**Fig. 1: Surface View of Two Adjacent Spot Cubes**

Legend: Each color-coded spot is a 2x2x2 cube of bits. A spot cube contains 8 spots, 4 of which are partially visible in this view. Electron spots (e-L; white) and right (R) and left (L) d quark (d) spots (r, red; g, green; b, blue). Mites (circles) and lites (arrows and stars). Stars are lites moving toward the viewer. Purple arrows indicate the direction of the three inter-dimensional strong bit operations within a spot, one of which is visible in each spot in this view.

**The EM Plane Wave**

Fig. 1 shows a schematic view of the surface of two adjacent spot cubes (left and right). Rows 2 and 3 of the schematic show right and left countercurrent spot unit channels respectively [1].

Consider 1-state bit motion due to the unconditional bit operation in these channels. For example, in row 2 of Fig. 1, imagine a 1-state bit at the leftmost mite with positive sign at X1, Y0, Z0. With each unconditional bit operation, this 1-state bit may move to the right one BM distance unit d. After four movements to the right, the 1-state bit arrives at another mite locus of positive sign. Thus, the wavelength of our EM plane wave is 4d, where d is the BM quantized length unit.

The macroscopic plane wave nature of EM radiation is thought to consist of orthogonal electrostatic and magnetic fields defining a plane which is perpendicular (orthogonal) to the direction of EM wave propagation. On the microscopic level of BM, there is an exact analogy to macroscopically observed EM plane waves.

In the example above, consider a 1-state bit at the Z0 coordinate and a plane perpendicular to its motion to the right.

**Fig. 2: Binary EM Plane Wave**

Legend: Perpendicular electrostatic (grey) and magnetic (blue) potentials form pulse-like square waves, out of phase, each with 50% duty cycles.

Starting with the mite, its 1-state is postulated to be an electrostatic potential (Table 2 in [1], grey line in Fig. 2), with effect over distance d in a direction perpendicular to the page of Fig. 1. That is, at X2 (not shown), above the surface of the page, is another set of spot unit channels, which are concurrent to those shown in Fig. 1. These concurrent spot units point in the same direction as rows 2 and 3 of Fig. 1, and hence, would look much the same. In short, an electrostatic (scalar) potential accelerates a like-signed mite in a concurrent spot unit. This is the electrostatic component of the plane wave.

When the Z0 1-state mite moves to the Z0 lite locus, this 1-state lite defines a magnetic (vector) potential (blue line in Fig. 2) which acts on a 1-state mite in the adjacent countercurrent spot unit in the X1, Y1, Z0 spot (Fig. 1). Hence, by definition, the magnetic potential is orthogonal to the electrostatic potential and to the direction of wave propagation.

As the 1-state bit moves to the next spot unit in this example at X1, Y0, Z1, both the electrostatic and magnetic components of the EM plane wave change sign, in exact agreement with conceptions of EM radiation at the macroscopic level. First, mite sign is now negative reversing the sign of the electrostatic potential. Second, since mite sign is negative at Z1, sign of the magnetic field also reverses.

The wavelength of this microscopic plane wave is completed at X1, Y0, Z2, where mite sign again reverses the sign of the EM components of the plane wave back to their values at X1, Y0, Z0. Textbook illustrations of the propagation of this plane wave typically show smooth sine waves at a 90 degree angle changing sign as the wave moves along each half wavelength of its propagation direction. At the most microscopic level possible in BM described above, these two smooth sine waves are seen to be more like square waves forming a sequence of EM potential pulses, but otherwise similar to the macroscopic EM plane wave concept.

This analysis in one dimension may be generalized. Each spot unit in any dimension X, Y or Z, may create scalar or vector potentials or both depending on the state of its mite and lite bits, which are postulated to be equivalent to electrostatic and magnetic potentials respectively (Table 2 in [1]). Further, each spot unit can affect only two adjacent spot units over BM distance d by accelerating their 1-state mite bits. If a spot unit has a 1-state mite (electrostatic potential), acceleration of a 1-state mite in its concurrent spot unit may occur. If a spot unit has a 1-state lite (magnetic potential), acceleration of a 1-state mite in its countercurrent spot unit may occur. In both cases, these accelerations are only possible if the affected spot unit has a 0-state lite locus, as required by the postulate that any bit locus may contain only one 1-state bit.

In summary, the macroscopic EM plane wave concept closely fits analysis at the BM microscopic level. BM started with a set of postulates including the EM bit operations which define time-development of BM states. A second step was to try a particular physical interpretation of BM space [3]. Combining these two information sets resulted in the rather clear mechanism underlying EM plane waves presented above. Macroscopic observations of the plane wave nature of EM radiation arise from numbers of 1-state bits propagated through BM space, summing or interfering to produce observed physical effects.

Similar analysis applies to right-to-left motion of a 1-state bit (row 3 of Fig. 1). Considering radiation in both directions in each of three spatial dimensions, a number of interactions may occur due to EM forces as defined in BM. Since electrostatic and magnetic potentials can result in changing the phase of single bits comprising the radiation, they can work to increase or decrease coherence.

Interference would be the most likely result given a degree of randomness (incoherence) in the radiation field (bit distribution). At reduced temperature, there is decreased bit motion due to EM forces by definition and hence, decreased likelihood of decreasing coherence (i.e., increased coherence), which may be one mechanism contributing to superconductivity [4].

**Ampere's Law**

Ampere's Law states that the magnetic field created by electric current is proportional to that current. 1-state mite motion per unit area per tick may define current, where a tick is the BM fundamental time unit t. Just as d

^{3}, where d is the BM fundamental length unit, is the

**absolute minimum volume**of a single bit locus, d

^{2}is the

**absolute minimum possible area**. Hence, BM predicts that calculations assuming smaller volumes or areas -- typical in calculus used with assumed continuous space-time, are likely to produce distorted results. This error factor is compounded with assumption of point-like (0-dimensional) particles, which violates the BM principle of absolute minimum volume.

Considering mite motion in the unconditional bit operation tick, each 1-state mite moves to the lite locus in a spot unit. The resulting 1-state lite is equivalent to a vector potential component of the magnetic field. If the mite is 0-state, so will be the resulting lite bit.

In short, the proportionality is perfect, in accordance with Ampere's law. Further, as understood by the author, Ampere was correct to posit that motion of the mite bit was a cause of a magnetic (vector) potential.

However, with quantized space-time in BM, the geometry is somewhat different than might be assumed under the regimen of continuous space-time. Hence, an appropriate homework assignment may be to show that this single-bit analysis provides equivalent results when averaged over a large number of spot units in a larger volume. In this context, it may be relevant that the resulting vector potential above, may accelerate a mite in the countercurrent spot unit in exactly the opposite direction. In this acceleration, the countercurrent mite motion is said to be induced by the magnetic (vector) force.

**Coulomb's Law**

The electrostatic (so-called scalar) potential is the presence of a mite with a like-signed charge in an adjacent concurrent spot unit (e.g., X2, Y0, Z0 not shown in Fig. 1). The BM scalar force is the product of the two concurrent mite states. If both are 1-state mites, the force evaluates to one. At this single-bit level of analysis, the distance r in Coulomb's law is always one BM distance unit d. Hence, Coulomb's law that the force F is proportional to kq

_{1}q

_{2}/r

^{2}, where q

_{1}and q

_{2}are signed charge values, k is Coulomb's constant and r

^{2}is one, is exactly the same as the scalar force at the single mite bit level, where k is also one. Namely, in both Coulomb's law and its BM equivalent, if the scalar potential (q

_{1}or q

_{2}) is zero, the force is zero.

Since adjacent concurrent mite bits are always like-signed, the effect of the scalar bit operation is to move like-signed mites away from each other. Note that mite bit motion due to the scalar bit operation is exactly in the direction of an opposite signed mite bit locus (Fig. 1). These two effects appear to correspond respectively to the rules that like-signed charges repel each other and opposite-signed charges attract each other.

However, again, the geometry is different with quantized space-time. Since the electrostatic (scalar) bit operation has an effect over only one BM distance unit -- that is, it is as local as it can be in BM space, the r

^{2}term in Coulomb's law must represent the average propagation of EM radiation resulting from these bit operations over greater distances, which provides another problem to solve as a homework assignment.

For this problem, consider that each spot unit may contain 1-state bits which are by definition scalar or vector potentials that can influence bit motion in exactly two other spot units -- the concurrent one for the scalar potential and the countercurrent one for the vector potential. That is, the bit states in the three spot units in a spot influence the behavior of bits in six adjacent spot units and so forth. The r

^{2}proportionality factor in Coulomb's law has provided an excellect approximation of the extent to which these effects propagate and average over large numbers of spots.

**Further Issues**

1. The electrostatic (scalar) and magnetic (vector) forces are simply summed in the Lorentz Force law. In contrast, the BM intra-dimensional bit operations -- unconditional, scalar and vector, do not commute and are thought to occur in a sequence of one tick each [5]. Hence, if the application of these bit operations is in fact sequential, then only one sequence can be correct. Further, when averaged on a macroscopic scale, the correct microscopic sequence of these time-development operations would be expected to converge on the Lorentz Force law which states that the EM forces occur simultaneously.

On the other hand, the correct sequence of the intra-dimensional bit opertions might reveal some new physics as experimentalists explore phenomena at increasingly microscopic levels where effects of the hypothesized sequential application of the forces may be observed. In addition, new phenomena may be observed due to the sequential pulsed nature of the single-bit binary plane wave as it propagates (Fig. 2).

2. Whether applied simultaneously or sequentially, the EM bit operations do not sum exactly at higher energies (bit densities). Consider a 1-state mite bit subject to both a scalar and vector potential. It can only be accelerated once to the lite locus in its spot unit in a BM bit operations cycle of 4 ticks -- the unconditional, scalar, vector and strong operators. Hence, the possible forces of the two potentials can result in only one mite motion, as evaluated by the resulting kinetic energy. In such a situation, the forces do not sum, an effect which might well be observable experimentally if sufficiently high energy densities can be achieved. In other words, experimental confirmations of SR may be viewed as supporting BM postulates at the microscopic, single bit level.

3. From SR by Einstein, the observed amplitudes of the electrostatic and magnetic components of the EM four-potential may vary as a function of a moving reference frame of the observer. BM was originally based on the relativistic quantum mechanical Dirac equation and as such has no specific quarrel with the dependence of measured values on the observer's reference frame.

While different observers may assess different amplitudes of the EM potentials as distributed between scalar (electrostatic) and vector (magnetic) components, it is not plausible to suppose that the postulated spot unit mechanisms underlying physical phenomena behave differently merely because of the presence of one or multiple observers, regardless of their reference frame. In this context SR may have more relevance to experimental methodology than to the physical mechanisms of the phenomena observed.

Consistent with SR, BM may explain, at least in part, why observed velocities may not exceed the speed of light in vacuum [6]. First, as described in item #2 above, at higher bit densities, BM predicts that the electrostatic and magnetic components may sum to a force less than predicted by the Lorentz Force law. Second, even if a scalar and vector force on a 1-state mite both evaluate to one, the mite cannot move to its spot unit lite locus if that locus is already in the 1-state, per the postulate that any bit locus can contain only one 1-state bit. This

**force-motion dissociation**between an EM force and whether a 1-state mite moves as a result, is more likely at higher energies (bit densities) and velocities, acting to limit maximum observed velocity. In short, BM may provide systematic descriptions of mechanisms underlying SR, starting with the simple postulate that each instance of absolute minimum volume may have only one of two states - 0 or 1.

**References**

[1] Keene, J. J. "Binary mechanics" July, 2010.

[2] Keene, J. J. "Electromagnetic bit operations revised" March, 2011.

[3] Keene, J. J. "Physical interpretation of binary mechanical space" February, 2011.

[4] Keene, J. J. "Superconductivity in binary mechanics" March, 2011.

[5] Keene, J. J. "Bit operations order" May, 2011.

[6] Keene, J. J. "Electron acceleration and quantized velocity" April, 2011.

© 2011 James J Keene