invasionof Delphi in 168 bc. Obviously, all the carved elaborations on this newer object would render ituseless as a reflective spatial inverter; in addition, the fact that the included angle at the apex isnot 90°, but a plateau, rules it out as a simple spatial inverter. The flat top of this Omphalosallegedly once held a small statue of two eagles (Flaceliere 1965); this was in reference to the twoeagles that myth states that Zeus released, and whose flight-paths crossed over at Delphi, thusdefining the centre of the World (Flaceliere 1965). So, we propose that the true significance of theshape of the ancient Omphalos was already lost by the time of the Roman invasion; or the seerswished to keep the significance of the shape a secret, and then without a written record itsubsequently became lost to history.

ANALOGIES AND MATHEMATICAL MYSTICISM

The putative loss of insight regarding the ancient Omphalos is reminiscent of the situationsurrounding the mechanical Antikythera device. It is a marvel of engineering ingenuity, with ~32

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gear wheels and mechanical representations of cycles of the motion of the moon and stars; it couldpredict dates of solar and lunar eclipses (Wright 2007). But prototypes of it have not been found.It is hard to imagine that such a sophisticated device sprang fully formed to its current realization.

Similarly, it is interesting to speculate on who in ancient Greece might have had the insight todevelop anamorphic art or geometry, quite apart from noting the special property of spatialinversion, and then to use this idea in a religious object. One possibility is Pythagoras (570–495bc), or a member of his following of mystics. The name, Pythagoras, suggests that his parentshoped that he would speak (Greek agor-) the truth no less than the Pythia (hence Pyth-), thusimplying a personal link to Delphi. In fact, according to Aristippus, his mother consulted thePythia at Delphi and was told that she would give birth to a man who was supremely beautiful,wise and beneficial to mankind (Riedweg 2005). Thus it is not beyond the realms of possibilitythat such a brilliant man as he, who gave us the ‘right-angle triangle theorem’, could havespecified the form of the Delphian Omphalos. If the inventor failed to explain the symbolism ofthe conoidal shape to those around him/her, they might have enjoyed the fact that it was their ownspecial secret; and the secret died with them!

CONSTRUCTION

The Omphalos is a distinctly axisymmetrical shape. It would have been much simpler to make astraight-sided cone, if some sort of cone was all that was desired. An understanding of thegeometry of conic sections (straight-sided cones) was already well developed by the time ofApollonius of Perga in the third century bc, so presumably there were progenitor studies beforethat. The curved surface of the conoidal Omphalos could have been made using a woodentemplate as a guide for the sculptor, thus achieving its axial symmetry. However, we assume thatthe design was entirely empirical, and merely virtual spatial inversion (or a symbolic represen-tation of it) was all that was sought; this is especially relevant because strictly circle-invertinganamorphoscopes must be calculated using modern methods of numerical integration (Kuchel1979, 2004; Hunt et al. 2000; Cargo et al. 2002; Sharp et al. 2011).

The marble from which the Omphalos is made is hard, and yet the grey matrix has worn awaymore than the interspersed seams of white stone (Fig. 1). There is little doubt that the surfacecould have taken a high polish that was suitable for forming a reflecting surface, especially if, asnoted above, oil or water was applied to it.

The position from which the Omphalos was viewed, to gain an impression of spatial inversion,can only be speculated upon here. There could have been a viewing platform above it, or aplane-metallic mirror could have been positioned overhead in the room, or over a colonnade inwhich it was placed, exploiting the second mirror to obviate the need to be physically positionedabove the Omphalos to see the effect of spatial inversion. On the other hand, as stated above, itcould simply have represented the centre of the World by geometrical abstraction, so that it wouldhave needed no special arrangements for its viewing.

A MODERN FACSIMILE

In order to test further the optical transforming properties of the Omphalos (should it actuallyhave been a mirror), a ~1/10 scale model was turned on a lathe from aluminium, and polished toa high finish (Fig. 3 (a)). The photograph shows a distorted image, akin to those seen in wavymirrors at fun parlours. The model was used in a series of optical experiments to determine howsimilar in operation it is to a true circle-inverting anamorphoscope. Figure 3 (b) shows the

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aluminium conoid viewed from directly above its apex, surrounded by red, green and blue discson the basal plane. The inversion of space is best appreciated by noting the smaller green disc atthe bottom edge of the large blue disc in the lower part of the picture. However, in the reflectionin the base of the conoid, the green disc is now near the inner edge of the image of the blue disc;thus object space has been inverted in image space. But from Figure 3 (b) it is clear that theOmphalos is not a pure-circle inverter (cf., Fig. 3 (d)); it nevertheless inverts space and invokesa one-to-one mapping of the real, object plane into the virtual, image field inside the circular baseof the conoid.

Figure 3 (c) shows that even when viewed from the side, which would most probably havebeen the simplest configuration for viewing in the temple, the image invokes spatial inversion.However, the inversion of the horizon at (almost) infinity into the centre of the image field isstrictly only evident when the conoid is viewed from directly above its apex.

Figure 3 (d) shows a different conoid made from brass and viewed from directly above its apex(Kuchel 1979). The conoid rests on the plane containing a graphical object that is a circle-inverted Union Jack, for which all curves are parts of the arcs of pure circles. The particularanamorphoscope is very close to being a true circle-inverting device, whereby a circle in theobject field is inverted to a pure circle (or straight line) in the image field. Failure to strictlyachieve this transformation is due to imperfections in the turning of the object on a lathe, in spiteof the use of a template that was based on numerical integration (Kuchel 1979).

A more graphical comparison of the shape of the Omphalos and that of modern circle-invertinganamorphoscopes is given in Figure 4. There, the left-hand outline of the Omphalos fromFigure 1 is overlaid by three curves generated numerically (Kuchel 1979) using Mathematica(Wolfram 2008), to yield true circle inverters. It is clear that the true circle inverters with thesame diameter and height as the Omphalos would lie within the Omphalos; in other words, theOmphalos is ‘fatter’ at every vertical position than the respective circle inverters. The latter werecalculated for three different viewing heights, twice the height of the conoid, 10 times the heightand effectively at infinity, axially above it.

0

1

10

Omphalos

Viewing heightinfinity

Viewing height10 times h

Viewing height2 times h

Radius (normalized)

Heig

ht (n

orm

alized)

2 = h

Figure 4 The left curved generating line of the Omphalos from Figure 1, compared with the corresponding lines forthree pure, circle-inverting anamorphoscopes of the same relative diameter and height. They were calculated for viewingheights, two, 10 and 100 000 (effectively infinity) times the height of the conoid.

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FINAL COMMENTS

In conclusion, we postulate that the ancient Omphalos at Delphi was a symbolic optical trans-former that represented the centre of the World by the theoretical notion of reflective spatialinversion. Either it could have been simply symbolic, or it could have been a mirror that gave avirtual image of space inverted from infinity inwards—but this latter effect could only have beenperceived if it was viewed from directly above, or via a secondary mirror positioned above it; orit could have been viewed from the side like mirrors in fun parlours, and the effect only partiallyrevealed (Figs 3 (a) and 3 (c)). To achieve a virtual image that shows complete inversion of all ofspace from infinity inwards, conoids need to have an included angle of 90° at their apex as, wepostulate, did the present Omphalos (Fig. 1). However, over time, whether by natural means orvandalism, the change to the Omphalos’ current truncated form may have led to an oversight ofits true significance; hence millions of visitors to Delphi, classical scholars and archaeologistshave missed the point regarding its deeper and cryptic geometrical meaning.

ACKNOWLEDGEMENTS

An Australian Research Council Discovery Project Grant funded mathematical work on shapetransformation in cells that lay behind the present analysis. Dr Wayne Mullen of the AustralianArchaeological Institute at Athens, University of Sydney, is thanked for valuable discussions,references and leads into the classical literature on Omphaloi. Professors Jelle de Boer and JohnHale are thanked for valuable comments and discussions. Professor Jim Hunt is thanked forcomments, key references and the preprint of his paper in The Mathematical Gazette. MerilynKuchel is thanked for critical comments and proofreading. Mr Max Francis is thanked for theloaning of the book by Flaceliere that started the whole adventure, and Mr Ces Delapaz is thankedfor machining the aluminium scale replica of the Omphalos.

REFERENCES

Broad, W. J., 2006, The oracle, ancient Delphi and the science behind its lost secrets, Penguin, New York.Cargo, G. T., Graver, J. E., and Troutman, J. L., 2002, Designing a mirror that inverts in a circle, The Mathematical

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(Greece), Geology, 29, 707–10.Flaceliere, R., 1965, Greek oracles, transl. D. Garman, Paul Elek, London.Hale, J. R., de Boer, J. Z., Chanton, J. P., and Spiller, H. A., 2003, Questioning the Delphic oracle, Scientific American,

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