Discovery of The Simulacrum System, and specifically The Unit Simulacrum, led to examining Unit Definitions of other symbols in our concensus reality. As mentioned in my blogs on The Simulacrum System and The Unit Simulacrum, when the Bottom Up Solution led to the Fibonacci numbers and Fibonacci ratios, then to the Phi ratio, I thought "What other transcendental symbols has man always thought about and pursued?" The first obvious answer was pi, π . So I constructed the Pi Unit Simulacrum. The Pi Unit Simulacrum is as follows:

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

(Possibilities to Observe: 1, π , (1/π), (π/1);{

Case 1: 1<=π, π >=1, (1/π)<=1, (π/1)>=1

1 ( 1 + 1/(1/π) )    or    π ( 1 + (1/π ) )

1 ( 1 + (π /1) )    or    π  ( 1 + 1/(1/π ) )

Case 2: 1>=π, π<=1, (1/π)>=1, (π/1)<=1

1 ( 1 + 1/(1/π) )    or    π ( 1 + (1/π) )

1 ( 1 + (π/1) )    or    π  ( 1 + 1/(1/π) )

})

The Pi Unit Simulacrum simplifies to two Case 2 solutions, 1 + (π /1) or 1 + 1/(1/π ).

So, we can examine two ratios, (π /1) or (1/π ). Pi divided by one or one divided by pi. So the question is: "What is the one?"

What is the Unit Definition?

It turns out that pi, π, has two Unit Definitions, one pi relative to one radius or one pi relative to one diameter.

The dictionary definition of pi is based on diameter. Pi is the ratio of a circle's circumferece to its diameter, or pi = C/d, π = C/d.

The common definition that we all learned as children is that a circle is two times pi times r, C = 2πr. The definition of pi based on radius is π = C/(2r). So the definition for pi is it is the ratio of a circle's circumference to two radii.

We learned the equations for the three dimensions of space based on radius, as circumference, C = 2π r; area, A = πr^2; volume, V = 4/3πr^3.

From the dictionary definition of π, we get C = πd. The equations for Area is A = π(d^2)/4; volume, V = π(d^3)/6.

That's the most we're ever taught.

To me, it was obvious that the 2nd and 3rd dimensions, π(d^2)/4 and π(d^3)/6, followed the same pattern, π(d^p)/2p.

But the first dimension, πd, does not follow that pattern. The dimension pattern shows duality in the denominator, which can be seen to represent the radii coming from the dimension number, d. All diameters have two radii, so it can be said that all dimensions based on diameter are inherently dual. Even the first diameter has two radii, but they are not reflected in the equation πd. So the question can be asked "What would it take to the obvious duality of the first dimension and make it in the same form as the other dimensions?" The answer was super simple, and leads to a new understanding of the Dimensions.

We can simply multiply πd times 1, πd *1 = πd, according to the identity law of mathematics. We can choose to make that 1 = (2/2). 

The First One is two twos. 1 =  (2/2). The First One is (2/2)πd. This now fits the form 2π(d^p)/2p. This is almost the same as the other dimensions, except that it has an extra "2" in the denominator. The common core in these equations is π(d^p)/2p, but the First One has an extra 2. The new equations for the three dimensions based on diameter are: 

C = 2π(d^p)/2p; A = π(d^p)/2p; V = π(d^p)/2p. 

Now, all equations show the number of radii in the denominator and reflect the duality of all diameters, and dimensions. This why we say "All are dual." This comes from the fact that every diameter has two radii, with the number of radii in each dimension given by the 2p in the denominator, where "p" is the dimension number, which represents the number of perpendicular diameters.