The Case for Whole PI and Alternative Equations for Space,
Mass, and the Periodic Table
W. C. Byrdwell
USDA, ARS, Beltsville Human Nutrition Research Center, Food Composition and Methods Development
Lab, 10300 Baltimore Ave., Beltsville, MD, USA. E-mail: firstname.lastname@example.org
A new series of equations for space, mass and the Periodic Table based on a common pattern is
presented. Three equations for circular space represent the circumference (C), area (A), and volume (V)
of a circle or sphere, which are mathematically equivalent to the conventional equations, specifically,
C=2r, A=r2, and V=4/3r3. The new equations incorporate a new understanding of pi, referred to as
Whole PI to distinguish it from the classic understanding. A new symbol for Whole PI, B, is presented
and explained. Using Whole PI, the equations for the dimensions of space become 2Bdp/2p for the first
dimension and Bdp/2p for the others. It is further shown that the second mass, helium, stands in relation to
the first mass, hydrogen, the same as the second dimension of space stands in relation to the first
dimension of space, specifically, H=2mp/2p and He=mp/2p, in which m equals the integer unit mass
(m=1), the power signifies the atomic number (and therefore the number of electrons), and the
denominator signifies the integer mass of the atom. Because of the similarity to the equations for
dimensions of space, the elements may be referred to as dimensions of mass. Using the new equations, it
is shown that the Periodic Table contains exactly ten dimensions of mass, and the other elements can be
considered deconstructions of the ten dimensions of mass. Finally, an approximation is introduced for the
accurate monoisotopic mass of hydrogen, specifically (B/3)1/6, which is 99.989% accurate to the observed
monoisotopic mass of hydrogen.
Keywords: pi; tau; phi; Unit Simulacrum; Whole PI; Periodic Table
This work arose from research into the use of liquid chromatography/mass spectrometry (LC-MS) to
describe structural characteristics of triacylglycerols (TAGs) in fats and oils. To facilitate understanding
of the ideas presented, it is helpful to retrace the evolution of the new concepts, reflected in the literature
trail, from a very specific and targeted LC-MS application, through an update of those concepts, followed
by generalization of the construct to be more widely applicable, then further generalization to
development of a new tool and universal function, all the way through to application of the new tool to
areas far outside the realm of LC-MS of TAGs.
In 2005, a construct called the Bottom Up Solution (BUS) to the Triacylglycerol Lipidome  was
developed to derive structural information, related to the nutritional value of TAGs, from LC-MS data
obtained on an instrument that employed atmospheric pressure chemical ionization mass spectrometry
(APCI-MS). The BUS used ratios of the abundances of protonated molecule ions, [MH]+, and
diacylglycerol-like fragment ions, [DAG]+, in APCI-MS mass spectra to determine structural
characteristics of TAGs. Three Critical Ratios were identified that provided the desired structural
information. The first Critical Ratio correlated the [MH]+/[DAG]+ ratio to the degree of unsaturation
(number of double bonds) in TAGs based on trends first reported by Byrdwell and Emken  in 1995.
This relationship was modelled using a sigmoid function . Polyunsaturated fats gave a high
[MH]+/[DAG]+ ratio, and saturated fats gave a [MH]+/[DAG]+ ratio of essentially zero. The second
Critical Ratio allows identification regioisomers of TAGs based on trends earlier reported by Mottram
and Evershed , and Laakso and Voutilainen . In other words, Critical Ratio 2 was used to identify
the fatty acid (FA) in the middle, or sn-2 position (using stereospecific numbering, sn). Knowledge of
regioisomers is important to human nutrition, since fat metabolism is regioselective, with preferential
removal of FAs in the sn-1 and sn-3 positions during digestion. Regarding the third Critical Ratio, for two
decades, no trends were reported for the [1,2-DAG]+ versus [2,3-DAG]+ fragments. However, the use of
Critical Ratios allowed new insights that made it possible for trends to be identified for the first time .
The factors primarily responsible for the abundances of the [1,2-DAG]+ and [2,3-DAG]+ fragment ions
were the degree of unsaturation and the grouping of unsaturated FAs either adjacent to each other or not.
Thus, the three Critical Ratios provided new information necessary for structural characterization and
quantification of TAGs by APCI-MS.
The great benefit of the Critical Ratios was that they also constituted a compact library of mass
spectra. Not only did they provide the structural information desired at face value, but since it took fewer
Critical Ratios to express the data than raw abundances in the mass spectra, they also represented a
compressed data set. When the ratios were processed through the BUS, the original mass spectrum could
be reproduced. Thus, the Critical Ratios also constituted a library of TAG mass spectra. The BUS from
APCI-MS was later generalized and updated, by noticing the similarities in the Case classifications and
simplifying them, to produce the Updated Bottom Up Solution (UBUS) . Next, the construct was
further generalized to apply equally well to atmospheric pressure photoionization (APPI) MS and
electrospray ionization (ESI) MS of TAGs .
As the name states, and as described above, the Bottom Up Solution  and the Updated Bottom Up
Solution  were developed from the bottom up, based on the foundation of Critical Ratios, and built up
from those to allow the original mass spectra to be reproduced from the ratios. Once the BUS and UBUS
were constructed in their entireties, the pattern behind these constructs could be seen and elucidated. It
was found that for every ratio that was constructed, there existed the inverse ratio that could have been
constructed, but was not. However, there are circumstances in which the inverse ratios might be more
desirable than the ratio that was constructed. For instance, in ESI-MS (in contrast to APCI-MS), TAGs
sometimes give only an ammonium or other adduct ions, with no [DAG]+ fragments (unless some up-
front collision induced dissociation (CID) energy is provided). In such cases, it would not be appropriate
to use the [MH]+/[DAG]+ ratio as Critical Ratio 1, since this could lead to division by zero, and an
irrational value. In such cases, the [DAG]+/[MH]+ would be the preferred Critical Ratio, so that the
construct would remain bounded and rational, but still provide the desired structural information and still
represent a compressed data set and compact library of mass spectra. These factors have been discussed in
detail elsewhere , but the important point is that it was desirable to have a “top-down” solution that
provided all options, whether selected or not selected, constructed or not constructed, real or unreal,
rational or irrational, so that the best alternative for a particular MS application could be selected.
Therefore, the BUS and UBUS were further generalized to produce the top-down solution that contained
all possibilities of ratios and their inverses, known as the Simulacrum System (SS) for mass spectrometry
of triacylglycerols .
A simulacrum is a construct that expresses the sum of two values (e.g. MS abundances) as a value
and a ratio, and when one value is 1 (a requirement of MS), the solution simplifies to depend only on the
ratio. When ratios are judiciously constructed (i.e. Critical Ratios) and processed in simulacrum solutions
that are nested one, two or three levels deep, they provide structural information about TAGs and also
produce compressed data sets like those described in the BUS  and UBUS [3, 6].
Note that in MS, ion abundances are usually expressed as percent relative abundances, and that
percent means “per hundred”, so 100% (100 per hundred) expressed as a pure ratio is 1. In mass spectra
expressed as percent relative abundance, one peak, the base peak, is assigned a value of 100% (=1), and
no ion can be greater than 100%. A simulacrum in which one value is 1 is called a Unit Simulacrum (US).
Interestingly, every simulacrum solution contains a Unit Simulacrum inside the parentheses. Thus, a Unit
Simulacrum is a fundamental component of all simulacra.
The process of identifying the “unit” in mass spectra over and over again, and the process of
generalizing from the Bottom Up Solution to the top-down Simulacrum System, led to further
generalization of the definition of any “unit”. One characteristic of the Simulacrum System was that it is a
function that applies not only to mass spectrometry of TAGs and other molecules, but also it is a universal
function that applies to any number, letter, name, symbol, emoji, scribble, or any other designation that
can be written in physical form. Therefore, for this report, we want to consider the nature of the “unit(s)”
associated with the symbol for, and meaning of, a “unit pi”, , as well as a “unit circle”, a “unit radius”,
and a “unit diameter”. Specifically, it is worthwhile to consider what is/are the “unit(s)” in the equations
commonly used for circular dimensions of space.
A dictionary definition of pi is the ratio of the circumference of a circle to its diameter . From this
comes the mathematical equality for the circumference of a circle, C = ·d. For a unit circle based on a
unit diameter, the unit circle circumference is C = ·d1.
There is another definition for a circle, based on its radius, which is commonly used. The
circumference of a circle based on radius is C = 2·r. For a unit circle based on a unit radius, the unit
circle circumference is C = 2·r1.
The two equations above show that there are two definitions for a unit circle: one based on diameter
and one based on radius. Thus, for a unit circle, C = ·d1 = 2·r1, or C = ·1d = 2·1r. Of course, whatever
the unit diameter is, a unit radius is ½ of a unit diameter based on the same units, since d=2r and r=d/2. In
the Unit Simulacrum solutions for mass spectrometry, the goal was always to classify the Critical Ratio to
determine which value was “carrying the unit”, so the “1” on the outside of the parentheses could be
cancelled out and ignored, leaving the simplified simulacrum solution. In the above unit circle equations,
the “1” can never be cancelled out and ignored, otherwise we arrive at the contradiction, or mathematical
paradox, that = 2, or 3.14159… = 6.283185…. The unit circle has two different definitions, one based
on “r” and one based on “d”. The name of the defining unit must accompany , otherwise one cannot
determine the unit that the circle is based on. See Supplemental Figure 1 for an explanatory analogy.
Another way to look at it is to consider the value of circumference = 10 (or any other specific
multiple of ). From the designation 10 alone, it is impossible to distinguish whether this is a circle
based on radii (radius of 5) or a circle based on a diameter (diameter of 10). Both are equal and true.
There are two possible definitions for the same symbolic value, 10.
We can look at this another way. The above discussion was about a unit circle, not a unit . For a
circle based on radius, the circumference is 2, so a single unit is half of a circle. In contrast, for a circle
based on diameter, the circumference is , so a single unit is a whole circle. Thus, a single unit
represents either a half a circle or a whole circle, depending on whether the defining unit is a radius or a
diameter. In other words, the symbol has two different unit definitions. The fact that one single symbol,
, represents two different entities, either half of a circle or a whole circle, and that without the defining
unit always accompanying we can arrive at the mathematical paradox 2 = , can be called the “pi
Of course, this paradox also plays out in calculations for the second dimension (r·r or d·d) and third
dimension (r·r·r or d·d·d). The equations based on radius are the well-known ·r2 and 4/3·r3. The
equations based on diameter are the less well-known ·d2/4 and ·d3/6. If one were to see only the value
for area of A = , one could not determine from this information alone whether it was the area of a circle
based on radius, with a radius of 1, or the area of a circle based on diameter, with a diameter of 2. Both
are mathematically equal, but represent different circles based on different units. Nevertheless, because of
the pi symbol paradox, these two cannot be differentiated without being explicitly told what the defining
unit is. Similarly, if one were to see only the value for volume, V = 36, one could not determine from
this information alone whether it was the volume of a sphere based on radius, with a radius of 3, or the
volume of a sphere based on diameter, with a diameter of 6. Both are mathematically equal, and give a
volume of 36, but represent different spheres based on different units. Because of the pi symbol paradox,
two different spheres based on different units give the same symbolic value for a volume. This report
discusses this issue in greater detail below, and presents an unambiguous solution to the pi symbol
Before proceeding, it is important to mention that for more than a decade there has been a movement
underway to implement a new definition, called , which represents a complete circle in terms of radians,
or = 2 = 6.283185…, which is the ratio of a circle’s circumference to its radius. As discussed in
Scientific American : “The crux of the argument is that pi is a ratio comparing a circle’s circumference
with its diameter, which is not a quantity mathematicians generally care about. In fact, almost every
mathematical equation about circles is written in terms of r for radius. Tau is precisely the number that
connects a circumference to that quantity… At its heart, pi refers to a semicircle, whereas tau refers to the
circle in its entirety.” The fact that 2·r is really two times a semicircle was summarized in a 2013 feature
on the PBS Newshour in a quote by Mike Keith : “It’s like reaching your destination and saying you
are twice halfway there”.
While I was not aware of the proposal of and its base of support (self-proclaimed “tauists”) before
using the Unit Simulacrum to reconsider the nature of and the “unit” associated with it, I separately
came to a similar conclusion about the value of considering the whole circle, instead of only half. While I
have not adopted the opinion expressed elsewhere that “pi is wrong” , I did arrive at a new
formulation that may prove beneficial for modelling the dimensions of space, mass, and the Periodic
Table, to complement, not replace, the classic understanding of .
The Unit Simulacrum for , SimSum(1,), which equals (1+), where the mathematical sum is
expressed as a value and a ratio, is shown in Figure 1. In this example, the “unit” is a unit radius, r = 1, or
r1. As was done in the case of MS, we choose the solutions that have the 1 multiplying outside the
parentheses, so that it cancels out, leaving the simplified simulacrum solutions: 1+(/1) or 1+1/(1/). The
Unit Simulacrum for can be called the Pi Unit Simulacrum (PiUS). Since has a well-known value of
3.14159…, which is 1, the PiUS is always Case 2 (note that it was previously demonstrated  that the
Simulacrum provides all solutions, whether selected or unselected, observed or unobserved, real or
unreal, rational or irrational). As with the BUS and UBUS for MS of TAGs, the simplified Unit
Simulacrum solutions always end up being 1+ratio or 1+1/ratio.
Let us further examine the ratios (/1) and (1/) that appear in the PiUS solutions. As stated above,
the value that equals 1, the unit, is r1. When we draw the unit and , Figure 2A, we see that it has r1 and
half of a circle. Thus, while Fig. 2A does show a unit radius, and a unit (one) , it does not show a unit
(one) circle; it shows a ½ circle. Since the BUS and UBUS for MS of TAGs were based on judiciously
Simulacrum Sum (1, ) = SimSum(1, ) =
Possibilities to Observe:
Figure 1. The Unit Simulacrum (PiUS), where one value is specified to be 1 and the other value is
specified to be . This is Case 2 by default, since 1. A simulacrum is composed of the Simulacrum
Sum, four Possibilities to Observe, two Cases, and eight solutions.
π,1 ,1 π:1 Case
π,1 ,1 π:2 Case
selecting (constructing) Critical Ratios, we can ask the question of : “Is there another ratio we could
Obviously, from the discussion of above, the answer is yes. The proponents of advocate
constructing the ratio of the unit radius, r1, to the unit circle, 2. If 2 is substituted into Fig.1 for , and
the solutions that contain the unit radius, r1, outside the parentheses are chosen as before, to allow the 1 to
cancel out and simplify the solutions, then the simulacrum solutions for SimSum(1,2) = (1+2)
become 1+(2/1) or 1+1/(1/2), taken from Case 2, since 2 1. Again, we can graphically visualize the
two ratios (2/1) and (1/2) as shown in Fig. 2B. The figure based on the constructed ratio 2 has the
advantage that it now shows a unit radius and a unit circle, but it does not show unit . Instead, it shows
2, which makes the unit circle. In this case, the circumference of the unit circle is 2·r1, or just 2, which
If instead of using 2, we substitute into the Unit Simulacrum for in Fig. 1, and again allow the
unit radius outside the parentheses to cancel out, we obtain the two Case 2 (since 1) solutions based
on the two ratios that were possible to observe. The two simulacrum solutions to the SimSum(1,) =
(1+) are 1+(/1) and 1+1/(1/). These ratios are also visualized in Fig. 2B. The figure based on
constructing the ratio of to r1 now has the benefits of showing a unit radius, a unit (one) , and a unit
circle. In this case, the circumference of the unit circle is ·r1, or simply , which equals 6.283185…
In Figure 3 we also have a unit circle, but now a unit diameter, d1, instead of a unit radius, r1. The
circumference of this unit circle is ·d1, or simply . Comparing Fig. 2 to Fig. 3 inescapably brings us to
the “pi symbol paradox”, which is that the symbol for pi has two different unit definitions based on two
different unit circles. Based on a unit radius, the unit circle is 2·r1. Based on a unit diameter, the unit
circle is ·d1. Thus, the same one whole unit circle is both 2·r1 and ·d1. Therefore, the units, r1 or d1,
can never be cancelled out or ignored, otherwise we obtain 2 = , or 6.283185… = 3.14159265…
Figure 2. Unit radius definitions. (A) Graphical representation of the relationship between a unit radius (r
= 1), r1, and , showing the ratios of to r1 and r1 to . (B) Graphical representation of the relationship
between a whole circle based on radii and a unit radius, r1, showing the ratios of 2 to r1 and r1 to 2, as
well as to r1 and r1 to . The whole circle in (B) has a circumference of 2, or 6.283185…, or .
Figure 3. Definition of based on diameter. Graphical representation of the relationship between a unit
diameter (d = 1), d1, and , showing the ratios of to d1 and d1 to . The whole circle has a circumference
of , or 3.14159265…
Thus, unless the “r” or the “d” always accompanies , it cannot be known with certainty which whole
circle refers to.
To solve the pi symbol paradox, a new symbol for pi was developed, based on a whole unit diameter
and a whole unit circle, which is called Whole PI. Figure 4 shows the components of the new symbol for
Whole PI. Since the symbol is intended to represent a whole unit circle based on a diameter, the symbol
contains a whole circle, K, and a diameter, H. However, to emphasize that the symbol represents all
three dimensions of space, 3-D, the symbol includes three diameters representing the three dimensions,
J, Fig. 4A. Furthermore, to make the symbol two-dimensional so it can easily be written, the diameters
are arranged in a ratio of 1:2, which also reflects the relationship between radii and diameters, which is
1:2, Fig. 4B. Thus, the symbol for Whole PI is B, Fig. 4C, and this clearly and unambiguously
differentiates it from the original, or classic pi, which can now be used exclusively for representing the
whole unit circle, 2, defined based on a unit radius. Having two symbols for the two different circles
based on two different unit definitions eliminates all uncertainty and solves the pi symbol paradox,
because the symbol itself carries an indication of the defining unit.
If we construct the Unit Simulacrum for B, we can select the two simplified simulacrum solutions,
1+(B/1) and 1+1/(1/B). The ratios (B/1) and (1/B) used in these solutions are represented in Figure 5.
From this figure, it is now perfectly clear that the unit B, based on the unit d1, cannot be confused with
from Fig. 2 or Fig. 3.
2.1. Dimensions of Space
Using Whole PI for the equations given in the introduction for the three dimensions based on
diameter, we have Bd, Bd2/4, and Bd3/6. It is easy to see from the second and third dimensions that they
obey a common pattern, which is Bdp/2p, shown in Figure 6B and Fig. 6C. In this pattern we can see that
the power is the dimension number, which is the same as the number of diameters in that dimension. The
denominator, 2p, reflects the numbers of radii in the area and volume, which are 4 and 6, respectively.
Figure 4. Construction of the new symbol for Whole PI. (A) It contains three diameters, J, representing
the three dimensions of space, and contains a whole unit diameter circle, K. (B) The ratio of the first
diameter to the two others, 1/2, is the same as the ratio of a radius to a diameter, providing self-similarity.
(C) The symbol for Whole PI is two-dimensional, B, (D) but the third dimension can be envisioned from
Figure 5. Unit diameter Whole PI circle. Graphical representation of the relationship between a unit
diameter (d = 1), d1, and C, showing the ratios of C to d1 and d1 to C. The whole circle has a
circumference of C, or 3.14159265…
The first dimension is unique. If we construct the equation in the same form as the second and third
dimensions, to raise “d” to the power of “p” and then show 2p in the denominator to reflect the number of
radii in the first dimension, we find that the equation for the first dimension has an additional “2” in the
numerator, to give 2Bdp/2p. We can think of the first of anything as being unique because it serves two
purposes: it defines a new category, and also represents the first unit in that category. Every other
occurrence of the defined thing is another unit in that category. Pragmatically, we know that the first of
anything is different and special, such as a first literature report, first car, first job, first house, etc. Thus, it
is not entirely unprecedented or unexpected that the First Dimension has a “2” in the numerator that
makes it unique, while the rest of the equation follows the same pattern as the other dimensions. Together,
we have the equations for the three dimensions as 2Bdp/2p, Bdp/2p, and Bdp/2p for the first, second, and
third dimensions of space. These simplify to being one equation for the first dimension, 2Bdp/2p, and a
slightly different single equation for the other dimensions, Bdp/2p.
As an example, we can consider a circle with a diameter of 12 units (so a radius of 6 units): the
circumference is C = 2Bdp/2p = 2B(12)1/2*1 = 12B. Or using the classic approach: C = 2r = 2(6) =
12. These two circumferences, 12B and 12 are mathematically equal, but can now be clearly
differentiated, and there is no ambiguity that 12B is the circumference of the circle based on diameter,
with d = 12; and 12 is the circumference of the circle based on radius, with r = 6. The uncertainty
discussed in the Introduction is eliminated, and the pi symbol paradox is unambiguously resolved.
Whereas having two unit definitions for classic pi led to the paradox 2 = , having two different symbols
for the two different defined circles leads to 2 = B, in which the two symbols carry the meaning that they
are based on two different “units”, but they both represent a whole unit circle.
Similarly, the area is given by A = Bdp/2p = B(12)2/2*2 = 36B using the new equations. This is
mathematically equal to, but clearly differentiated from the classic approach, which equals A = r2 =
(6)2 = 36. Thus, the two approaches give equivalent numerical solutions that are clearly
Figure 6. Whole PI equations for dimensions of space. (A) Graphical representation of, and equation for,
the relationship between any diameter, d, and Whole PI, C, for the First Dimension of space. (B)
Graphical representation of, and equation for, the relationship between any diameter, d, and Whole PI, C,
for the second dimension of space. (C) Graphical representation of, and equation for, the relationship
between any diameter, d, and Whole PI, C, for the third dimension of space. Notice that the number in the
denominator (2p) represents the number of radii in each dimension. The First Dimension is unique, and
has a two in the numerator, 2Bdp/2p. The equations for the other dimensions are identical, Bdp/2p,
differentiated only by the value of p, representing the dimension number, which also equals the number of
(A) (B) (C)
or or or
Likewise, the volume for the same circle is also given by V = Bdp/2p = B (12)3/2*3 = 288B using the
new equations. This is mathematically equal, but clearly differentiated from the classic approach, which
equals V = 4/3r3 = 4/3(6)3 = 288. Thus, the two approaches give equivalent numerical solutions that
are clearly distinguishable as one being based on diameter, the other being based on radius.
The alternative equations for dimensions of space based on Whole PI have several advantages and
convey more information at face value. First, it is advantageous that it is very easy to see the pattern
behind the equations 2Bdp/2p, Bdp/2p, and Bdp/2p, especially since the equations for the second and third
dimensions are identical. Second, the first dimension is designated as unique compared to all other
dimensions, by having a factor of 2 to distinguish it from the other dimensions. Third, all equations for
dimensions now reflect the duality that every diameter contains two radii, and the number of radii in each
dimension is explicitly given in the denominator of every equation, in the factor “2p”.
2.2. Dimensions of Mass
It soon became apparent that the second dimension of space stands in relation to the first dimension of
space in the same way that the second mass in the Periodic Table, helium, stands in relation to the first
mass, hydrogen. That is to say that an equation for hydrogen can be put in the same form as the equation
for the first dimension of space, by replacing “d” with “m”, shown in Figure 7, while an equation for
helium can be put in the same form as the second dimension of space by replacing “d” with “m”. Integer
masses are considered first, which do not require the use of Whole PI, while an approximation for
monoisotopic masses that incorporates B and/or is discussed further below. Using the same pattern as
the dimensions for space, the first mass is 2m1p/2p, while the second mass is mp/2p, where “m1” is a unit
integer mass, m1, representing a single proton or neutron, the “p” represents the atomic number, and the
denominator represents the total integer mass. Or more precisely, the total integer mass is the inverse of
the equations given below. Thus, 2m1/2 (= m1/1) and m2/4 are similar in form to 2Bd1/2 (= Bd1/1) and
Figure 7. Dimension model of mass. (A) Equation for the first unit mass, m, which contains the atomic
number as the power, p, and the integer mass as the denominator. (B) Equation for the second mass, m,
which contains the atomic number as the power and the integer mass as the denominator. (C) Equation for
the third mass, m, which contains the atomic number as the power and the integer mass as the
denominator. (D and E) Equations for the fourth and fifth masses, which are deconstructions. (F to H)
Equations for sixth, seventh, and eighth elements, which are the third, fourth, and fifth dimensions of
(F) (G) (H)
Based on the same form of equation for dimensions of space and mass, we could hypothesize that
masses that are in this form of equation may be considered “dimensions of mass”. In the same way that
the dimensions of space contain the number of subcomponents (radii) in the denominator of all equations,
the dimensions of mass contain the number of subcomponents (protons + neutrons) in the denominator of
all equations. Using the dimension form of equations, there are exactly ten dimensions of mass in the
Periodic Table, enumerated below.
The third mass, lithium, incorporates another principle discussed in the report of the Simulacrum
System for mass spectrometry. A Unit Simulacrum, based on 1+ratio, is the same as a ratio+1. Adding 1
to something can said to be incrementing that something. Conversely, adding 1 to the denominator of a
ratio, such as (1/1), can be said to be decrementing that something, as previously discussed . Thus, the
first increment of 1 is 1+1=2, and the first decrement of 1/1 is 1/(1+1) = ½. The first decrement is also
referred to as the first deconstruction. From the sections above, we can see that a radius, r, is the first
deconstruction of a diameter, d, or r = d/(1+1), r = d/2.
Since m1p/2p is analogous to a dimension of mass, just as Bdp/2p was a dimension of space, then the
first increment in the denominator is a deconstruction of that dimension of mass (again, considering
integer masses, with a factor incorporating B discussed further below). Lithium follows the form of
equation m1p/(2p+1), indicating that it is a deconstruction of the previous dimension of mass. In fact,
lithium, beryllium, and boron all follow the form m1p/(2p+1). The principles in Fig. 7 are followed
through the penultimate dimension of mass, sulfur, which is m1p/2p, where p=16, as follows:
Li2-1: m13/(2p+1), Be2-2: m14/(2p+1), B2-3: m15/(2p+1)
9-8: m117/(2p+1,3), Ar9-9: m118/(2p+4)
Since the power (exponent) is the atomic number, it includes both dimensions and deconstructions.
To differentiate the dimension number and number of deconstructions, a nomenclature can be adopted
that distinguishes the dimensions from the deconstructions, specifically 2 dash 1, 2-1, for lithium, which
specifies the dimension number followed by the deconstruction number. This nomenclature is shown as
the subscript for each element symbol in the list above. For example, using this nomenclature sulfur is the
9th dimension of mass and contains 7 deconstructions, 9-7, while calcium, element 20, is the 10th (and
final) dimension of mass and contains ten deconstructions, 10-10. The element number, the single number
for the power of the unit mass, can be replaced with the dimension-deconstruction notation, and the
atomic number is the sum of dimensions and deconstructions. But for this first report, the atomic number
is listed as a single value in the powers shown.
Chlorine9-8 and argon9-9 exhibit different patterns, while potassium9-10 is again a simple
deconstruction, m119/(2p+1). Chlorine is unique, since it has a molar mass of ~35½, being the isotope-
weighted average of m117/(2p+1) and m117/(2p+3), and is the first element to exhibit such a large second
isotope that differs by two masses from the monoisotopic mass (~1/3 of primary isotope). Argon is
unique, since it is the first anomaly in the periodic table, being the first element to have a mass larger than
the following element. Before addressing these elements, it is worthwhile to mention the shortcomings of
the simple model presented here.
While the simple model for dimensions of mass and deconstructions provides information about the
dimension number, deconstruction number, atomic numbers, and masses of elements, it does not
incorporate any factors related to several other physically observed phenomena, specifically: 1) isotope
distribution, 2) ferromagnetism, 3) radioactivity and others (metals, etc.). Because of these factors, the
simple model presented here is only a starting point for thinking of a fuller theoretical description of
atomic elements, not a final all-encompassing model. Of course, there is no other extant model that
presents masses in the same form as dimensions of space, and certainly not one that incorporates the
above factors, so the dimension model of mass (DMM) does have benefits, despite its shortcomings.
Chlorine is not a simple (2p+1) deconstruction, since it has two abundant isotopes, m117/(2p+1) and
m117/(2p+3) in a ratio of ~3:1. The isotope (2p+3) could be expressed as (2p+2f(x1)+1), where f(x1) is a
different function (with first value equal to 1), not the atomic number, but related to some other
phenomenon, such as those mentioned above (e.g. isotope distribution). Chlorine has a molar mass of
very close to 35 ½, represented by m17/(2p+1+(1/(1+1))). Thus, because chlorine is the first element to
exhibit such a substantial amount of a higher order of deconstruction than simply 1/(2p+1), it is not
included with the other simple 1/(2p+1) deconstructions.
Argon similarly exhibits a higher level of deconstruction, but now as its primary isotope, having an
equation of m118/(2p+4). The deconstruction 1/(2p+4) can be expressed either as: 1) 1/(2p+2f(x2)), where
f(x2) is the same function as mentioned for chlorine, but with a value of 2, or 2) 1/(2p+1+(2f(x1)+1)),
where 2f(x1)+1 is the same function of deconstruction as for chlorine, or 3) 1/(2p+1+(2f(y1)+1)), where
2f(y1)+1 is a different function of deconstruction related to some other characteristic or phenomenon
(ferromagnetism, radioactivity, etc.), with its first value equal to 1. For now, it is indeterminate which of
these possibilities (or another not listed) gives rise to the 1/(2p+4), so argon is listed in Figure 8 simply as
Figure 8. The Periodic Table of the Elements described using the dimensions of mass model, with the ten dimensions of mass in bold boxes. The power to
which the integer unit mass, m1, is raised represents the atomic number and the denominator represents the integer mass. When multiple isotopes are
abundant (>~20%), the major isotope is listed first, with the second most abundant in smaller text. Anomalous masses are marked with asterisks. In the
same way that r = d/(1+1) shows that the first increment in the denominator is a deconstruction, the values in the denominator indicate the level of
After argon, potassium is the last of the 1/(2p+1) deconstructions, being m119/(2p+1). Based on the
template equation for dimensions, m1p/2p, calcium is the tenth and final dimension of mass, with m120/2p.
All other elements after calcium are deconstructions of the tenth dimension of mass, which are shown in
Figure 8. The deconstructions become increasingly large, indicating increasing index values for f(x) and
the likely presence of other functions, such as f(y), both related to characteristics or phenomena that are
not incorporated into the simple initial model based only on dimensions of mass and first deconstructions,
1/(2p+1). The table in Fig. 8 was not extended past xenon since additional understanding of the
phenomena that lead to larger values of deconstruction (e.g., f(x1), f(y1),…) is needed. Nevertheless, the
dimension model of mass does provide new insight into a previously unreported pattern behind the
Periodic Table that reflects a similarity between mass and dimensions of space. Even if used only for
integer masses, the DMM provides new insights into the anomalies in the Periodic Table, as well as other
trends, discussed below.
It is possible to use the same pattern given for space and mass above, incorporating 2xp/2p, xp/2p, and
xp/(2p+1), to further extend the model to provide approximations for observed monoisotopic masses.
2.3. The 99.989% Solution
While the equations for mass and the periodic table above did follow the form of dimensions and
deconstructions and did provide information about the atomic number and integer mass (or the integer +
½ in the case of chlorine), they did not approximate the observed accurate monoisotopic masses. They
also did not include B or ; which is to say, they did not reflect circularity. The simplest whole number
ratio to B, raised to a simple power that produces the closest value to the accurate monoisotopic mass of
hydrogen is (B/3)1/6 (based on spreadsheet calculations). This constant has a value of 1.00771588137…,
which is 99.989% accurate to the monoisotopic mass of hydrogen, which is 1.00782503223(9) .
This represents a difference of -108.3 ppm. However, simply multiplying each mass (2pm, 2pm+1, etc.,
where pm is the dimension+deconstruction atomic number) times (B/3)1/6 does not provide a good enough
model for all other masses. A better model, given below, is found by nesting the exponent, 1/6, into the
same form of equations described for the first dimension, other dimensions, and deconstructions of mass,
described in Fig. 7.
The better model is found by nesting the exponent (1/6) into the general form of dimension and
deconstruction equations: 2xp/2p, xp/2p, and xp/(2p+1), where x = (1/6). The (1/6) may represent the three
dimensions of space in which mass operates, d1p/2p3 = 13/6, or other factor including both space and mass,
discussed further in the Electronic Supplementary Material. All possibilities proposed for the numerators
below give a factor that reflects both mass and the three dimensions of space in which mass operates,
resulting in a space-mass function. Then, the space-mass function is divided by the equation for each
integer mass equation, 2mp/2p, mp/2p, and mp/(2p+1), as given in Fig. 7 and Fig. 8, to yield the model
monoisotopic mass for each element. For instance, for the first mass, hydrogen = H1, the power, 1/6, is
incorporated into the first dimension of mass equation as follows:
where the extra 2 in the numerator of the power is because this is the first dimension of mass, pm=1, as
described above (see Fig. 7). Since all masses operate in three dimensions of space, ps always equals 3,
and (d1p/2ps) always equals 1/6. This expression shows the basic form of a hypothetical space-mass
function, and is 99.989% accurate to the observed monoisotopic mass of hydrogen. The second mass then
has the (1/6) incorporated into the dimension equation for the second dimension, and divided by the
equation for the second dimension of mass, as follows:
where pm = 2 for the second dimension of mass, helium. This model mass is 99.967% of the accurate
observed monoisotopic mass of helium, 4.002603254136 . Equation 2 is the DMM form of equation
used for all dimensions of mass from 2 through 10 (p=2, 6, 7, 8, 10, 12, 14, 16, 20). Analogously, masses
that are deconstructions of dimensions of mass in Fig. 7, up to the 10th dimension of mass, follow the
related DMM form of equation that includes (2pm+1) for deconstructions, instead of simply 2pm, (except
Ar9-9, which is 2pm+4, and only the first isotope of Cl9-8) as follows:
where pm = 3 for the third mass. The 2pm+1 form of equations applies to pm = 3, 4, 5, 9, 11, 13, 15, 17, 19.
The calculated masses from the 99.989% Solution and observed monoisotopic masses are shown in
Table 1, along with the approximation error values. The dimensions of mass give less error than the
corresponding deconstructions. Based on this model, the calculated monoisotopic mass for carbon is
12.00000098846…, which is 100.000008% of the observed monoisotopic value of 12.0000000000000,
for 0.1 ppm error.
The average error for the first 20 elements, representing the ten dimensions of mass, is 694.0 ppm,
with lithium and beryllium showing the greatest deviation.
As mentioned, different possible equations behind the term (B/3)1/6, as well as the equivalent term
(/3)1/6, are provided in Electronic Supplementary Material. However, since these cannot be definitively
differentiated, and the body of this report is sufficiently challenging, those details are not included here.
Nevertheless, the 99.989% Solution presents a new starting point for considering the pattern behind mass
and the Periodic Table, using a single form of equations, 2xp/2p, xp/2p, and xp/(2p+1), that are nested into
each other to produce approximate monoisotopic masses, in addition to integer masses.
Table 1. Dimension model for mass incorporating Whole PI, using the pattern from dimensions of space.
Element Number Dim.-Dec. Mass Acc. Mass (A.M.)a Calc. from Modelb % A. M. ppm
H 1 1-0 1 1.007825032239 1.0077158813690 99.989 -108.3
He 2 2-0 4 4.002603254136 4.0012812495224 99.967 -330.3
Li 3 2-1 7 7.016003436645 7.0002135106505 99.775 -2250.6
Be 4 2-2 9 9.01218306582 9.0000355846361 99.865 -1347.9
B 5 2-3 11 11.0093053645 11.0000059307626 99.916 -844.7
C 6 3-3 12 12.000000000000 12.0000009884602 100.000 0.1
N 7 4-3 14 14.003074004432 14.0000001647434 99.978 -219.5
O 8 5-3 16 15.9949146195717 16.0000000274572 100.032 317.9
F 9 5-4 19 18.9984031627392 19.0000000045762 100.008 84.1
Ne 10 6-4 20 19.992440176217 20.0000000007627 100.038 378.1
Na 11 6-5 23 22.989769282019 23.0000000001271 100.045 445.0
Mg 12 7-5 24 23.98504169714 24.0000000000212 100.062 623.7
Al 13 7-6 27 26.9815385311 27.0000000000035 100.068 684.2
Si 14 8-6 28 27.9769265346544 28.0000000000006 100.082 824.7
P 15 8-7 31 30.973761998427 31.0000000000001 100.085 847.1
S 16 9-7 32 31.972071174414 32.0000000000000 100.087 873.5
Cl 17 9-8 35 34.96885268237 35.0000000000000 100.089 890.7
Ar 18 9-9 40 39.962383123724 40.0000000000000 100.094 941.3
K 19 9-10 39 38.963706486449 39.0000000000000 100.093 931.5
Ca 20 10-10 40 39.96259086322 40.0000000000000 100.094 936.1
aMonoisotopic mass from NIST: http://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl?ele
bCalculated from Byrdwell model with (B/3)^(((1/6)^p)/2p) or (B/3)^(((1/6)^p)/(2p+1)), except
first dimension of mass, H, which uses (B/3)^((2*(1/6)^p)/2p), and Argon, (B/3)^((2*(1/6)^p)/(2p+4))
The results above come from a reassessment of the nature of the “unit” that is pi and how it relates to
the unit radius that is commonly used in equations for circumference, area, and volume that most people
learned as children (2r, r2, 4/3r3), and how those relate to a whole unit circle. Proponents of the use of
the symbol to represent 2 have long recognized that a single unit of is only half a circle, and that a
whole circle is 2. In this regard, is similar to Whole PI, because both of these represent a whole unit
circle. However, the important distinction between and B is that has a value that is numerically
different from , because it is still based on a unit r. B has the benefit that it is the exact same value as
that has been known for centuries, only the understanding of it is different. The figures herein attempt to
make it clear that a single symbol, , has been associated with two different units, a unit radius and a unit
diameter. Because of this, the unit radius or unit diameter had to accompany the symbol for , as in 2r1
or d1, otherwise a mathematical paradox could result, such that 2 = . The solution to the problem of
having one symbol for two different units is straightforward, though unconventional. By adopting the
classic symbol for pi to represent based on a unit radius, and Whole PI to represent B based on a unit
diameter, all ambiguity is eliminated, and the symbol alone clearly identifies the unit on which pi is
It is important to differentiate the goals of tauists from the reasons for introducing Whole PI as a
hypothesis. Tauists generally seek to simplify equations in physics and other areas by eliminating the “2”
that often accompanies , in the form of 2. Their goal is to eliminate extraneous 2’s that must be
followed through complex equations. In contrast, the reason for introducing Whole PI is not to simplify
the nomenclature, but to reveal patterns behind the dimensions of space. The purpose is exposition rather
than abbreviation. Thus, some tauists may take exception to the fact that the first dimension now has two
2’s in 2Bdp/2p. But examination of the equations above reveals that the 2’s come from different places
and have different meanings. The two in the denominator reflects the duality of all dimensions, while the
two in the numerator identifies the first dimension as unique. If the 2/2 is cancelled out, the indications of
duality and the first dimension disappear, and less meaning is conveyed in the simplified equation Bd,
and the common pattern behind all three dimensions is no longer evident.
In looking for a symbol for Whole PI, most Greek letters that contained a circle were already used for
other ideas and concepts, such as , which represents the Golden Ratio . Therefore it was appropriate
to develop a new symbol, and this provided the opportunity to reflect the meaning of the symbol with the
symbol itself. In fact, the symbol also represents the integer value of B, since the value of B is ~3, and the
symbol contains three diameters, J. Furthermore, the minimum projection of the symbol is one diameter,
the defining unit, when viewed from above. The minimum projection of the symbol viewed from the side
is two diameters, and when viewed from the front it also exhibits the 2:1 ratio that reflects the fact that
each diameter is composed of two radii. The inherent duality of the dimensions is reflected in the
equations that use B, since the “2p” in the denominator of every dimension calculates the number of radii
present in each dimension. Also, the six radii of three-dimensional space can be seen in the symbol B,
because of the way the diameters intersect each other and the circle.
A major hurdle to an entirely new symbol is that it is not included in existing fonts. However, these
days producing a new font is straightforward and not very time-consuming, as demonstrated by the Whole
PI TrueType font used throughout this report, which took only a couple of hours to produce in a form that
is recognized by most programs commonly used. This font is freely available in the electronic
supplementary materials and will be updated over time at:
Another convention has been used here to differentiate and B. The name for can appear as pi or
Pi, depending on whether it is at the beginning of a sentence or not, or in a title. Therefore, the name for B
has been used as all-capital PI, so that it can be distinguished from pi without having to explicitly state
Whole PI. Thus, pi and PI have the same spelling and the same numerical value, but a different
understanding, in the same way that the symbols have the same value but reflect different definitions of
the underlying unit.
Based on the new understanding of PI, new equations were developed that make the pattern behind
the equations very clear. In contrast, it is not easy to see that pattern behind the equations that most of us
learned as children. 4/3r3? Where did the 4/3 come from? Why? If one learned integral calculus, it
became possible to see where this came from, but it is not obvious to the vast majority of people who did
not learn calculus. On the other hand, the pattern behind the equations using B is obvious: there are only
two forms, one for the first dimension, which is unique, and another for both of the other dimensions, and
all forms are identical except for the extra “2” in the equation for the first dimension. Thus, B not only
solves the pi symbol paradox, but also it reveals the pattern behind the dimensions directly.
If the new, straightforward equations for the dimensions of space were the only outcome of the new
understanding of Whole PI, that would be sufficient reason for adoption of the new symbol, B. But the
fact that equations for masses can be constructed in the same form as the dimensions of space provides
additional benefit. In the same way that the equations for dimensions of space contained the dimension
number as the power, and the number of radii (the subcomponents of the unit diameter) as the
denominator, the equations for mass contain the element number as the power and the number of
subcomponents, protons and neutrons, in the denominator. Thus, the new equations reflect the inherent
duality in mass. This allows us to think of mass in a new way, as dimensions of mass that follow the same
form of equations as dimensions of space.
Many chemists have not considered the fact that the periodic table is based on an empirical definition
of the “unit”. Originally, the unit was a hydrogen atom set as 1 mass unit. Later, 1/16 of an oxygen atom
became the defining unit of the periodic table. And finally, 1/12 of a carbon atom has become the defining
unit  (https://en.wikipedia.org/wiki/Atomic_mass_unit). The 99.989% Solution based on B and/or
provided here is not a complete and all-encompassing model for mass to replace the empirical basis, since
it does not include factors for isotope distribution, or account for negative mass defects, or other factors
mentioned above. It is a converging series that converges with diminishing mass defect. Therefore, it
requires additional terms to incorporate the factors discussed above. Nevertheless, it does provide a new
foundation for thinking about mass in new ways, and a starting point for developing a non-empirical
model for the mass scale
The new model also allows us to consider the anomalies in the periodic table , Fig. 8, where an
element with a higher atomic number has a lower mass than the preceding element. We can ask “are the
masses of elements such as argon anomalously high, or is the mass of the neighboring element
anomalously low?”. In the case of argon, it is easy to see that it is anomalously high, having a large
deconstruction in the equation, m18/(2p+4). In the case of cobalt and nickel, it is less clear. The model
reveals that cobalt is m27/(2p+5) in Fig. 8, which is similar to neighboring 23V, 25Mn, and 29Cu, whereas
the primary nickel isotope is m28/(2p+2), which is lower than the other transition metals. Thus, the model
reveals that the mass of nickel is anomalously low, instead of cobalt being anomalously high. On the
other hand, the most abundant isotope of 34Se, mass = 80, arises from m34/(2p+12), which is anomalously
high compared to the major isotope of 35Br, mass 79, which is m35/(2p+9) that is similar to surrounding
elements. Similarly, 52Te, mass=130, m52/(2p+26), is anomalously high compared to 53I, mass 127, which
is m53/(2p+21), similar to neighboring elements.
It is worth mentioning that the dimension model of mass (DMM) and the 99.989% Solution showed
three sequential deconstructions of mass that started at a location that was two masses (i.e. at lithium)
prior to the observation of the first of the three “p” electronic orbitals that begin with boron. Similarly, the
DMM and 99.989% Solution exhibited five alternating unit deconstructions (2p+1), excluding chlorine as
discussed above, prior to the beginning of the five “d” orbitals. The last unit deconstruction occurred at
element 19, potassium, two elements before the observation of the first “d” orbitals that begin with
scandium. Thus, it appears that deconstructions in the DMM occurred prior to observation of patterns in
electronic energy levels. It might be anticipated that the higher levels of deconstructions that occur with
chlorine and argon (chlorine gave 2p+1 and higher 2p+3 than all prior elements, argon gave 2p+4), may
precede and be related to other phenomena that occur later in the table.
Presented here is a new hypothesis based on Whole PI to complement, not replace, the classic
understanding of pi and the possibly more useful tau. In his “Tau Manifesto” , Mike Hartl makes a
good argument for the virtue of using instead of 2. It does make sense to refer to a whole circle,
instead of half of a circle, as does, especially since 2 appears so often in physics and mathematics, and
fractions of are equal to fractions of a circle. However, since is still based on radii, it is also useful to
consider the whole circle based on the diameter. While the radius-based circle is commonly used in
physics, geometry, trigonometry, and other areas, the equations behind the diameter-based whole circle
allow the patterns behind the dimensions to be seen without resorting to integral calculus. Thus, it is
appropriate to have three symbols relating to the circle: 1) classic used in the classic equations for
circumference, area, and volume; 2) to refer to the whole circle based on radius and eliminate constantly
referring to 2, and 3) B to refer to the whole circle based on diameter, which makes the pattern behind
the dimensions more evident at face value.
Since the models for space and mass above both contain the pattern 2xp/2p and xp/2p for the first and
following dimensions, respectively, we can consider the possibility of whether these might apply to other
variables. For instance, both E=mc2 and E=hc/ contain the constant “c”, the speed of light in a vacuum.
The speed, or velocity, is space/time. Since we already constructed the model for dimensions of space,
we could conjecture that if there were such a thing as dimensions of time, they might similarly follow the
pattern of 2tp/2p and tp/2p for the first two dimensions. Finally, every definition, term, equation, and
concept discussed in the history of man is the result of thought, so we could also speculate whether there
exist dimensions of thought, and construct the appropriate equations using the model presented here.
Funding: This work was supported by the USDA Agricultural Research Service. This research did not
receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
The author owns the trademarks for the names and symbols of the Unit Simulacrum and Whole PI.
The author grants the full and unrestricted rights to use the names and symbols of the Unit Simulacrum
and Whole PI for all purposes educational, commercial, public, private, and otherwise.
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Electronic Supplementary Material
The Case for Whole PI and Alternative Equations
for Space, Mass, and the Periodic Table
William Craig Byrdwell
USDA, Agricultural Research Service, Food Composition and Methods Development Lab,
10300 Baltimore Ave., Bldg. 161, Beltsville, MD 20705 USA
See Supplemental Figure 1 below to demonstrate this analogy.
Imagine a wife calls her husband at work as says: "Honey, would you please pick up a jug of
milk on the way home?"
He responds, "Sure, no problem." He stops by the store on the way home and picks up a half-
gallon jug of milk.
When he gets home, she says "No, No, I needed a GALLON of milk!"
The next day, the wife calls and says: "The kids already went through that jug of milk you
bought yesterday, would you buy a GALLON of milk this time?"
He responds, "Sure, no problem." He stops by the store on the way home and picks up TWO
half-gallon jugs of milk.
When he gets home, she says "Yes, you brought a gallon of milk home, but I needed it in one jug
instead of two jugs, because we wanted to use the empty jug to make a terrarium later."
He responds: "Honey, you told me to bring home a GALLON of milk, so I brought home a
gallon of milk. You didn't tell me what units to bring the milk home in. A gallon is a gallon,
whether it is made up of two half-gallons or one whole gallon. They are entirely equivalent, and
two half-gallons make one gallon every day of the week."
There are times when we want gallons of milk in half-gallon jugs, and there are times when we
want gallons of milk in whole gallon jugs. The term jug needs to be specified based on the unit
on which it is based (half-gallon jug or gallon jug). Seeing the word "jug" alone is not sufficient
to tell which jug is being referred to. Similarly, a gallon can be obtained from two jugs or one
jug, depending on the unit definition of the jug. To simply say that "a gallon is a gallon,
regardless of the units it comes in, and they are entirely equivalent" belies the fact that there are
times when one unit is more useful than the other, and although we can easily convert from one
unit to the other (2 x "half-gallon jug" = "gallon jug"), they are not completely functionally
equivalent (it is easier to make a one gallon terrarium in a gallon jug than two half-gallon jugs).
Supplement to section 2.3.
It may be instructive to briefly examine the form of (B/3)1/6, where B is divided by three dimensions of
space, 3d1, raised to the power of d1p/2ps. to give
where again ps = 3, for the three dimensions of space in which mass operates. In this case, B can be used
instead of , because B is being divided by three diameters, which are proportional to the three dimensions
of space. This starting point only incorporates space, but when the power (1/6) is nested into the dimensions
of mass equations and divided by the dimension+deconstruction forms of equations for mass, it becomes a
An alternative way to model the unit mass in the 99.989% Solution is to use (/3)1/6. Although it was
at first empirically determined, the constant (/3)1/6 can be modeled using the principles presented in the
body of this report. As a first approximation, (/3)1/6 is mathematically equal to the first dimension of mass,
2mp/2pm, times the third dimension of space, Bdp/2ps, times the factor 2/B, all raised to the power of
2mp/2pm times dp/2ps as follows:
where the factor 2/B is incorporated because, for the defining unit, we are interested in a single unit mass,
proportional to ‘pm’, instead of the proton + neutron pairs represented by ‘2pm’. We went to great lengths
to differentiate based on sub-components (radii) and B based on duality and the sum of sub-components
(diameters) in the text. Thus, since is directly proportional to radii, and not diameters, as seen in Fig. 2a,
it appears that may be appropriate to be proportional to one sub-component (one proton) that constitutes
the unit mass, rather than B, which might be thought of as proportional to the sum of sub-components, a
proton + neutron. Furthermore, this form of model for the unit mass has the advantage that it incorporates
both the first dimension of mass (2mp/2p) and three dimensions of space (Bdp/2p), such that it is a
“1mass3space” function. There are other conceivable ways to model the 99.989% solution, but they all lead
to the same value for the unit “1mass3space” of 1.00771588137…