THE BEAUTIFUL RATIONALITY OF Π,
√2,
& Φ

PI Rational PI

Where π = + (4/1) + (4/-3) + (4/5) + (4/-7) + (4/9) + (4/-11) + (4/13) + (4/-15) + (4/17) . . ., π is the sum of an infinite repetition of the four-square successively divided by the number of squares on the leading edge of an infinitely expanding square of squares beginning with one-square, which number of squares––though always odd itself––is negative when not the leading edge of the square of an odd number of squares.

Every partial sum of this series is itself a rational number: (4/1) = 4, (4/1) + (4/-3) = 8/3, (4/1) + (4/-3) + (4/5) = 52/15, (4/1) + (4/-3) + (4/5) + (4/-7) = 304/105, (4/1) + (4/-3) + (4/5) + (4/-7) + (4/9) = 1052/315, (4/1) + (4/-3) + (4/5) + (4/-7) + (4/9) + (4/-11) = 10312/3465, (4/1) + (4/-3) + (4/5) + (4/-7) + (4/9) + (4/-11) + (4/13) = 147916/45045, (4/1) + (4/-3) + (4/5) + (4/-7) + (4/9) + (4/-11) + (4/13) + (4/-15) = 407712/135135, (4/1) + (4/-3) + (4/5) + (4/-7) + (4/9) + (4/-11) + (4/13) + (4/-15) + (4/17) = 7471644/2297295, . . ..

And the denominator of a rational
partial sum is the product of the *absolute* *values* of the last *
x* denominator(s) in the
π
series to that point, where *x* successively equals 1, 2, 3, 3, 3, 4, 5, 5,
6, 7, 8, 9, . . .*.*
The product of the *absolute values* of the numerator(s) and
denominator(s) of the last *x* ratios in the
π
series at the point of a rational partial sum divided by the denominator of that
rational partial sum = 4*x*,
where *x* successively equals 1, 2, 3, 3, 3, 4, 5, 5, 6, 7, 8, 9 . . ..

Compare the 4-numerator of this
simplest of the infinite series equal to π with the
notion that "the
constant identity of the system of reference, the identity of the system of
reference as the center, is the numerator" (D.G. Leahy, *Foundation:
Matter the Body Itself* [Albany, 1996], pp. 485ff.), and with the 4-fold
structure of the hyperlinear vertex that is the unique hypercubic rational
constitution of the absolute dead center cube (The
Deep Epidermal Surface: The Cornerstone Construction Order, Minimum Order
Tetrahedron Hypercube, & Absolute Dead Center Hypercube), whose definition
is (*x*/6)^{4}/*x*^{3 }= 8*x*^{3}/*x*^{4}
(= [√2×9]^{-1} = √162^{-1}) (where
φ is the division in extreme and mean ratio, this definition of the absolute
dead center cube = [(φ×10^{162})^{1/162}]^{1/[162}^{(√162/2)}^{2]}/10
× π/4 accurate to 1 part in 100 million):^{1}

Reality is essentially
rational. π is the area of the circle circumscribing the square whose area is 2
on the diagonal of the square whose area is 1. The *ratio* 2 : 1 is
the foundation of all irrationality. Hence the rational derivation of
√2 here
illustrated:

Like π,
√2 is perpetually
rational, as illustrated in the above series, each element of which is
positive (+) where the number of numbers in the denominator is even and
negative (–)
when that number is odd. Indeed, as here illustrated

qua
inscribing : inscribed areas, π : 2
:: 2 : 1, the proof of which proportion is precisely

Like π and √2, φ, the division in extreme and mean ratio, where AB : AC :: AC : CB, is also a function of the 2 : 1 (here, the 1-square transformed to a triangle, linear) ratio and perpetually rational as here illustrated:

Each element of this series is positive (+) unless it has an immediate predecessor whose denominator is the lesser and the difference between the two is 1 less than the sum of all the preceding denominators including that of its immediate predecessor, in which case it is negative (–).

**Note**

** ^{
1}
**
It is uniquely true for the absolute dead center cube that [(