This report presents functions to scale physical measurements of length to BM fundamental distance units and inversely, to project distance measurements in BM space to experimental measurements in meters. In a possible milestone for the theory of BM,

**these scaling and inverse projection functions may absorb no less than two fundamental physical constants**.

**Space-Time Calibration.**The present working hypothesis is that

**some physical constants pertain to the scaling or calibration between space-time as reckoned in experiment and in BM**. The BM length unit

*d*and time unit for a single tick

*t*may be expressed as functions

d = f

_{length}(d'); t = f

_{time}(t') [Eqs. 1]

with d and d' in meters, t and t' in seconds, where d' and t' are experimental measurements and d and t are multiples of BM length

*d*and time

*t*units respectively.

An example may clarify these symbols and functions.

First, a classical calculation of electron angular momentum or spin requires an estimate of electron radius in BM length units

*d*. Consider that the geometric center of an electron spot is the point which is the shared vertex of the three spot unit mite sub-cubes of dimension

*d*(Fig. 1 in [2]). Assume the center of each mite sub-cube may be used to estimate electron radius. Then, the estimated radius is the distance from the shared vertex at spot center to mite sub-cube center {0.5, 0.5, 0.5} in BM length units

*d*. Hence, the distance is sqr(0.5

^{2}+0.5

^{2}+0.5

^{2}) = sqr(0.75) = 0.866

*d*in BM length units (Table 1).

Second, choose a preferred estimate of electron radius d' based on physics theory or experiment. Apply f

_{length}in Eqs. 1. If all is correct, the result d should be 0.866.

**Space-Time Projection.**Functions f

_{length}and f

_{time}in Eqs. 1 must perform a one-to-one mapping and hence, are invertible. Let g be the inverse functions (g = f

^{-1})

d' = g

_{length}(d); t' = g

_{time}(t) [Eqs. 2]

Eqs. 2 assert definite relations -- the g functions -- between computed BM distance and time units and physical measurements. However, these functions are obviously not simply the identity one. That is, if electron and proton radii are tabulated by consistent methods in BM space, the radii ratio is not in agreement with most estimates based on current physics theory or experiment. This apparent disparity must be be resolved by projection function g

_{length}where the metric of BM space is projected or mapped into the space of our physical experience and experimental measurements.

Consider that the physical distance between bits in RAM memory in a computing system may have no particular relation to the spatial arrangement of pixels in an image on the system's display device. In contrast, Eqs. 1 and 2 define a definite mapping between the two spaces -- BM space-time and physical measurements of length and time.

The present task is to incorporate as many fundamental physics constants as possible into the calibration and projection functions f and g respectively.

**Methods and Results**

**Test Data Set.**Data points from physical measurements may include approximate Compton nucleon wavelength 1.32E-15 meters [3] and a purported upper limit for electron radius of about 1.0E-22 meters [4]. The Bohr radius 5.29E-11 for a ground state hydrogen atom will be used as a third point (d'_Ex in meters column in Table 1).

**Table 1: Length Scaling/Projection between Experimental (d'_Ex) and BM (d_BM) Spaces**The d_BM column in Table 1 uses the electron r = 0.866 calculated above. The nucleon r was approximated by the radius of a sphere inside the cube spanned by the seven baryon bit loops described previously [5]. This cube is 3x3x3 spots. Each spot is 2

Length d_BM d'_Ex r_e units Scaled d_BM Projected d'_Ex

Electron r 0.866 1.00E-22 1.00E+00 8.68E-01 9.87E-23

Nucleon r 3.000 1.31E-15 1.31E+07 3.00E+00 1.35E-15

Bohr r 4.370 5.29E-11 5.29E+11 4.37E+00 5.18E-11

*d*where

*d*is the BM distance unit. Hence, the sphere diameter is 6

*d*with radius 3

*d*. The d_BM value for the Bohr r for the hydrogen atom is an estimate based on two facts. First, the electron spot within the proton spot cube is thought to function to form neutrons, not hydrogen atoms. Second, there is no theoretical need to waste BM space. Thus, the electron spot required to bind an electron to a proton to form ground-state hydrogen is most probably in a spot cube which is a close neighbor to the spot cube occupied by the proton. This hydrogen Bohr radius was therefore set to 4.370 to include the proton spot cube and an electron in an neighboring spot cube.

Next, in the r_e column of Table 1, the d'_Ex estimates in meters are expressed in electron radius units, where each value is the estimated radius of the object (d'_Ex) divided by the nominal electron radius used in this exercise (Eq. 3). The r_e units, Scaled d_BM and Projected d'_Ex columns of Table 1 were calculated using real10 (10-byte floating number) precision although usually four most significant digits are used in the following text.

**Length Calibration Function f.**Eq. 3 tries the natural logarithm ln() of the r_e ratios for a f

_{length}calibration function.

d_BM = a * ln(r_e) + b [Eq. 3]

Fig. 1 shows a nearly linear relation between the BM length estimates d_BM and experimental results for electron, nucleon and hydrogen radius estimates d'_Ex (Table 1) as expressed in ln(r_e) (Eq. 3).

**Fig. 1: ln(r_e) vs BM distance units**

Using linear regression, the a and b constants in Eq. 3 may be replaced with the resulting numerical estimates in Eg. 4, which may be used to estimate length in BM space based on physical measurements in our experimental space and experience (Scaled d_BM in Table 1).

d_BM = 0.1300 * ln(r_e) + 0.8678 [Eq. 4]

r_e in Eq. 4 and Table 1 is a unitless ratio. Using the original values, Eq. 4 becomes

d_BM = 0.1300 * ln(d'_Ex/d'_r_e) + 0.8678 [Eq. 5]

Eq. 5 is an example of the f

_{length}function, where d_BM is d and d'_Ex is d' in Eqs. 1 and d'_r_e is approximate electron radius determined experimentally, scaling or calibrating experimental measures to their equivalent values in BM space.

**Inverse Length Projection Function g.**The inverse function g

_{length}in Eqs. 2 is expected to project length measurements in BM distance units to corresponding experimental length measurements. Solving for d'_Ex, the only intended variable in Eq. 5, the product operand of the natural logarithm may be expressed as the sum of ln() values:

d_BM = 0.1300 * (ln(d'_Ex) + ln(1/d'_r_e)) + 0.8678 [Eq. 6]

Substituting values in Eq. 6 with their constants and rearranging,

d_BM = 0.1300 * (ln(d'_Ex) + ln(1/d'_r_e)) + 0.8678

d_BM = 0.1300 * (ln(d'_Ex) + 50.657) + 0.8678

d_BM = 0.1300 * ln(d'_Ex) + 6.577 + 0.8678

d_BM = 0.1300 * ln(d'_Ex) + 7.445

(d_BM - 7.445)/0.1300 = ln(d'_Ex) [Eqs. 7]

Use of the exponential function completes an expression for experimental measurements predicted by length results in BM space,

d'_Ex = e

^{((d_BM - 7.445)/0.1300)}[Eq. 8]

which is a prototype for g

_{length}in Eqs. 2. The Projected d'_Ex values using Eq. 8 based on the d_BM estimates were in good agreement with the measured d'_Ex values (Table 1).

**Discussion**

**Conversion functions between BM space and physical experiment.**This report presents functions to scale physical measurements of length to BM fundamental distance units (Eq. 4) and inversely, to project distance measurements in BM space to experimental measurements in meters (Eq 8). These scaling and projection functions have both practical and theoretical significance.

First, an important trend in physics research is investigation of the very small, such as trapped single particles or atoms and the whole field of nanotechnology. If the reported conversion functions between BM space and experimental measurement are correct, investigators can now simulate phenomena in BM space to study its exact time-development and project results into familiar lengths as measured in physical experiments. In addition, experimental results can be length scaled to study their underlying mechanisms using BM simulation software [6]. In short, BM may provide a powerful new tool in physics research.

Second, concerning the theoretical significance, the proposed conversion functions may absorb at least two physics constants represented by the two numerical constants in Eqs. 4 and 8. This finding suggests that some physical constants may pertain to how the underlying mechanisms for physical phenomena as described by BM are projected into our world as science is able to measure and understand it.

This situation is as if science studied the screen image of a 3D computer game. With the fantastic mental powers of humans, no doubt a science with trust-worthy physical constants would be created based on examination of the changing screen images. And that science would be a marvelous feat, considering that the sensory systems and measurement devices of the scientists consist of the same sort of pixels as occur on the screen. The reader knows the ending of this story, namely that the mechanisms underlying the impressive screen images are to be found in the computing system that produces them.

**Methodological Issues.**The major point of the present article is that accurate conversion functions are feasible between BM space and space as conventionally known and measured in physics, enabling use of BM as a research tool. However, the exercise presented does have weaknesses.

For example, a reported upper limit to electron radius was used as the radius, which could introduce serious error in the calculations presented. Further, all the radii values may be considered as somewhat variable or subject to change under varying conditions, both for the BM (d_BM) and experimental/theoretical (d'_Ex) values. For the BM radii values, the number of bits in the respective bit cycles would be expected to change the effective radius of an object such as an electron or proton according to the principles of BM. This sort of variation may correspond to the perceived fuzzy nature of particle wave functions based on quantum mechanical conceptions, as measured by collider particle scattering cross sections.

Nonetheless, considering such possible error factors, the ability to calculate quite accurate estimates of experimentally determined lengths based

**only**on BM postulates and two numeric constants which may represent selected physics constants is impressive (Projected d'_Ex in Table 1). Likewise, the inverse process produced good estimates of BM distances based

**only**on experimental measurements and the equivalent two numeric constants which may represent the same selected physics constants (Scaled d_BM in Table 1).

The accuracy and physical interpretation of the numeric constants in the space mapping Eqs. 4 and 8 may be examined from every angle as investigators explore alternative, but mathematically equivalent, forms for the expressions and apply the conversion functions to different or more accurate data sets.

**The BM length constant**The present preliminary data suggest a value for

*d*.*d*, using r_e values where d_BM = approx. 0.866 and d'_Ex = approx. 1.00E-22. Hence,

*d*= 1.00E-22/0.866 = approx. 1.15E-22 m [Eq. 9]

One implication is that the mechanisms producing physical phenomena as described by BM appear to occupy less physical space than might be presently thought based on experimental length measurements. This is like saying the computing mechanisms producing screen images can be much smaller than their screen displays, a fact which can be easily determined by anybody who has taken apart a flat screen display device or DVD player and found the entire circuit board area to be a small fraction of screen area.

**References**

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

[2] Keene, J. J. "Physical interpretation of binary mechanical space" J. Bin. Mech. February, 2011.

[3] NIST. "Fundamental physics constants -- complete list" 2010.

[4] Demhelt, Hans. "A single atomic particle forever floating at rest in free space: new value for electron radius" January, 1988.

[5] Keene, J. J. "Binary mechanics electron, positron and proton" J. Bin. Mech. July, 2010.

[6] Keene, J. J. "Binary mechanics simulator updated" J. Bin. Mech. March, 2011.

© 2011 James J Keene