The Larson Research Center

As discussed in the last post, it seems like the only consistent way to produce the natural numbers is via a natural progression of points; that is, the 0D mathematical series

1⁰, 2⁰, 3⁰ …

must be actually

1*2⁰/2⁰, 2*2⁰/2⁰, 3*2⁰/2⁰, …

because, when, starting with space and time only, there are no “things” to count, which implies that the natural series,

1¹, 2¹, 3¹ …

is mathematically incorrect, as an initial condition in a space|time progression, since

1*2¹/2⁰, 2*2¹/2⁰, 3*2¹/2⁰…,

is a natural progression of double magnitudes (one in each “direction”) not single magnitudes. Therefore, as a space|time progression, the natural 1D mathematical series necessarily begins with 2, not 1, and increases by 2, 1D, magnitudes, not 1:

2*1¹, 4*1¹, 6*1¹…,

while the natural series,

1², 2², 3² …,

is also incorrect, because

1*2²/2⁰, 2*2²/2⁰, 3*2²/2⁰, … 

is the natural mathematical progression of area, which begins with 2² = 4, 2D, magnitudes, not 1, or 2, increasing the base of the series by a factor of 2:

4*1², 16*1², 36*1²….

Finally, the natural 3D series:

1³, 2³, 3³ …

is also incorrect, as a space|time progression, because it is actually,

1*2³/2⁰, 2*2³/2⁰, 3*2³/2⁰, …, 

which is the natural progression of volume, its magnitudes beginning with 2³ = 8, 3D, magnitudes, not 1, not 2, not 4, again increasing the base of the previous series by a factor of 2:

8*1³, 64*1³, 216*1³ …

All of this means, among other things, that the algebra of these numbers begins with the pseudoscalar value of an n-dimensional progression (2ⁿ), not its scalar value (2⁰); that is, each series begins with the corresponding right side of the tetraktys, not the left side. This is because one line has two directions, and one area has four directions, not two, and it is therefore incorrect to write the progression of 1D magnitudes beginning with the scalar magnitude 1 (2⁰), or to write the progression of area beginning with the 2¹, or 1D, pseudoscalar magnitude. Likewise, one volume has eight directions, not two, and not four, and therefore the natural volumetric series must begin with eight cubic scalars, not one. To accurately denote this, we need to rewrite the 1D progression as

(1*2)¹/(1*2)⁰, (2*2)¹/(2*2)⁰, (3*2)¹/(3*2)⁰, …,

the 2D progression as

(1*2)²/(1*2)⁰, (2*2)²/(2*2)⁰, (3*2)²/(3*2)⁰, …,

and the 3D progression as 

(1*2)³/(1*2)⁰, (2*2)³/(2*2)⁰, (3*2)³/(3*2)⁰, …. 

It’s important to recognize that, when the uniform 3D progression is measured from a given point (2⁰ = 1), at t_n - t₀, the apparent one-dimensional interval characterizes the expanding volume by its 1D radius. However, to calculate the true 1D interval, which is the diameter of the volume, the radius must be doubled; to calculate the true 2D interval, the doubled radius, the diameter, must be squared, and to calculate the 3D interval, it must be cubed:

2*1¹ = 2r = d,

4*1² = d²,

8*1³ = d³

However, this brings us face to face with the age old problem of the quadrature, or of “squaring the circle,” because the 2D space component of the 3D space|time expansion must expand geometrically over time, or circularly, and the 3D component must expand spherically, while the algebraic square and the algebraic cube are necessarily rectilinear, and therefore an issue of 2D and 3D numerical integration, or quadrature, and cubature, as it’s sometimes referred to, arises. 

That this problem is related to the foundations of quantum mechanics is indicated, when it’s recognized that only one point on the surface of an expanding circle, or sphere, can be measured at any given time. Special relativity makes it impossible to simultaneously specify t_n at any more than one point on the 2D, or 3D, surface of the expansion, because points on these surfaces are always moving apart. Therefore, we are brought back to consider the physical enigma of point/wave duality, and the mathematical dilemma of quadrature, and the logical challenge of unifying the concept of the discrete numbers of algebra with the concept of the smooth functions of geometry.

In the next post, we will discuss how the ancient way of dealing with these fundamental issues turns out to be remarkably congruent with our ideas of the space|time progression; that is, what has been called the “mediato/duplatio” (halving/doubling) method of ancient reckoning, intimately associated with the notion of the tetraktys, turns out to be our “factor of 2,” playing in the space|time progression series, as described above, and we will discuss the correspondence between them next time.

This topic is very interesting as it relates the modern concept of rotation, implemented with complex numbers, to our new concept of 3D expansion, implemented with scalars and pseudoscalars, which is a crucial point to understand, I believe.