Because the element hit is black, an object must be produced that allows information from the block at this step to pass through.

Traditional mathematical methods began to be applied to it in the second half of the 1800s, particularly through the development of psychophysics. … But mainly through a desire to use traditional mathematics, these implementations have tended to be implicitly restricted to using elements with various linearity properties—typically leading to rather unconvincing results.

In 1644 René Descartes proposed that space might initially consist of an array of identical tiny discrete spheres, with motion then occurring through chains of these spheres going around in vortices—albeit with pieces being abraded off. … Models that involve discreteness have been proposed—most often based on spin networks—but there is usually still some form of continuous averaging present, leading for example to suggestions very different from mine that perhaps this could lead to the traditional continuum description through some analog of the wave-particle duality of elementary quantum mechanics.

In a system described by linear equations, there is always a simple superposition principle, and waves just pass through each other unchanged. … (This is analogous to what happens for example in classical diffraction theory, where there is an analog of the path integral—with ℏ replaced by inverse frequency—whose stationary points correspond through the so-called eikonal approximation to rays in geometrical optics.) … In the quantum case, a sign of confinement would be exponential decrease with spacetime area of the average phase of color flux through so-called Wilson loops—and this is achieved if there is in a sense maximal randomness in field configurations.

And in 1918 Hermann Weyl suggested that this could happen through local variations of scale or "gauge" in space, while in the 1920s Theodor Kaluza and Oskar Klein suggested that it could be associated with a fifth spacetime dimension of invisibly small extent. … Consistency also typically required the basic spacetime to be 10-dimensional—with the reduction to observed 4D spacetime normally assumed to occur through restriction of the other dimensions to some kind of so-called Calabi–Yau manifold of small extent, associating excitations with various particles through an analog of the Kaluza–Klein mechanism.

And indeed the previous page shows that if one looks at the evolution of a one-dimensional slice through each two-dimensional pattern the results one gets are strikingly similar to what we have seen in ordinary one-dimensional cellular automata.

But the correspondence with one dimension becomes much more obvious if one looks not at the complete state of a two-dimensional cellular automaton at a few specific steps, but rather at a one-dimensional slice through the system for a whole sequence of steps.

In some cases, one structure essentially just passes through another with a slight delay.

And indeed, to the contrary, our perception is that different parts of the universe seem to evolve in parallel and progress through time together.

One might imagine that with only a single active cell being updated at each step different parts of the system would inevitably be perceived to progress through time one after another.