Object Convolution

The arithmetic product is well known and takes a geometric form on the real numbers and the complex numbers. By generalizing the point product to objects which contain a set of points we arrive at object convolution:

Define an object O as containing a series of points:

   O = p1, p2, p3, ..., pn .

Two such objects O1 and O2

   O1 = p11, p12, p13, ..., p1n
   O2 = p21, p22, p23, ..., p2m

can be consistently joined by arithmetic product as

   O3 = O1 O2
      = p11 p21, p12 p21, p13 p21, ... p1n p21,
        p11 p22, p12 p22, p13 p22, ... p1n p22,
        p11 p23, p12 p23, p13 p23, ... p1n p23,
        ...
        p11 p2m, p12 p2m, p13 p2m, ... p1n p2m .

This concept has been studied briefly for the complex plane (P3 in the polysign system) and yields the following:

The line segments to the right are convolved to yield the shaded area to the left. Curvature is exposed in the area of the resultant. To expose the underlying structure of these 2D segment convolutions a sparse animated version has been compiled. The procedure of object convolution is nearby to integration since in both cases a dimensional increase is observed. While integration is often considered a summation operation there is a product inherently involved as well. A connection may exist between these forms. That curvature can be found here as a resultant of flat sources is suggestive. The resulting image has exceeding complexity just for the 2D case. The coverage of the convolution will become problematic for area computation.

This topic is covered by Farouki and is introduced on page 6 of his paper with Hass on geometric convolution where this style of convolution is labeled the Minkowski product.

Such products bear no direct resemblance to reality as we know it. Perhaps in the development of object convolution a tier of physical correspondence will be discovered. Polysign behavior indicates that spacetime is tied to the arithmetic product. Furthermore such products can be defined in any dimension. I encourage you to investigate further.

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