The Story of The Unit Simulacrum and Whole PI

© 2004-2010

The names “The Bottom Up Solution”, “The Unit Simulacrum”, “Whole PI” and all equations, figures and schemes are copyrighted in 2004-2010 by William Craig Byrdwell.  No portion of the equations, figures or schemes may be used without acknowledging the copyright holder, William Craig Byrdwell. or



The First Increment and the First Decrement

I noticed that the unit simulacrum, (1+A), which is 1 plus any ratio, is the same as ‘the first increment’ of A, which is A+1. It turns out that the math of the Unit Simulacrum is the math of the first increment. The first increment is the Ratio plus 1. This equals the Unit Simulacrum, which is 1 plus a Ratio. If we are dealing with a variable called Ratio, and that ratio equals One, then the first increment of One would be One + 1 = 2. Thus, the first increment of one is two. This is equal to the ratio One operated in the Unit Simulacrum to get S(1+One) = 1·(1+(One/1)) or 1·(1+(1/(1/One)) which, if we ignore the multiplying 1, simplifies to 1+(One/1) or 1+(1/(1/One)) = 2 = Two.

When the ratio One can vary, so that One does not equal the numerical value 1 anymore, then one or both of the two numbers that makes up the ratio can be greater than or equal to one, or less than or equal to one (yes there is a reason I wrote it that way). If either A or B is One, then the answer can be found by the Unit Simulacrum, and the answer to (A + B) will be given by (1 + Ratio) or (1 + 1/Ratio), where the Ratio is (A/B)* or (B/A)*. Whenever A or B is One, the solution can be found using the Unit Simulacrum. In cases where the One becomes a ratio less than or equal to 1, it is solved as Case 1, in cases where the One becomes a ratio greater than or equal to 1, the solution is found using Case 2. In Case 1, the maximum simulacrum sum is 2. In Case 2, there is no maximum. At the Critical Limit, where One = (1/1), both Cases and all Eight Possibilities for Observation are equal. This shows that there are two possibilities, or Cases for any Ratio: Case1) the Ratio is less than or equal to One; if the Ratio < 1, it is a decrement of One, and is solved using Case 1 Observations and the maximum simulacrum sum is Two. Or Case2) the Ratio is greater than or equal to One; if the Ratio > 1, the Ratio is an increment of One and it is solved using Case 2 Observations; It has no specified maximum simulacrum sum; It is open-ended.

In Case 1 solutions, the answer can always be found using the unit inverses based on perception, or in another word, observation. Was the (One/1) ratio observed or was (1/One) ratio observed? The sum of possibilities boils down to two: S(1+One) = 1·(1+(One/1)) or 1·(1+(1/(1/One)) = 1+(One/1) or 1+(1/(1/One)) = 2. We can see from the Pattern of the Construct that there are Four Possibilities for Observation in each Case, and Two Cases in the Aligned Unit Simulacrum. This would give a maximum Sum of Possibilities for the Aligned Unit Simulacrum of Eight Possible Proper Observations.

Based on the way things were defined above, and by defining the ratio (Correct/Aligned) = 1, in other words defining the term ‘correct’ in simulacrum terms as being Aligned, with the Cases Aligned with the Many and the One. The ratio is constructed such that Correct can go to zero, so the Ratio can go to Zero. If, in the sum of all possibilities, there exists the possibility that the Cases could be incorrectly observed, as demonstrated by my first labeling the Aligned Case 2 as the Original Case 1, then this possible perception would be embodied in the ratio (Correct/Aligned). The (Correct/Aligned) ratio can have only two possible values, 0 or 1. Since the (Correct/Aligned) ratio provides two more possible Cases for solutions in the Sum of All Possibilities, there exists the possibility for observing Before-Alignment Cases. I have made a choice. I have chosen the terminology Before-Alignment case to reflect my own personal observation timeline. I discovered the Before-Alignment Unit Simulacrum before I aligned it with the Many and the One, in which the Many is greater than or equal to 1. The reason behind the reason for choosing to define the term Before-Alignment is that it can be represented by only the designation B for Before-Alignment, while Aligned can be represented as by only the designation A. Thus, an A-Unit Simulacrum is an Aligned Unit Simulacrum, and a B-Unit Simulacrum is a Before-Alignment Unit simulacrum. An A-Unit Simulacrum is based on the assumption that the Ratio in the Unit Simulacrum is expected to be greater than or equal to one. That is to say the Many is expected to be larger than the One. The Case numbers are aligned to go from 1 to 2 with an increasing Ratio, or Totality. This is the Macro World. The Case 1 solution is less than the Case 2 solution. The label of the Case is correlated with the magnitude of the Ratio. A Case 1 Ratio is less than a Case 2 Ratio. I thought this might also align with Energy; the Case 1 solution would be the lower-energy solution, and Case 2 would be the higher-energy solution. This type of Unit Simulacrum is observed when you define a Ratio based on a Unit Definition, and increment the Unit Definition, and get a new sum greater than one. Again this corresponds to the Macro World.

The B-Unit Simulacrum had Case 1 used to represent t the case where One was expected to be the larger value (based on mass spectrometry) and all ratios were expected to be less than or equal to One. Case 1 was set to correspond to the expectation that the ‘natural case’ would correspond to the [MH]+ ion being larger than the SDAG, with all ions having abundances less than One. Case 2 was used to describe ratios greater than or equal to One. There was nothing wrong with that construct, it was true and self-consistent. That is why the terms Aligned and Before-Alignment are better terms than proper and improper, or correct and incorrect. The Before-Alignment Unit Simulacrum could be called the simulacrum for the Micro World. It is based on the expectation that that the abundance of the ion from the one variable in the denominator of the Ratio is equal to One (=100%=1.00), and the simulacrum sum of the other ions is less than the abundance of the one ion in the numerator of the ratio. The sum being larger than the One was set as the unexpected, or Case 2 solution. Since the Before-Alignment Unit Simulacrum was based on the Case 1 solution being defined as expecting a ratio less than One, this simulacrum could be called the Simulacrum for the Micro World.

Unless otherwise specified throughout the rest of this book, I have used the Aligned Unit Simulacrum. The Unit Simulacrum shown in Figure 30 is Aligned. Unless otherwise specified, it is automatically assumed that we will be using the Aligned Unit Simulacrum, so it will be called simply The Unit Simulacrum. I will not do anything else with the idea of a B-Unit Simulacrum right now. I am only making sure that I can account for the Sum of All Possibilities for Observation, Aligned or Not Aligned.

I digressed; I was talking about the First Increment of a Ratio. Where there is a first increment, there should be considered to be a first decrement. The question is to figure it out. If A is a ratio, 1/1, such that its first increment is two, then either of the two terms in the ratio could be incremented to produce the new ratio. If the numerator of 1/1 is incremented, we get a (1+1)/1, or 2/1 = 2. This is the expected First Increment of the ratio 1/1.

On the other hand, if the denominator of 1/1 is incremented (the first increment of the denominator), it gives 1/(1+1), which is 1/2 = 0.5 The First Increment in the Denominator is the First Decrement of the Unit ratio of One, 1/1. You can say we have specified a relationship between 1/2 , 1 and 2. This relationship starting with the variable one is that 2 is the first increment of the variable One, while 1/2 is the first decrement of the variable One. The terms 1/2, 1 and 2 share the relationships: first decrement, variable, first increment.

PI in the Unit Simulacrum the First Time

When a variable other than One is to be considered, the variable can be substituted, One variable for Another, or alternatively the Another variable can multiply Times the One variable. In the case of substitution, PI may be substituted into the (1/1) ratio instead of the top one, 1, to give a (p/1) ratio. Just as 1/2, 1, 2 came from 1/1, the first increment and decrement of p gives (p+1)/1 and p/(1+1), p, which equal p/2 , p , 2p. These three terms are the first decrement, or deconstruction of p, the original p, and the first increment of p.

Instead of substituting p into the (1/1) ratio for the top One, the (1/1) and its First Increment and First Decrement, ((1+1)/1) and (1/(1+1)), can be kept together, and multiplied times a variable. When PI is multiplied times the One ratio, it becomes: p·1/1. The first increment, then becomes p.·(1+1)/1 = p · 2 = 2 p. The first decrement would be p.·1/(1+1) = p/2. Thus, the first decrement, the Unit and the first increment of PI times One are: p/2 , p , 2p. You can see that this is entirely analogous to the 1/2, 1, and 2 that came from 1/1. I didn’t do anything with this observation at that point, I just wrote it down and moved on. But it did get me to think more about the Unit Simulacrum being the First Increment, and the inherent deconstruction of the First Decrement leading to an inherent three-level deep nesting of variables, including p.

So, I started with p/2 because is the smallest and closest to 1, of the three possible ratios with PI. PI is larger than One, and therefore, all increments are larger than 1, so this puts them all in Cases of being greater than One, and therefore Case 2. All integer increments of PI, including 1 are ³ One. The First Increment of the First One times PI is 2·p, called the Second One times PI. The Second Increment would be the Third One times p, and be 3·p, etc. The Third One equals the One That Always IS plus Two Times the First Increment, which equals One plus the (Second Increment/UnitInc) = S(1+(SecondInc/FirstInc)) = S(1+(2/1)) Times p = 3 · p. This continues on, with ((1+n)/1)·p always ³ One.

Chapter Six first posted to the Web on  Monday, January 8, 2007


Mending the Sacred Hoop—

The Meaning of Whole PI


is in the process of editing.  It is not yet in its final form.

Chapter Four - The First Increment