Magnitude Sweep Generator Functions

In polysign it is possible to use sign as a means to yield dimensional behaviors. The magnitude sweep technique exposed here relies upon iterated functions. In the polysign domains an iterated function which introduces a sign modification within the iterated loop yields structured results. We can define a function:

    z(n+1) = - z(n) z(n) @ m

where m is a magnitude and z is a polysign number and the '@' symbol is superposition or summation. The input basis is extremely simple. It is merely a swept magnitude m. These instances start with z(0) = 0 for simplicity. No limits need be applied to the sweep. Large values will scatter off to infinity rapidly. No conditional operations are performed to yield this structured set of points. While the quantity of iterations is limited it is apparent that coherency is obtained. This is a way to build some fascinating dimensional data from nearly nill.

This is a 2D projection of a 5D space. Occlusion is not taken into account. Larger magnitude points show through in this 2D projection since they were plotted last. No depth has been considered in the projection. The color is magenta at first then yellows with iteration. The image is actually very flat for many projections so I believe that the shape is largely disklike with interesting protrusions. This shape generally recurs in any dimension for the given function though it has fascinating variations. The density of points in m is so high in this first image that some of the underlying structure is obscured. At lighter density(stepping 2e-4) more structure is revealed

This instance is merely a starting point for a clan of such functions. A relativist might wish to explore the magnitude sweep as if it were in a different reference frame. Many approaches look promising and some of them will be explored on this site in the future. The relationship to fractals is quite slim. The swept function is an iterated procedure but beyond that no conditionals are in use here. Rather than a test of every point in space the points which are graphed are generated directly from magnitude. The motivation for this construction is to generate multidimensional data from as small a construction as possible. Magnitude can be mapped in to any signed domain and this step is of crucial importance here though it comes transparently. These graphics are possible in the complex plane so even if you do not have a polysign library it is possible to do some work with traditional tools.

I am still troubled as to the believability of iterated functions as a plausible basis of reality. Without a sound physical paradigm such graphics are only pretty mathematics. The quality of an individual function and why that particular function ought to be given attention is not clear to me. Here a most simplistic function with dynamics is exposed. If we consider an individual function to be instantaneous then we witness a point series as a function of magnitude m which is nicely connected to its neighbors in m. There are transitions, connections, periodicities, and stable points which need to be explored. Some of that I have worked on but it is challenging to get to a comfortable conclusion. So instead this is presented as an open area of research. Rather than let this fester on my hard drive any longer I share it with you in the hope that you might pursue something similar or find it helpful in your own search. There is no end to the number of images that can be generated here so instead of burdening myself with so many I give you just as few as a proof of concept. This is not the only interesting production; rather, there are an overwhelming series of possible graphical results that are too numerous to prioritize yet.

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