The graphics on this page were created using LibertyBasic CAD software. While it is not incorrect to call them animations, one can animate anything, including fantasy. These were designed to display the results of iterating through mathematical expressions, so that there is an isomorphism between 2D pixel position and 3D coordinates. In some, a proprietary transformation was used that allows 32K pixels in a 32x32x32 grid to all be displayed without overlap. This eliminates the usual problem of nearer pixels covering farther pixels. It means the picture is not a smooth analog image, but it allows us to see internal structure without cutaway images. If there is demand, they can be converted to free-standing applications that allow user interaction to select parameters and change the perspective. While analog transformations provide fine control of the image, coarse control is accomplished using 4D symmetry coordinates. These allow 24 different 3D perspectives by simple permutation of the indices of the coordinates.

Spectral decomposition using symmetry functions will be added later. While similar to a Fourier decomposition, the 4 quadrant analog multiplication required by sinusoidal primitives is replaced by simple addition and subtraction. And since the primitives are a finite set of digital waveforms, there is no infinite string of harmonics. Large data sets can be processed with a first-order approximation in much reduced times. For example, it is true that the complex plane cannot be ordered like a number line. But to within a few permutations, I have developed an algorithm that orders points in 3-space (or higher). The 32K grid can be threaded by a string that uniquely identifies the position of each point with its digital harmonic number. The unit ramp through these 32K points requires only 15 spectral components to recreate the sequence in 3 dimensions. A truly random distribution may require all 32K spectra, but the process helps to identify the polynomial order of the information in the grid, and sorts it according to symmetry. For example, a distribution that is aligned with a 45 degree diagonal slice through the cube will have identical spectra in two directions using normal techniques. But symmetry decomposition has only 1 spectrum composed of primitives that share the 45 degree diagonal symmetry, and the normal axial projections have 0 amplitude. There is a remarkable connection between the symmetry functions and the Sierpinski tetrix, as well as the Cayley tables for hypercomplex numbers, which have been shown to be applicable to the Standard Model of particle physics.

For now, we focus on the Lorentz transformations in Minkowski and eigenvector coordinates. Usually, a general set of coordinates is projected through a transformation, and we compare an arbitrary input to its output. These simulations are based on the proposition, why not take an entire array of uniformly spaced points and project all of them, so we can get a sense of what a Lorentz transformation does to a region of spacetime, instead of a point. As is customary, relative velocity is aligned with an axis, so that the other two are invariant, and contain no information about the transformation. So instead of a 3D grid of points in space, we depict a 2D grid of spacetime. This gives us the extra degree of freedom to use the normal direction as the rapidity coordinate of hyperbolic trigonometry. Instead of velocity being the slope of a line in the spacetime grid it can be applied to an entire slice, giving us a set of individual slices which correspond to different relative velocity observers. Then we can trace the evolution of the grid points as the remote observer’s relative velocity changes. Take a look.