The gnomon in the greek theater of vision, I

In the post Theatron as an eye I proposed the Greek Theater, or Theatron (as opposed to the “theater in a box”, or Cartesian Theater, see further) as a good model for   vision.

Any model of vision should avoid the homunculus fallacy. What looks less understood is that any good model of vision should avoid the scenic space fallacy. The Cartesian Theater argument against the existence of the homunculus is not, by construction, an argument against the scenic space. Or, in the Cartesian Theater, homunculus and scenic space come to existence in a pair. As a conclusion, it seems that there could not be a model of vision which avoids the homunculus but is not avoiding the scenic space. This observation is confirmed by facts: there is no good, rigorous  model of vision up to date, because all proposed models rely on the a priori existence of a scenic space. There is, on the contrary, a great quantity of experimental data and theoretical partial models which show just how complex the problem of vision is. But, essentially, from a mathematician viewpoint, it is not known how to even formulate the problem of vision.

In the influent paper “The brain a geometry engine”  J. Koenderink proposes that (at least a part of) the visual mechanism is doing a kind of massively parallel computation, by using an embodiment of the geometry of jet spaces (the euclidean infinitesimal geometry of a smooth manifold)  of the scenic space. Jean Petitot continues along this idea, by proposing a neurogeometry of vision based essentially on the sub-riemannian geometry of those jet spaces. This an active mathematical area of research, see for example “Antropomorphic image reconstruction via hypoelliptic diffusion“, by Ugo Boscain et al.

Sub-riemannian geometry is one of my favorite mathematical subjects, because it  is just a  particular model of a metric space with dilations.  Such spaces are somehow fundamental for the problem of vision, I think. Why? because there is behind them a purely relational formalism, called “emergent algebra“, which allow to understand “understanding space” in a purely relational way. Thus I hope emergent algebras could be used in order to formulate the problem of vision as the problem of computing with space, which in turn could be used for getting a good model of vision.

To my surprise, some time ago I have found that this  very complex subject has a respectable age, starting with Pythagora  and Plato!  This is how I arrived to write this blog, as an effort to disseminate what I progressively understand.

This brings me back to the theater and, finally, to gnomon. I cite from previous wiki link:

Hero defined a gnomon as that which, added to an entity (number or shape), makes a new entity similar to the starting entity.

In the greek theater, a gnomon sits in the center of the orchestra (which is the circular place where things happen in the greek thater, later replaced by the scene in the theater in a box). Why?

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