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topological analysis can help to explain the ritual meaning of some neolithic signs and artifacts
|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 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
THE TOPOLOGICAL CLASSIFICATION OF CURVED LINES AND SURFACES
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
|The second is of the type also 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.
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 ëfrom fold to foldí.
Image courtesy of the University of California
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:
cited by Joseph Campbell, (note 6)
ABOUT VOIDSThese 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
- back to last page on neolithic signs
- next page is on the application of our analysis of signs to design theory