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1

Electronic Supplementary Material

The Simplified Case for Whole PI and Alternative Equations

for Space, Mass, and the Periodic Table

Wm. Craig Byrdwell, Ph.D.

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 a convenience 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. The convenience store only sold half gallon jugs. 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).

2

2x=2x=

2x“1jug”=1gallonUnitCircle

==

2x“1jug”=“1jug”=1gallonUnitCircle

C=2∙r1C=∙d1

2x“1jugHG”=“1jugG”2r1=d

r1

2

r1

2

r1 d1

1

The Simplified Case for Whole PI and Alternative Equations

for Space, Mass, and the Periodic Table

Wm. Craig Byrdwell, Ph.D.

Abstract

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. This report shows a common pattern behind

space and mass, and provides new insight into the anomalies in the Periodic Table.

Keywords: pi; tau; Unit Simulacrum; Whole PI; Periodic Table

2

1. Introduction

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 (Byrdwell

2005) 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 (1995). This

relationship was modelled using a sigmoid function (Byrdwell 2015b). 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 (1996), and Laakso and Voutilainen (1996). 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

3

for the first time (Byrdwell 2015b). 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) (Byrdwell 2015b). Next, the

construct was further generalized to apply equally well to atmospheric pressure photoionization (APPI)

MS and electrospray ionization (ESI) MS of TAGs (Byrdwell 2015a).

As the name states, and as described above, the Bottom Up Solution (Byrdwell 2005) and the

Updated Bottom Up Solution (Byrdwell 2015b) 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

4

desired structural information and still represent a compressed data set and compact library of mass

spectra. These factors have been discussed in detail elsewhere (Byrdwell 2016), 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 (Byrdwell 2016).

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 (Byrdwell 2005) and UBUS (Byrdwell

2015b, 2015a).

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”,

5

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 (Mirriam-

Webster 2019). From this comes the mathematical equality for the circumference of a circle, C = ꞏd. For

a unit circle based on a unit diameter, the unit circle circumference is C = ꞏd1, which is shown in Figure

1B.

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 = 2ꞏr. For a unit circle based on a unit radius, the unit

circle circumference is C = 2ꞏr1, depicted in Fig. 1A.

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 = ꞏd1 = 2ꞏr1, or C = ꞏ1d = 2ꞏ1r. 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…, shown in Figure 2. 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.

6

Figure 1. Conventional constructions of unit circles based on (A) a unit radius and (B) a unit

diameter. The circumference of a unit circle based on radius is 2ꞏr1 = 2 and a unit circle based

on diameter is ꞏd1 = .

r1

d1

UnitCircle

Circumference=

2r1

UnitCircle

Circumference=

d1

2∙1

∙1

2

(A) (B)

7

Figure 2. Calculated circumferences of unit circles based on (A) a unit radius and (B) a unit diameter.

The circumference of a unit circle based on radius is 2 = 6.28318530… and a unit circle based on

diameter is = 3.14159265…. These two circles are identical in appearance, but one has a circumference

of 2 and the other has a circumference of .

UnitCircle

Circumference=

6.2831…

UnitCircle

Circumference=

3.1415...

(A) (B)

8

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, Figure 3A. In contrast,

for a circle based on diameter, the circumference is , so a single unit is a whole circle, Fig. 3B. 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, as shown in Fig. 3. 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

(Fig. 3), and that without the defining unit always accompanying we can arrive at the mathematical

paradox 2 = (Fig.2), can be called the “pi symbol paradox”.

Of course, this paradox also plays out in calculations for the second dimension (rꞏr or dꞏd) and third

dimension (rꞏrꞏr or dꞏdꞏd). The equations based on radius are the well-known ꞏr2 and 4/3ꞏr3. The

equations based on diameter are the less well-known ꞏd2/4 and ꞏd3/6. 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

paradox.

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.

9

Figure 3. The two different definitions of based on (A) a unit radius and (B) a unit diameter. based on

a unit radius is ½ of a circle (a whole circle based on a unit radius is 2), and the value for is

3.14159265…. Based on a unit diameter, is a whole circle, and the value for is 3.14159265…. Thus,

the same symbol, , based on two different units (radius or diameter) has the same numerical value but

two different meanings (1/2 of a unit circle or a whole unit circle). This can be called the pi symbol

paradox.

3.1415...3.1415...

=½Circle

3.1415… 3.1415...

=Circle

Samenumber

Samesymbol

Differentmeanings

ThePi

Symbol

Paradox

(A) (B)

10

As discussed in Scientific American (Bartholomew 2014): “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 2ꞏr is really two times a

semicircle was summarized in a 2013 feature on the PBS Newshour in a quote by Mike Keith (Jacobson

2013): “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” (Palais 2001), 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 .

2. Results

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 4. 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 (Byrdwell

2016) 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 5A, we see that it has r1 and

11

Simulacrum Sum (1, ) = SimSum(1, ) =

Possibilities to Observe:

1

or

or

or

or

Figure 4. 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.

1

π

π

1

π 1

π-1-0-2

; 1

π

1

, 1

1

π

π,1 ,1 π:1 Case

1

π

1

1 π

1

π

1 1

π

1

1 π

π

1

1

1 1

; 1

π

1

, 1

1

π

π,1 ,1 π:2 Case

1

π

1

1 π

1

π

1 1

π

1

1 π

π

1

1

1 1

12

Figure 5. 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 .

r1

(A)

1

π

r

π

1

π

1

π

r1

or

(B)

=

r1

1

τ

r

τ

1

2π

r

2π

11

τ

1

τ

r

2π

1

2π

r11

or

13

half of a circle. Thus, while Fig. 5A 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

selecting (constructing) Critical Ratios, we can ask the question of : “Is there another ratio we could

use?”

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. 5B. 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 2ꞏr1, or just 2, which

equals 6.283185…

If instead of using 2, we substitute into the Unit Simulacrum for in Fig. 4, 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. 5B. 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 ꞏr1, or simply , which equals 6.283185…

In Figure 6 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 ꞏd1, or simply . Comparing Fig. 5 to Fig. 6 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 2ꞏr1. Based on a unit diameter, the unit

circle is ꞏd1.

14

Figure 6. 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…

d1

1

π

d

π

1

π

1

π

d1

or

15

Thus, the same one whole unit circle is both 2ꞏr1 and ꞏd1. Therefore, the units, r1 or d1, can never be

cancelled out or ignored, otherwise we obtain 2 = , or 6.283185… = 3.14159265…

Thus, unless the “r” or the “d” always accompanies , it cannot be known with certainty whether refers

to ½ of a unit circle or a whole unit circle.

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 7 shows the components and construction 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 diameters, 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. 7A. 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. 7B. Thus, the symbol for Whole PI is B, Fig. 7C, 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 and two different meanings 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 8.

From this figure, it is now perfectly clear that the unit B, based on the unit d1, cannot be confused with

from Fig. 5 or Fig. 6.

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 9B and Fig. 9C.

16

Figure 7. 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

(A), G.

(A)

(C)

(B)

(D)

Whole PI

17

Figure 8. 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…

d1

C

1

d1

CC

CC

1

d1

or

18

Figure 9. 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

diameters.

C

d C

d d

d

d

(A) (B) (C)

or or or

C

d

2Bdp

2p

Bdp

2p

Bdp

2p

2Bd1

2

Bd2

4

Bd3

6

=Bd

19

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.

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, in this case dimensions. 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.

20

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

distinguishable.

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 10, 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 in this first report, which do not require the use of Whole PI, while an

approximation for monoisotopic masses that incorporates B and/or has been developed, but is not

presented here, since this report is already sufficiently groundbreaking.

21

p

p

1 1

2

p

p

p

3

4

5

p

p

p

6

7

8

Figure 10. Dimension model of mass based on unit mass, m1. (A) Equation for the first mass, Hydrogen,

which contains the atomic number 1 as the power, p, and the integer mass 1 as the denominator. (B)

Equation for the second mass, Helium, which contains the atomic number 2 as the power and the integer

mass 4 as the denominator. (C) Equation for the third mass, Lithium, which contains the atomic number 3

as the power and the integer mass 7 as the deconstruction in 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 mass.

(A) H (B) He

(C) Li (D) Be (E) B

(F) C (G) N (H) O

22

Using the same pattern as the dimensions for space, the first mass is 2m1p/2p, Fig. 10A, while the

second mass is mp/2p, Fig. 10B, where “m” 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.

Thus, 2m1/2 (= m1/1) and m2/4 are similar in form to 2Bd1/2 (= Bd1/1) and Bd2/4.

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, Fig. 10C, 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 (Byrdwell 2016). 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 first). 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. 10 are followed through the penultimate dimension of mass,

sulfur, which is m1p/2p, where p=16, as follows (and shown in Figure 11):

H1: 2m11/2p

He2: m12/2p

Li

2-1: m13/(2p+1), Be2-2: m14/(2p+1), B2-3: m15/(2p+1)

23

C3-3: m16/2p

N4-3: m17/2p

O5-3: m18/2p

F

5-4: m19/(2p+1)

Ne6-4: m110/2p

Na

6-5: m111/(2p+1)

Mg7-5: m112/2p

Al

7-6: m113/(2p+1)

Si8-6: m114/2p

P

8-7: m115/(2p+1)

S9-7: m116/2p

Cl

9-8: m117/(2p+1,3)

Ar

9-9: m118/(2p+4)

K

9-10: m119/(2p+1)

Ca10-10: m120/2p

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.

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 an isotope-weighted average molar mass of

~35½, being the isotope-weighted average of m117/(2p+1) = m117/35 and m117/(2p+3) = m117/37, and is the

24

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, masses of elements, reflects the inherent

duality in all masses, and shows a common pattern behind space and mass, 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 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 show a unified pattern behind space and masses in the Periodic Table better

than any other model presented, despite its shortcomings.

Chlorine is more than just 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 primary isotope is a normal (2p+1) deconstruction.

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) related to some other phenomenon, such as those mentioned above (e.g. isotope

distribution).

Argon exhibits a higher level of deconstruction as its primary isotope, having an equation of

m118/(2p+4). There are several possible functions that could be envisioned for the higher deconstruction

number. The deconstruction 1/(2p+4) could 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 (metals,

25

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 11 simply

as m118/(2p+4). It is worth noting that just as the first deconstruction, lithium, appeared three elements

prior to the three p-orbitals appearing in the electron configuration of carbon, the higher deconstructions

appear just prior to the appearance of d-orbitals in metals. It appears that perhaps the pattern in the

equations for mass (based on nucleons) preceded the appearance of patterns in the electron configuration.

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

Fig. 11. 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. 11 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.

However, since this report is already sufficiently challenging, that refinement will be saved for later. For

now, the new series of equations that shows the common pattern behind space and integer masses is

sufficient.

26

Figure 11. 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.

H

2m1

2p

Li

m3

2p+1

Na

m11

2p+1

K

m19

2p+1

Rb

m37

2p+11,13

Ca

m20

2p

Sc

m21

2p+3

Ti

m22

2p+4

V

m23

2p+5

Cr

m24

2p+4

Mn

m25

2p+5

Fe

m26

2p+4

Co

m27

2p+5

Ni*

m28

2p+2,4

Cu

m29

2p+5,7

Zn

m30

2p+4,6

Ga

m31

2p+7,9

Ge

m32

2p+10,8

As

m33

2p+9

Se*

m34

2p+12,10

Br

m35

2p+9,11

Kr

m36

2p+12,14

Sr

m38

2p+12

Y

m39

2p+11

Zr

m40

2p+10,14

Nb

m41

2p+11

Mo

m42

2p+14,12

Tc

m43

2p+13

Ru

m44

2p+14,16

Rh

m45

2p+13

Pd

m46

2p+14,16

Ag

m47

2p+13,15

Cd

m48

2p+18,16

In

m49

2p+17

Sn

m50

2p+20,18

Sb

m51

2p+19,21

Te*

m52

2p+26,24

I

m53

2p+21

Xe

m54

2p+24,21

Al

m13

2p+1

Si

m14

2p

P

m15

2p+1

S

m16

2p

Cl

m17

2p+1,3

Ar*

m18

2p+4

B

m5

2p+1

C

m6

2p

N

m7

2p

O

m8

2p

F

m9

2p+1

Ne

m10

2p

He

m2

2p

B

m4

2p+1

Mg

m12

2p

27

3. Discussion

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

based.

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

28

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 (Wikipedia 2019). 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. These three diameters were arranged in the ratio of

1:2 to reflect the same ratio as the ratio of a unit radius to a unit diameter.

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 at (right clickSave Link

As): [redacted for review].

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

29

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 based on

Whole PI 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.

There are two differences between the equations for space and the equations for mass. The first is

that, since only integer masses are considered in this initial report, the equations for mass herein do not

include B. A refinement of the model that approximates the accurate monoisotopic masses that includes B

has been developed, but is beyond the scope of this initial report. The second difference is that the two

radii that make up a diameter are indistinguishable, whereas the two nucleons that make up masses are

different: protons and neutrons. Equations for both space and mass reflect the inherent duality in

diameter-based space and nucleon-based mass, but the first mass is unique. Since only protons have

corresponding electrons, the integer mass of the first mass is given in the denominator of the equation

after cancelling out the upper and lower 2’s, as in Fig. 10A. This is similar to cancelling out the upper and

lower 2’s for the first dimension of space to obtain Bꞏd1, Fig. 9A. So in this way, the first mass is unique.

30

The new DMM allows us to consider the anomalies in the periodic table (Firsching 1981), Fig. 11,

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), which is larger than any surrounding elements. In the case of

cobalt and nickel, it is less clear. The model reveals that cobalt is m27/(2p+5) in Fig. 11, which is similar

to neighboring 23V, 25Mn, and 29Cu, whereas the primary nickel isotope is m28/(2p+2), which is lower than

all 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.

4. Conclusions

Presented here is a new model 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 (2018) 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 based on radius. 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 based on radius; 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.

31

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