Tuesday, July 24, 2007

The symmetry between repulsive and attractive force

The relational perspective map (RPM) algorithm in its core simulates a multi-particle system on a torus surface in which the particles exerts repulsive forces to each other. Some people has asked the question why just repulsive force? Why not use attractive forces like other force directed mapping methods?

A simple answer to this question is that we prefer to use a simpler dynamic system if it can solve the problem. Just like physicists who try to reduce the number of fundamental interactions, I would prefer to avoid using attractive force if it is not absolutely needed.

Classical multidimensional scaling (MDS) methods have to use both types of forces (as can be derived from their stress function) because their base information space is the infinite open Euclidean space: without attractive force their configuration will quickly degrade to infinite size; and without repulsive forces their configuration will shrink to a single point. With RPM method, the closed manifold (i.e. the torus surface) confines the configuration into a limited size.

Another not-so-obvious answer to above question is that on a closed manifold the repulsive and attractive forces are the manifestation of the same thing, from certain point of view at least. To see this, let use consider the simplest case of 1-dimensional curled space (i.e. the circle). As be shown in the following picture, imaging that we have two ants living on the circle and there are two positively charged particles on the circle which can move freely on the circle but are confined on the circle.

From the point of view of the ant on the left side, the two charges exert repulsive force to each other according to coulomb's law.

However, from the point of view of the ant on the right side, the two charges attract each other. This is a little contra-intuitive for us as observers from the 3-dimensional space. But, we need to remind us that these ants are living in the 1-dimensional space. The ant on the right side has no way to see the two charges moving apart from each other. This ant can only move on the circle, particularly the right arc of the circle, to measure the distance between the two charges. What this ant would find out is that the same force as indicated in the picture will seemly press the two charge closer to each other; and larger the charge, the stronger the force. Thus, the ant on the right side would claim that two charges attract each other according a law similar to the coulomb's law (the anti-coulomb's law?).

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