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 ëneckí i.e. the ëprotectioní motive of the convex curve becomes the ëofferingí  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. 

Michelangelo once confessed that  the secret of beauty lies in the alternation (or inflection) 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 ëjumpsí 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 ìDas bildnerische Denkenî  (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 ënon dimensional place of cosmo genesisí and for Deleuze it signifies the ëeternal returní. 
 

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.

 THE TOPOLOGICAL CLASSIFICATION OF CURVED LINES AND SURFACES

"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.." 

this image of a shell, an example of a three dimensional logarithmic spiral, was  generated on a computer by Øyvind Hammer

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

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 - http://www.ucmp.berkeley.edu
 
 
 

It is the locus of vortices, sponges, mazes, meanders and labyrinths. ë..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 ìtakes its flight or risks to crash upon usîí (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)

ABOUT  VOIDS

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.