IMAR MARIUS BULIGA

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Simulating spaces

     
See this working version of "Computing with space: a tangle formalism for chora and difference". If you want to dig more into the maths then follow the links below.
     
Let's look at some examples of spaces, like: the real world, a virtual world of a game, mathematical spaces as manifolds, fractals, symmetric spaces, groups, linear spaces ...
     
As a set of points, a space is uninteresting. Is better to look at it from a computational viewpoint.
     
Indeed, all spaces given as examples may be characterized by the class of algebraic/differential computations which are possible, like: zoom into details, look from afar, describe velocities and perform other differential calculations needed for describing the physics of such a space, perform reflexions (as in symmetric spaces), linear combinations (as in linear spaces), do affine or projective geometry constructions and so on.
     
Such computations are finite or virtually infinite "recipes", which can be implemented by some class of circuits made by very simple gates (as in boolean computing, where transistors are universal gates for computing boolean functions).
     
(Computation in) a space is then described by emergent algebras, or by deformations of groupoids, which are inspired by the considerations about a formal calculus with binary decorated planar trees in relation with dilatation structures :
     
A - a class of transistor-like gates, with in/out ports labelled by points of the space and a internal state variable which can be interpreted as "scale". I propose dilations as such gates (basically these are idempotent right quasigroup operations).
     
B - a class of elementary circuits made of such gates (these are the "generators" of the space). The elementary circuits have the property that the output converges as the scale goes to zero, uniformly with respect to the input.
     
C - a class of equivalence rules saying that some simple assemblies of elementary circuits have equivalent function, or saying that relations between those simple assemblies converge to relations of the space as the scale goes to zero. This is the emergent point of view. Note that the example of deformations of groupoids and the one concerning braided spaces with dilations show that at least some relations are not emergent.
     
     
Seen like this, "simulating a space" means: give a set of transistors (and maybe some non-emergent operations and relations), elementary circuits and relations which are sufficient to generate any interesting computation in this space.
     
For the moment I know how to simulate affine spaces, sub-riemannian spaces, some symmetric spaces and braided spaces with dilations (and work in progress).
     
This point of view is loosely related to Leibniz' mill and Koenderink' Brain a geometry engine.