REAL TRINARY LOGIC, THE IMAGINARY NUMBER
*
i*,

& THE CYCLIC SERIES OF COMPLEX NUMBERS

The mathematical
equation *e ^{i}*

The imaginary number *
i* is the generator in the cyclic series of complex numbers under
multiplication by complex numbers.

The analogy between the cyclic series of imaginary numbers and the
fourfold of real trinary logic elements, *i *: 0 (not nothing) ::
−1 :
(not not nothing) :: -*i* : 1 (not not not nothing) :: 1 :
**1**, is illustrated, together with further logical, ontological,
and geometric analogues, in this table where p = no negative and q =
ordered/non-evental:^{2}

Note that an even
exponent reverses the 0, 1 order and the sign of 1, as an odd exponent
reverses the 0, 1 order and the sign of *i*.

In ordinary mathematics
it is true that *x*^{2} + 1 = 0, whether *x* = *i
*or *x* = -*i*.

*i*^{2} =
−1. *i*^{4} = −1 × −1 = 1. *i* × *i*^{3}
= *i* × -*i* = *i*^{4 }= 1.

At this point the analogy with real trinary logic breaks down.

Where *i*^{2}
= −1 = ,
*i*^{4} = −1 × −1 = =
**1**,

Where *i* = 0 and
*i*^{3 }= -*i* = 1, *i* × *i*^{3} =
*i* × -*i* = 01 =
= *i*^{2} = −1, that is, ≠
,
≠ *i*^{4}, ≠ −1 × −1, ≠ **1**.

In
real trinary logic the participation of
is
indispensable to the production of **1** involving 0 or 1. 01 =
=
**1**. 01 (*i* × *i*^{3}) =
(*i*^{2})
= −1. [01 (*i* × *i*^{3}) =
(*i*^{2})
= −1] ×
(*i*^{2}) = 01 (*i*^{4})
= (*i*^{4})
= **1**.

00
= 1 on condition that 00 = 0()
= *i* × *i*^{6} = *i*^{3} = 1.

11
= 1 on condition that 11 = 1()
= *i*^{3} × *i*^{4 }= *i*^{3} =
1.

00
= **1** on condition that 00 = ()()
= *i*^{12} = *i*^{4} = **1**

11
= **1** on condition that 11 = =
*i*^{8} = *i*^{4} =
**1**

These equations call
attention to the fact that where *x* is the exponent of *i*,
and *y* is the numerator of the remainder in the division *x*/4,

*i ^{x}*/

But then the proper understanding of the multiples of the imaginary
number *i* is that they are elements in a geometric series whose
underlying structure is the infinite repetition of the mathematical
analogue of the real trinary logic minimum order, **1**, 0,
,
1,^{3} that is, *i*^{0},
*i*^{1}, *i*^{2}, *i*^{3 }. . .,
here illustrated:

Where the infinitely recurring imaginary number sequence is construed as
the proportion, *i*^{0} : *i*^{1} :: *i*^{2}
: *i*^{3} (*= i*^{4} : *i*^{5} :: *
i*^{6}
: *i*^{7}, etc.), the linear product of this proportion
= *i*^{2 }(),
the rational product = *i*^{2 }(),
and the integral product = *i*^{4} ()
= **1**.^{4}

^{ }

Real trinary logic provides the synthetic analysis of the otherwise
apparent simple cycle in the powers of *i* so as to bring to the
fore the difference between the sequence of the first three linear
dimensions and the per se volumetric fourth dimension. While a line
times a second line perpendicular to the first = the first squared =
depth, and a third line perpendicular to both the first and the second =
the first raised to the third power = the vertical = the base for a
volume bounded by depth, still, a volume not simply bounded by depth,
but rather bounded by a volume bounded by depth, is not the product of
breadth and the base for a volume bounded by depth—not the product of
the first and third powers of the first line—but rather is a volume that
= depth squared, clarifying the fact that *i*^{4} is not
indifferently last in the sequence of the four dimensions.

Real trinary logic constrains the recognition that the power of *i*
apparently last in each cycle, Omnipotence (*i*^{4}, *i*^{8},*
i*^{12},* i*^{16}, . . .), is not in fact last
in the sequence, but always = *i*^{0}, always in fact
preceding Immediacy (*i*^{1},* i*^{5}, etc.),
Mediation (*i*^{2},* i*^{6}, etc.), and
Beginning (*i*^{3},* i*^{7}, etc.), so that
(not the fourth but) the second is the product of the first and the
third,^{5} while the fourth is not directly that product but directly the
product of that product squared. Mediation raised to the second power (*i*^{4
}= )
= the product of Immediacy raised to the second power (*i*^{2}
= )
and Beginning raised to the second power (*i*^{6} =
)—that
is, *i*^{4} = *i*^{2 }× *i*^{6} =
*i*^{8} = *i*^{0}, that is, Omnipotence (*i*^{0
}= **1**).

The cyclic series of complex numbers under multiplication by complex
numbers has been shown to be in fact a geometric series whose underlying
structure is the infinite repetition of the mathematical analogue (*i*^{0},
*i*^{1}, *i*^{2}, *i*^{3}) of the
real trinary logic minimum order (**1**, 0,
, 1). Raising sequentially the infinitely cycling elements of
this minimum fourfold order to the powers of the series of the three
real trinary logical digits, 0, ,
1,^{6} likewise
infinitely repeating, immediately produces the real trinary logic
geometric series whose infinite sequential summing is the matrix of the
mathematical geometric series and the series of perfect numbers,^{7} in such a way that the real trinary logic
powers of **1 **(**1**^{0},^{ }**1**,
**1**^{1},^{ }**1**^{0},^{
}**1**^{
}, **1**^{1},
. . . = *i*^{0}, *i*^{4}, *i*^{8},*
i*^{12},* i*^{16}, *i*^{20}, . . .)
mark off the infinitely repeating sets of the bottom, middle, and top
rows of the real trinary logic Cornerstone,^{8}
as here illustrated:

**
**

Since
*e ^{i}*

In two
dimensions, **
1**^{0}
0
^{1
}1^{0} **1**,. . .^{ }: 1
0
1
0. . . :: sine wave : cosine wave:

Note
also that sin(*i*)^{2} + cos(*i*)^{2} : 1 :: **
1**^{0}
0
^{1
}1^{0} **1**. . . + 1
0
1
0. . . : **1**.

**The
Logical Double Helix, Transdecimal Calculation, & the Linear-Rational
Differential of the DNA Molecule**

When
the two series are seen in three dimensions as the double helix — (*e ^{i}*

This
double helix relates the sequence of the logical powers of the real
trinary logic analogue of the cyclic sequence of complex numbers** **
(**1**,^{ }0, ,^{
}1,^{ }**1**,
. . . = *i*^{0}, *i*^{1}, *i*^{2},*
i*^{3},* i*^{4}, . . .) to the real trinary
logic geometric series (1, 0, ,
1, 0, . . .), deriving, at once, the latter from the former.

In this
outworking of the foundational analogy existing between the imaginary
number *i* and the real trinary logic 0 (= not nothing) attention
is called to three foundational relations.

First,
the substantial analogy between the transdecimal calculation of
numerical identity (linear product, rational product, integral product)^{9}
and the structure of the molecule containing the genetic information for
all stable living organisms. As the transdecimal calculation of the
integral product of the
recurring fourfold cycle of the powers of *i* construed as a
proportion [(*i*^{2})(*i*^{2}) = *i*^{4}]
is produced by identifying the linear product (*i*^{0 }× *
i*^{1} × *i*^{2} × *i*^{3} *= i*^{2})
with the rational product (*i*^{0 }/ *i*^{1} ×
*i*^{2} / *i*^{3} *= i*^{2}),^{10}
so in the basic arrangement of the genetic information of stable living
organisms (the DNA molecule) the
double helix consists in identifying the linear order (G, C, T,
A, C, A, G, G, . . .) with the rational order (the pairing G:C, C:G,
T:A, A:T, C:G, A:T, G:C, G:C, . . .).

Second,
as in the double helix structure of the DNA molecule 3 hydrogen
bonds hold the rational G:C pairs together and 2 hydrogen bonds hold
the rational A:T pairs together, so in the logical double helix formed
above there are 3 logical bonds between the **1**’s in the series
corresponding to the geometric series of the powers of *i* **(1**^{0}
0 ^{1
}1^{0} **1**
. . .) and 2 logical bonds between the 1’s in the derived real
trinary logic geometric series (1 0
1
. . .).

Third,
the number of logical bonds between the logical powers of the **1**’s
in one complete cycle of the logical double helix
(**1**^{0}
0 ^{1
}
1^{0}
**1**
0^{1}
^{0
}
1 **1**^{1}
0^{0}
^{ }1^{1}
**1**^{0}) is 9, corresponding to the number of
atoms in each of the purine bases, linear A and G, while the number of
logical bonds between the 1’s in one complete cycle (1
0
1
0
1 0
1)
is 6, corresponding to the number of atoms in each of the pyrimidine
bases, linear T and C.
Likewise (5C + 4C = 9C) : (4N + 2N = 6N) is the ratio of total carbon
atoms to total nitrogen atoms in each of the rational G:C and A:T pairs.^{11}

Such is the logical structure of the linear-rational differential underlying the stability of the DNA molecule.

**Notes**

^{1} For various
relations of the imaginary number *i* to the dimensions of the real trinary
logic triple-cube, see on this web,
The Diagonal Logic of the Triple Absolute Dead Center Cube & Certain Fundamental
Mathematical
& Physical Constants. For an introduction to complex
numbers, cf. "Dave's Short Course on Complex Numbers." Online.
Available at:
http://www.clarku.edu/~djoyce/complex. July 2013,

^{2} Cf. D.G. Leahy, *Beyond Sovereignty: A New Global Ethics and
Morality *(Aurora, 2010), pp. 88, 203, 287f., 311, et passim. Cf.,
also, D.G. Leahy, *Faith and Philosophy: The Historical Impact *(Aldershot
and Burlington, 2003), pp. 123ff., et passim.

^{3} Cf. D.G. Leahy, *Foundation:
Matter the Body Itself *(Albany, 1996; eBook, 2009), Section III.1.
Also, Leahy, *
Beyond Sovereignty* and *Faith and Philosophy*, loc. cit.

^{4 } See, also, below. For linear,
rational, and integral products, see, on this web,
Transdecimal
Calculation of Number Identity: A Note
on *Integral Product* & Related Terms. See, also, on this
web,
Absolute
Particularity: Beyond the Problem of Universals.

^{5} Cf. Leahy, *
Beyond Sovereignty*, pp. 153ff.
and 176ff., et passim.

^{6} Cf. Leahy, *Foundation*, Section III.1.

^{7} See, on this web,
Real Trinary
Logic Geometric Series Matrix of the Numeric Geometric Series & the Series of
Perfect Numbers, and D.G. Leahy, *The Cube Unlike All
Others *(CreateSpace, 2010), Section III.1.

^{8} Cf. Leahy, *Foundation*,
Section III.1.

^{9} See, above, note
4. The transdecimal nature of this method of calculating numerical
identity is evident from the fact that the integral product of 82944 (= 8
10000's + 2 1000's + 9 100's + 4 10's + 4 1's) = (8×2×9×4×4)(8/2 × 9/4 × 4) =
(8×9×4)^{2}. For more on this and other integral product
calculations, cf. Leahy, *Foundation*, Section III.6-7, et passim.
For a vivid example of the significance of integral product calculation, see, on this web,
Measure Beyond
Beyond Reach, and, also, Leahy, *The Cube Unlike All
Others*,* *Section I.2.

^{10} See above.

^{11} Where,
following Chargaff's rule, base amounts are such that A ≈ T and C ≈ G, the ratio
of total percentages of base composition in human DNA, (A+T) : (C+G), ≈ (30+30)
: (20+20) ≈ 60 : 40 ≈ 3 : 2. For the
ultimately foundational role of the 3:2 ratio, cf. Leahy, *Foundation*, Section III.1, Afterword 2,
et passim.
Also, on this web,
Note to *Faith
and Philosophy* Further to the Ontology of Real Trinary Logic; also,
Leahy, *Beyond Sovereignty*, Backnote 1.