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Given two imaginary points a and b moving within a finite three-dimensional volume.  To any observer, they can be observed as moving towards each other along an imaginary straight line i.

There are seen to be three conditions associated with this puzzle: convergence, coincidence and divergence.

The instant of coincidence is denoted as t₀. It is the instant points a and b coincide with each other at m. Since any two points traveling at any speed require no time to occult each other, since they are infinitesimally small, their passage is instantaneous. Accordingly t₀ is an instantaneous event.

During convergence, the last instant before the points coincide, while they are separated, is denoted as t_-1.

Similarly, the first instant points separate after coincidence, is denoted as t₊₁.
 
 

Therefor, by definition, there are no intermediate instants between t_-1 and t₀ nor between t₀ and t₊₁.

Accordingly, all motion from one of these instants to the next is seen to be discontinuous. For example, representative points a and b are either distinctly apart or in complete conjunction, such as at t₀. Because points are so small, taking up no space at all, they are either separate or coincident;  no other conditions being possible.

Because the observer can be fixed relative to imaginary line i in both terms of position and orientation, because both motions are collinear to line i, and because no mechanism has been declared which might alter such motion, either in its scalar magnitude or its direction, the respective motions V₁ and V₂ will always appear invariant to the observer, which is very much in keeping with Newton's First Principle, that left to itself, velocity will not change.  This principle, developed by Galileo and Newton, normally deals with rolling bodies.

Upon their convergence, once both points coincide, a paradox manifests.

Because the the respective points a and b are indistinguishable from each other during coincidence, there is no mechanism assuring the retention of their previous association with their respective motions before coincidence to their association during and after coincidence.

Because coincidence during t₀ is instantaneous, thus excluding any decision making process, and because no specific mechanism governing this choice exists, it can only be assumed to be fifty-fifty, or a matter of chance.

Paradoxically, from the observer's point of view, fifty percent of the time the observer will witness two points converging and then passing through each other as though uninterrupted in flight, and fifty percent of the time, the observer will witness two points upon their coincidence, suddenly reverse directions as though seemingly undergoing some kind of elastic rebound, albeit instantaneous.

This paradox is typical of the nature of the subject matter in this school of thought, but not nearly as complicated as the relationships between lines and surfaces.  (See IDDI.)  However, though there is the tendency by the student to associate inertial behavior to point forms, once liberated from their traditional education dealing with the static geometry of Euclid and modern geometry and calculus, it may be realized that any mechanism of coherency causing permanent association between motion and point, such as some kind of inertial embodiment, is untenable geometrically speaking.  (Revised 12/6/2000)