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From study of vector space approach to geometry, it was determined that volume (triple) and area (double) may be considered independently of the metric properties of length (single).
Mathematical logic conclusively suggests that the degree of physical invariance is higher for volume than it is for area. That of area is higher than it is for length. For example changes of length as a triangle is deformed would still allow its area to remain constant even if any two of three sides are stretched to infinity. The change in space-time (universal expansion) and location of matter (position and velocity) would still allow the size and volume (mass and density) of each individual particle to remain constant within relativistic limits.
This latter implication together with a principle of least action led to a surprisingly simple conclusion to the question: why an electron and conceivably all other elementary particles appear point-like, minimizing their physical volumes as the highest triple invariance of nature.
Equivalently, the volume determinants for both positive and negative is equal to the scalar triple product of vector analysis: A´B×C where A, B, and C are vectors defined in Euclidean 3-space. Furthermore, the same can be extended to 4 dimensions space-time or to 11 dimensions superstring. In fact, it can extend arbitrarily to infinite dimensions. However, all these higher dimensions are not really necessary for the conservation of volume triple invariance of perfect symmetry. It is a fact of life that no baseball hitters ever hit a quadruple unless playing on a pentagon instead of a diamond or a strong golfer scoring triple eagle on a par 5 course.
Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: ¶a(t)·¶r(t)=c²
Re: triple double not single -
02-08-2008, 12:01 PM
Linear Mode
