The feedback I'm getting privately indicates that people do not understand the concept illustrated in figure 1 of my essay, in several respects. Given the constraints of the contest rules, I found it very difficult to include all the explanatory detail I would have liked to have included.
One of the difficulties has to do with background. There is no background. There are two, reciprocal aspects of one component, motion, which are 3D space and 3D time, but since time has no direction in space, and space has no direction in time, one of the reciprocal aspects of motion is always zero-dimensional.
However, just as, strictly speaking, natural numbers and energy are zero-dimensional measures, and, to be useful, they have to be regarded as one-dimensional, so also the 0D aspect of motion has to be one-dimensional.
In the case of numbers, any number raised to the zero power, n0, is equal to 1, because any number divided by itself is equal to 1, by virtue of the law of exponents. Hence, n/n is actually n1/n1 = n(1-1) = n0.
In the case of energy, any magnitude that does work has to be one-dimensional, because magnitudes that have no direction are scalar (i.e. 0D), while magnitudes with direction are vectorial (i.e. 1D). Energy per se is scalar. It has no specific direction in space, but in its form as work, it must have direction, and therefore it is regarded as a one-dimensional quantity.
In the geometric constructions of figure 1 in my essay, we see the discrete (digital) and the continuous (analog) expansion of 3D space (or 3D time), after one unit of 0D time (or 0D space) has elapsed. Though the 0D time (space) aspect of the expansion cannot be represented geometrically in any direct fashion, the 1D radius of the inner circle is equal to the time duration, in the same sense that a time line on a space and time graph, or the 1D sweep of an oscilloscope, represents a duration of time.
With this much understood, the radii of the inner circles are 1D representations of the 0D time (space) of the expansion. Now, the digital and analog representations of the two, inverse, expansions in the figure, each contain the 1D, 2D and 3D components. The digital representations are contained in the 2x2x2 stack of one-unit cubes, while the analog representations are contained in the ratio of the two balls that are determined by the stack of cubes.
In the digital case, the 1D magnitude is the width (or the height, or the depth) of the stack, while the 2D magnitude is one of the faces of the stack, and the 3D magnitude is the volume of the stack.
In the corresponding analog case, the 1D magnitude is the ratio of ball diameters, the 2D magnitude is the ratio of the spherical surfaces (or else the cross-sections of the balls), and the 3D magnitude is the ratio of the volumes of the balls.
The reason that the analog magnitudes are taken as the ratio of the two balls is because the magnitudes of the inner ball are necessarily less than the magnitudes of the stack, while the magnitudes of the outer ball are necessarily greater than the magnitudes of the stack. It turns out, however, that their ratios are integers and square roots of integers, which means that the digital magnitudes can be directly related to the analog magnitudes, not just approximated!
As the expansion continues beyond the one unit elapsed time stage, the digital and analog magnitudes follow their respective geometric progressions, part of which is explained in the previous post above. The key point is that these units can be arranged in the customary mathematical forms called groups. The usual 1D digital groups of integers and rational numbers apply, plus a new group with the square root of 2 as the 1D unit, instead of the number 1 as the 1D unit, applies.
However, this new group contains 2D and 3D elements as well as 1D elements, which provides for 1D, 2D and 3D scalar algebras called division algebras, which have all the properties an algebra needs to have to be used in physics.
Ultimately, this means that the 3D oscillations of these units, and their mathematical combinations and relations between their combinations, can be used as building blocks, called preons, to build the particles of the standard model of physics, at least in part. The hope is that the entire model will eventually emerge, including gravity. If this turns out as hoped, it will be very strong evidence that the physical universe consists of one component, motion, existing in three dimensions, in discrete units, with two reciprocal aspects, space and time.
I hope this helps.