randform

Last time when I was in Göttingen I found a poster at the math department documenting an art science collaboration between mathematics professors William Thurston, Kazushi Ahara and Sadayoshi Kojima on one side and a team around clothing designer Issey Miyake, notably including chief designer Dai Fujiwara of Issey Miyake (here a link to a partial version of the poster, see also absnews article by Jenny Barchfield). A result of this collaboration is that the Issey Miyake Fall-Winter 2010-2011 ready-to-wear collection is inspired by the geometrization conjecture.

From the poster:

In the mid-October of 2009, Prof. Thurston showed us the detail drawings of the “8 Geometry Link models as Metaphor of the Universe” They inspired us to make the collection based on them, accompanying design study with rope and toile. Considering the body itself as the Universe, we have added our own interpretation of beauty to them. The new perception of the body shared by all the members of the team resulted in the discoveries of new lines and forms, which were then applied to textile, color and detail studies. Thus the new collection has taken shape steadily, revealing its whole picture eventually. To sum up the exchange with Prof. Thurston led us to find a completely new kind of beauty and embody it in clothing. This mission was, as it were, an odyssee to explore the Universe with infinite imaginations.

The geometrization conjecture roughly says (I am not an expert on this) that a three dimensional volume form without boundary (a two dimensional analog of such a form would be for example the surface form (i.e. the “skin”) of a ball or the surface form of a doughnut) can be decomposed into “pieces” which have one of 8 characteristic “geometric structures”, which means roughly that in a small neighbourhood of any such “piece” there is – out of only 8 characteristic ways – one specific way to measure length. A theorem states that any three dimensional (oriented) volume form without boundary can be obtained by cutting a “thick” (that is instead of a rope take a ribbon) link out of a three dimensional sphere. Thus you can characterize special types of three dimensional volume forms (here: “the pieces”) by assigning a link to them. This is – by what I understood sofar- why there are 8 links (or link models) on the poster – they characterize the 8 types of possible “pieces”, which built up three dimensional volume forms without boundary.

Why do they call these 8 links “Metaphor of the Universe”? I can only make wild guesses, which sound rather like science fiction than science: Maybe if you imagine the space of the universe to be eventually such a three dimensional volume then by cutting it into pieces (may be along black hole horizons huh?!) and “measuring distances” (determine a metric) one could make deductions about the actual form of the universe? Or – reversely by making assumptions about the form of the universe (like e.g. that its space is a three sphere) one may get informations about what could be inside black holes…given that one finds all black holes…(this is just a funny joke).

But joking aside – I think they call it Metaphor of the Universe because these simple 8 links may be used to describe quite complicated things.

->wikipedia link math and fiber arts