Certainly particles are not shaped like anything. However shapes themselves are informative.(adsbygoogle = window.adsbygoogle || []).push({});

If particles were spheres, then they would trace out tubes in 4-d. After this though, we have lost the particle, so we have to chop the tube back up to pieces. However this piece has now become a "hypervolume" instead of a volume. This piece I'll call a bullet as a quantum of hypervolume out of the world tube. It is 4-d object, spheroidic in space but rectangular or parallelogramical by other cross-sections. It is a tube bound by two oblate spheroids as end pieces.

It must be easy enough to fashion a bullet to be

1) an invariant quantum of hypervolume

2) a shape which recovers kinematic variables

It seems I can formulate this shape as necessary. As I use hyperbolic functions, I omit the angle notation.

major axis = D

minor axis = D sech

bullet 4-height = i D cosh

bullet hypervolume = [ i D cosh ] x ( bullet end volume ) = = i ( pi / 6 ) D^4

So this shape does not reside in Euclidean 4-space. Hypervolume here is reckoned to require an imaginary component. Furthermore, velocity = c tanh, momentum = mc sinh, and energy = mc^2 cosh are familiar and all spaned by the inner geometry. The ellipse aspect provides an interfocal distance proportinal to the velocity. The "i" provides an inner transverse diagonal proportinal to the momentum.

So beyond this, I then have the question whether "hypervolume" requires the imaginary component "i", or if there is such a thing as actual hypervolume, or if this is a special kind of hypervolume described by a mathematical classification.

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# Hypervolume Quantum Shape

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