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updated: april 2004

ur signs

topological analysis can help to explain the ritual meaning of some neolithic signs and artifacts

testo italiano


Thus far I presented a system for the classification of two dimensional signs, as inscriptions on rocks or artifacts, as a pictographic or logo graphic kind of writing. How applies this to three dimensional forms? 
Surfaces present a transition from the second to the third dimension and it is easily seen that often they can be reduced to their two dimensional issue. E.g. a fold presents a line, a rippled surface corresponds to parallel lines or, if circular like the effect of a stone thrown into water, it represents the dilatation of a circle from a center. 

Artifacts and technology influence topologically closed surfaces more strongly than their 2D counterparts, signifying some functionality even in ritual artifacts (note 1). A convex plane for example presents a protecting function as in a shield, whereas a concave plane has a containing function such as in a bowl. Both have as a 2D correspondent the curved line. A form generated by rotation of a concave curve represents, if the inside is open towards the outside as a cup, an offering i.e. connecting the community to the Deity, and, when the cavity is directed towards the person that holds it as in a shield, a protection from harm. 

In the case of a vase this protective function is naturally extended to the contents originating the meaning of inside/outside, as in the circle motive. Ideally the 2D circle motive becomes in 3D the sphere. The opening in the case of a vase, represents the channel or passage of communication from inside to outside and vv. The form of this channel can be enhanced in different ways. If the vase e.g.. has the form of a bottle the convex curve of the container becomes concave towards the ‘neck’ i.e. the ‘protection’ motive of the convex curve becomes the ‘offering’  motive of the concave curve. A transformation of the volume as in the case of the neck of a bottle can be explained as the expression of an elongation or stretching towards the outside. It corresponds also to the sinusoid or serpentine line.

Petals of flowers,  plantleafs and innumerable other membranes of living creatures are wonderful  examples of alternating curved surfaces in nature, also called 'saddle surfaces'. In mathematics they represent the 'minimax function' of all points that converge towards a central area of equilibrium between the maximun value of one function and the minimum of another.
A further example of alernating curvature in space is the 'Moebius strip', known in topology, that curiously  is characterized by the fact that, the 'ouside' being not distinguable from the 'inside', it has only one side! 

Michelangelo once confessed that  the secret of beauty lies in the alternation of convex and concave forms and today we may observe how much of modern design, such as car design, is an application in three dimensional space of the serpentine line. We will try to understand it somewhat better in the following. 

The famous french philosopher Gilles Deleuze (1925-) (note 2.) points out how Paul Klee analyzed this curve or fold and especially, like Leibniz (1646-1716) before him, the (metaphysical) inflection point where the radius ‘jumps’ from inside to outside and vv. Paul Klee with his formidable insight in visual and psychological phenomena has effectively anticipated many explanations of the signification of visual signs in his “Das bildnerische Denken”  (note 3), a loose collection of transcriptions from his lessons at the Bauhaus (1920-1933). The inflection point of the sinusoid line, says Klee, is the ‘non dimensional place of cosmo genesis’ and for Deleuze it signifies the ‘eternal return’. 
a fold by Paul Klee Klee illustrated the inflection point in a line with these three figures in his "Theory of modern Art". 

The first figure shows a line as the path of a point that changes direction an inflection (or folding) point. 

In the second figure the line is playful accompanied by another line. 

In the third figure he shows a shadow on the 'outside' of the convex parts of the line. The shadowed area changes side at the inflection point. 





A topological analysis of curved lines and forms characterized by inflection points is possible if we consider them as originated by different types of transformation. The classification of the underlying transformations into three types, mentioned by Bernard Cache (note 4) , helps us to better understand them: 

The first type is exemplified by the serpentine line and, depending both on the morpho genetic field underlying the inflection point and (minimal but decisive) outside influences, by other lines:  twofolds, circles, umbilici and ogives. Each of these lines are in the third dimension to be considered as sections of planes or, more in general, surfaces. A very special case is, significantly, the euclidean space geometry with its straight lines, triangles, squares, polyhedrons, etc. where any 'edge' is also a locus of inflection points. The generalized meaning of all these geometrically well defined forms can be considered in a neolithic perspective, as we have seen before, as signs that represent the connection between our world and the divine. This link is both dangerous and comforting because of the nature of the almighty triple Bird Goddess. It was advisable to maintain good relations with her: it is known she was honored sometimes by sacrificing human beings. A convincing report on the serpents magic can be read in Campbells "The Masks of God: Occidental Mythology", (p.9) that we mentioned on the last page
    "The wonderful ability of the serpent to slough its skin and so renew its youth has earned for it throughout the world the character of the master of the mystery of rebirth - of which the moon, waxing and waning, sloughing its shadow and again waxing, is the celestial sign. The moon is the lord and measure of the life creating rhythm of the womb, and therewith of time, through which beings come and go: the lord of the mystery od birth and equally of death - which two, in sum are aspects of one state of being. The moon is the lord of tides and of the dew that falls at night to refresh the verdure on which the cattle graze. But the serpent, too, is a lord of waters. Dwelling in the earth, among the root of trees, frequenting springs, marshes, and water courses, it glides with the motion of waves.." 

The second type is described by W.DArcy Thompson, Waddington, Rene Thom (note 5) and E.C.Zeeman, as projections determined by a field of parameters that allows for infinite variations such as we find in nature as membranes like cells, shells, horns and generally surfaces of minimal tension. Some examples are: folds, cusps and hyperbolic, elliptic and parabolic umbilici. 
a shell this image of a shell, an example of a three dimensional logarithmic spiral, was  generated on a computer by Øyvind Hammer 


courtesy of:

The meaning we can affix to these literally manifold forms as signs is their earthly representation of the divine Goddess. 

The third topological transformation originates lines or planes with infinite variable curvature or fluctuations ‘from fold to fold’. 
the maze  Image of a sea urchin plate taken by 
the UCMP Environmental Scanning Electrom Microscope

Image courtesy of the University of California 
Museum of Paleontology 
web site -

It is the locus of vortices, sponges, mazes, meanders and labyrinths. ‘..the line effectively folds itself into a spiral differing the inflection in a movement suspended between heaven and earth, that distances or approaches a center of curvature indefinitely, and, any instant “takes its flight or risks to crash upon us”’ (Deleuze, Op.cit. , p.23) . These forms can be interpreted as follows citing a passage from Jackson Knight: 

    The maze form - which is an elaborated spiral - gives a long and indirect path from the outside of an area to the inside, at a point called the nucleus, generally near the center. Its principle seems to be the provision of a difficult but possible access to some important point. Two ideas are involved: the idea of defense and exclusion, and the idea of penetration, on correct terms, of this defense. 
    cited by Joseph Campbell, (note 6)


These spongiform structures can be classified from a different point of view: the topology of empty spaces. As a matter of fact any physical object has, as a counterpart, the space not filled by it. Recently in mereology the morphology of voids or holes has been partially analysed. (note 7)
Since Felix Klein (1849-1925),  holes play a fundamental role in topology in the classification of three dimensional forms (such as a sphere, a torus, a moebius strip  and others). After Conrad Hal Waddington, D'Arcy W.Thompson and René Thom have outlined a general geometry of natural objects, the systematic analysis of physics of intuitive, naive physics, including empty objects or cavities became feasible. But a closer look at these three dimensional entities having shapes but no physical existence became necessary  only in our age, in relation to artificial intelligence, especially in its application in robotry because these machines must learn to identify voids in the same way as they detect physical objects.
We met already these elusive objects in our Mental Design Model where, in the case of the cutting operation, we can substitute the subtraction with the addition of such a negative object to the 'host' object. 

Zen philosophy, of course, meditating the void or the absence of an object as a means to achieve illumination, leads oriental thought, especially in art, architecture and design, to assign equal value to both objects and voids.
We can thus condense the sense of all morphological transformations as symbols of sacred wisdom, of good and evil, of life and death. 

links to similar pages:

note 1 André Leroi-Gourhan L’homme et la Matière Albin Michel, Paris, 1943  
note 2 Gilles Deleuze Le Pli. Leibniz et le Baroque, p.20ff. Les Editions de Minuit, Paris, 1988  
note 3 Paul Klee Das bildnerische Denken (Visual Immagination)  Benno Schwabe & Co., Basel, 1956, Teoria della forma e della figurazione Feltrinelli, Milan, 1959
note 4 Bernard Cache and Michael Speaks Earth Moves: The Furnishing of Territories, p.85 MIT Press, 1995  
note 5 Rene Thom Stabilité structurelle et Morphogénèse. Essai d’une théorie générale des modèles InterEditions, Paris, 1977 (1972)  
note 6 Joseph Campbell The Masks of God: Primitive Mythology Viking Pinguin, England, 1991 (1959) p.69   
note 7 Roberto Casati and Achille C.Varzi Holes and other Superficialities MIT, Cambridge, 1995
Roberto Casati and Achille C.Varzi Parts and Places. The Structures of Spatial Representation MIT, Cambridge, 1999
andries van onck