Trinary Logic Geometric Series

REAL TRINARY LOGIC GEOMETRIC SERIES
MATRIX OF THE NUMERIC GEOMETRIC SERIES &
THE SERIES OF PERFECT NUMBERS

x_logic ϵ (0, , 1) → x_math ϵ (2, 4) → (x_math = 4²) → (x_math = 3)

__________________________________________________________________________

Now, infinitely iteratively sum sequentially, according to the rules governing the relations of the trinary logic digits,² the initial infinitely repeating series underlying the L series,³ 1, 0,

, 1, 0,

, . . . , itself the trinary logic analogue of the geometric series, x⁰,

, x¹, x⁰,

, x¹, . . ., where x = any of the three logical digits.⁴ In the illustration above the second row is the sequential sum of the first, the third of the second, the fourth of the third, etc. (Initial 1 is not repeated automatically since in the trinary logic there is no implicit zero equal to 'nothing'.) Notice the triangles filled with 1's. Only three in this picture share the long diagonal edge of the picture. If that long edge is examined, it will be found to consist of 1,

, 0; 1, 1, 1, 1 (the shared edge of the first triangle),

, 0,

, 0; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 (the shared edge of the second triangle),

, 0,

, 0; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 (the shared edge of the third triangle). There follows in this infinite expansion a set of 64 sets of

, 0's (not shown for lack of space). This is the beginning of a double series of 1's and

, 0's, the arithmetic transcription of each of which series, according to the logical rule that

and 0 each = .5, is 1 + 4 + 16 + 64 . . ., the ordinary sequential summing of which produces the exponents in the odd-numbered rows of the lattice above mentioned, while the two series added together give 2 + 8 + 32 + 128 . . ., the sequential summing of which produces the exponents in the even-numbered rows.⁵

This series of triangles of 1's sharing the infinite edge of the infinitely expanding triangle created by the infinite sequential summing of the logical geometric series is the series of the edges of the triangles composed of the number of logical 1's in the geometric series x⁰, x¹, x², x³, . . ., where x = 16, that is, the series, 1, 16, 256, 4096, . . . (see illustration above). The triangles composing the series occupy the odd numbered places relative to the intervening groups of logical digits touching the edge of the major triangle. And each triangle is internally an expansion of logical 1's according to the ratio of the odd numbers, 1, 1 + 3 + 5 + 7, 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 + 31, . . ., that is, according to the ratio of the distances traversed in successive equal intervals of time in Galileo's law of universal constant gravitational acceleration. Inspection of the pattern of triangles in this series shows that the longest edge of each of these triangles one of whose edges shares a portion of the infinite edge of the infinite triangular arrangement of digits is the first third of a three-digit column none of whose rows are composed of two or more consecutive 1's. The numbers of these three-digit columns correspond to the sequential summing of the series, x⁰, x¹, x³, x⁵, . . ., where x = 2, the series, namely, 1, 1 + 2, 1 + 2 + 8, 1 + 2 + 8 + 32, . . ., or 1, 3, 11, 43, . . .. The defining longest edges themselves of the edge-sharing triangles are the single-digit columns whose places are determined by the sequential summing of the series, [(xy)⁰]^½, [xy⁻¹]^½, [xy⁰]^½, [xy¹]^½, [xy²]^½, [xy³]^½, . . ., where x = the area of the 576-entry-square of the infinite structure of existence⁶ and y = the area of that square's central square, 16, the series, namely, 1, 1 + 6, 1 + 6 + 24, 1 + 6 + 24 + 96, 1 + 6 + 24 + 96 + 384, . . ., or 1, 7, 31, 127, 511, . . .. This is the equivalent of determining the column number of the longest edge of any such triangle by calculating 1 plus the product of 1.5 and the sequential sum of the number of 1's contained in the edges of the edge-sharing triangles shared along the infinite edge up to and including the triangle under investigation. The column number of the defining longest edge = the number of 1's constituting that edge. Since the longest edge is the first third of a three-digit column none of whose rows are composed of two or more consecutive 1's, the following is the discernible analytic structure of the digits constituting each triangle in the series of edge-sharing triangles of 1's: 1 + 0, + 0 (the merely implicit triangle of 1's inaugurating the infinitely triangular order); 1 + 8, + 15; 1 + 224, + 255; 1 + 3968, + 4095; . . ., where 1 plus the second number subtracted from the total number of 1's in each triangle is the number of 1's in that triangle's longest edge, and 1 plus the third number is the total number of 1's in each triangle. The series which is composed of the sums of these elements of the successive edge-sharing triangles is 1, 24, 480, 8064, . . .. Where E_l is the number of 1's constituting the longest edge of an edge-sharing triangle and E_s is the number of 1's constituting the edge of such a triangle shared with the infinite edge of the triangle created by the infinitely iterative summing of the trinary logic geometric series, E_s = (E_l + 1)/2, and E_l × E_s = the total number of digits in the infinite expansion up to and including the longest edge of the triangle under consideration. This total number of digits minus in sequence, beginning with the first edge-sharing triangle of 1's (shared edge = 4), the corresponding power of the series, x¹, x², x³, x⁴, . . ., where x = 4, = the sum of the elements of each edge-sharing triangle as analyzed, thusly, 28 - 4 = 24, 496 - 16 = 480, 8128 - 64 = 8064, . . ..

The infinitely expanding triangle created by the infinite sequential summing is a right-angled triangle each of whose legs is composed of the identical number of digits. This major triangle and each of its constituent triangular sets of 1's (for the two kinds of the latter see below) is thus a half-square. When the horizontal and vertical legs of this triangle are arranged so as to occupy the equal space corresponding to the equal number of digits in each, the infinite edge shared with the edge-sharing triangles -- the logical hypotenuse -- is to its geometric equivalent in the ratio 1 to 2^½, taking the number of digits as its logical length. At the point of expansion defined by the longest edge of an edge-sharing triangle let that longest edge be the vertical leg, a, of the total triangular arrangement of digits. Then the arithmetic value of a = 2^½ times the square root of the sum of .5 plus the sum of the elements of the edge-sharing triangle as analyzed above. The arithmetic value of the horizontal leg b = 1 + [2(a - 1)/3] and that of the hypotenuse c = b + [(a - b)/2].

The constant ratio of the total arithmetic value of the perimeter of the infinitely expanding triangle to the total number of logical digits occupying it is, at the point defined by the longest edge of each successive edge-sharing triangle of 1's,

to 1, which is at once the ratio of the arithmetic value of the three rows of the cornerstone⁷ computed as end-overlapping sets to the total arithmetic value of those rows calculated as discrete units, as well as 10 times the ratio of the successive edges of the squares in the infinite structure of existence.⁸ The difference between the total arithmetic value of this perimeter of the infinitely expanding triangle of digits and the total number of digits on the perimeter at the point defined by the longest edge is the number of 1's in the shared edge of that triangle minus 1, beginning with the first edge-sharing triangle (shared edge = 4) in accord with the series, x¹ - 1, x² - 1, x³ - 1, x⁴ - 1, . . ., where x = 4. The difference between the total arithmetic value of the digits interior to that total perimeter and the total number of interior digits increases, beginning with the first discernible edge-sharing triangle (shared edge = 4), according to the series, x¹ - (x¹ −1), x¹ + x³ - (x² - 1), x¹ + x³ + x⁵ - (x³ - 1), . . ., where x = 4.

In addition to the infinite expansion of edge-sharing triangular sets of 1's, the infinitely expanding triangle created by the infinite sequential summing of the logical geometric series also contains right-angled non-edge-sharing triangular sets of 1's distributed between the occurrences of the members of the edge-sharing set. Where t₁ is the total number of 1's forming a triangle of either kind and x is the number of 1's in the longest row of 1's in the triangle, then for non-edge-sharing triangles t₁ = x + .5x(x−1) and for edge-sharing triangles t₁ = x². If in these formulas the extreme and mean ratio, φ, is allowed to equal x, then the ratio of [x + .5x(x−1)] to x² = COS 36 (the central angle of the φ-structured regular decagon) = φ/2.

Finally, the absolute discontinuity of the continuum of the geometric series constituted by the series of edge-sharing triangles of 1's is demonstrated in the following way. All horizontally or vertically contiguous 1's in the array of digits produced by the infinite iterative sequential summing are members of a triangular arrangement of 1's. The edge-sharing triangles of 1's are exhaustively composed of such. But the digits between the triangles of 1's which share the infinite edge of the major expanding triangle can only be divided into triangles which reach from the latter's hypotenuse, c, to its horizontal leg, b, that contain horizontally or vertically contiguous 1's without containing the complete minor triangle of which the latter are elements. This feature of the infinite logical array, the perfect breaking of the continuum of the geometric series, is evident when one closely observes the earliest stages of the expansion, as here illustrated:

The total number of 1's in any group of digits lying between any two nearest triangles of 1's sharing the infinite hypotenuse of the infinitely expanding major triangle = 3 times the total number of digits in the expanding major triangle at the point marked by the longest edge of the smaller of the two edge-sharing triangles, while the total number of 0's and

's in any such group = 8 times the total number of 1's constituting that smaller triangle.¹⁰ In any such group the number of 0's = half that total of 0's and

's minus 1, while the number of

's = half that total plus 1. The 1's arranged in right-angled non-edge-sharing triangles appear in triangles whose bases are composed of numbers of 1's found in the sequential summing of the series x¹, x⁰, x¹, x², x³, x⁴, x⁵, . . ., where x = 2, beginning with x for the group between the initial 1 of the major expanding triangle and the first discernible edge-sharing triangle of 1's, and accumulating in each successive group by the addition of triangles whose bases = the next 2 places in the sequentially summed series, the numbers of triangles of each base in each group determined by size in descending order as multiples in order of the series, x⁰, x¹, x², x³, x⁴, x⁵, . . ., where x = 3 (see opening illustration). The total arithmetic value of any such group (where E_s−1 is the number of 1's constituting the shared edge of the smaller of the two edge-sharing triangles) = E_s−1(10E_s−1 - 3).¹¹ At infinity the ratio of the total arithmetic value of any two larger and smaller successive groups of digits lying between edge-sharing triangles of 1's is 16 to 1, the ratio of the successive edge-sharing triangles of 1's beginning with 1. At infinity the ratio of discontinuity = the ratio of continuity. The ratio of the total arithmetic value of the digits between any two nearest edge-sharing triangles of 1's to the total arithmetic value of the expanding major triangle =

, so that the total arithmetic value of the larger edge-sharing triangle of 1's equals the product of 16/10 and the total arithmetic value of the group of digits lying between it and the smaller edge-sharing triangle of 1's.¹²

Where E_l is the number of digits constituting the longest edge of any edge-sharing triangle (whether composed exclusively of 1's or composed of 0's,

's, and 1's) and E_s is the number of digits constituting the edge of such a triangle shared with the infinite edge of the triangle created by the infinitely iterative summing of the trinary logic geometric series, E_s = (E_l + 1)/2, and E_s × E_l = the total number of digits in the infinite expansion up to and including the longest edge of the triangle under consideration. This cumulative total = the product of the total number of digits in the longest edge of such a triangle times the number of digits that share the edge of the major triangle. Thus the infinitely iterative summing of the real trinary logic geometric series generates the matrix of the series of perfect numbers. The perfect numbers -- 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216, . . . -- = the cumulative total of digits up to and including such a triangle where the number of digits in the latter's longest edge is a prime number = the shared edge times the longest edge of that triangle, which, beyond the exceptional and unique 6 whose precipitating prime = even number 2 and the quotient of the addition of whose factors divided by their product (1+2+3)/(1×2×3) = 1, is always one of the series of edge-sharing triangles of 1's as here illustrated

The arithmetic transcription of the logical digits (see above) comprising the edge shared with the infinitely expanding major triangle by the very first edge-sharing triangle (composed of two

's and a 0 and a 1) is

0 = .5 + .5 = 1 = n−1. That of its longest edge is

10 = .5 + 1 + .5 = 2 = n. Thus the exceptional E_s × E_l real trinary logic form of this very first edge-sharing triangle founds precisely the exception (n = 2) of the perfect number (2ⁿ⁻¹)(2ⁿ−1) = 2 × 3 = 6.

The structural economy of the logical form of this triangle -- whereby the arithmetic transcriptions of the digits in the shared and longest edges the product of whose numbers of digits equals the perfect number at once respectively equal n−1 and n -- is unique among all edge-sharing triangles in the matrix of perfect numbers and founds the unique structural economy of the first such number 1+2+3 = 1×2×3 = 6.

In the thinking now occurring for the first time, where there is no nothing and 0 = not nothing,¹⁴ the body of bodies that is the universe is understood to be an infinitely flat surface¹⁵ whose structure is precisely a circular line or 'string' twisted upon itself and infinitely so, so that at infinity it is at once an absolute edge whose constituents are 'points' perfectly discontinuous, a line or 'string' composed of non-spatial non-tangent 'points'/surfaces, i.e., places, sans area, discrete, but without interstices––singularities in lieu of nothings. This absolute edge at once absolute singularity¹⁶ may be pictured abstractly as here illustrated (qua string the beginning could be made with a square,¹⁷ or, indeed, with any polygonal or non-polygonal shape enclosing an area, but the circle permits the simplest illustration), where the infinitely straight line produced by the expansion of the infinite geometric series of twisted circles (here shown in its relation to the perfect numbers together with which it is grounded in the infinitely expanding triangle produced by the iterative sequential summing of the real trinary logic geometric series) is, at the limit, the infinite repetition of the foundational fourfold¹⁸ actually beginning with the third twist in the fourth place:

where the bottom line is the absolute 'punchline' that is the body, the absolute multiplication that is the existing universe––the infinitely punctual line/the absolute(ly spaceless) time line—absolute singularity.

The infinite iterative sequential summing of the real trinary logic geometric series is the foundation of the numeric geometric series where x = 2, 2², and 2⁴(see infinitely expanding triangle first above).

2 and 3 are the only primes not a sum including some combination of two or more primes or reducible thereto. 2 = 1 + 1 = 0 + 2. 3 = 1 + 1 + 1 = 0 + 1 + 2. 2 is the perfectly compact base prime composed of no prime other than itself. 3 is the imperfectly compact base prime composed of 1 and a prime other than itself.

The singularity of the even prime is a function of its perfect compactness. The multiplicity of odd primes is a function of the odd base prime's imperfect compactness.

The perfectly and imperfectly compact base primes are intrinsic factors in the real trinary logic foundation of the numeric geometric series (ibid.) in the following ways:

Where x = the perfectly compact base prime 2 the foundational geometric series begins once and for all (perfectly): 1, 2, 4, 8, 16, 32, 64, . . . (as do likewise the series where x = 2² and x = 2⁴).

Where x = the imperfectly compact base prime 3 the foundational geometric series begins an infinite number of times (imperfectly): 1; 1, 3, 9; 1, 3, 9, 27, 81; . . .. This series does not perfectly begin and does not infinitely end, i.e., is never complete. This series as such does not actually exist foundationally.

Perfect numbers must be prime number multiples of a member of a complete (actually existing) foundational geometric series (see infinitely expanding triangle last above).

The longest edge of an edge-sharing triangle as defined above is the analogue of the logical depth or comprehension of a term and the shared edge defined above is the analogue of the logical breadth or extension of a term as these are understood by C.S. Peirce in "Upon Logical Comprehension and Extension," where logical area or information is understood to be logical breadth × logical depth.¹⁹

Concerning logical breadth Peirce says "that which can be predicated of whatever you please . . .has essential breadth," and "that of which you are at liberty to predicate what you please . . . has essential depth."²⁰ In the case of the infinitely expanding triangle as delineated 2ⁿ⁻¹ can be predicated of any member of the geometric series where x = 2, while a number of things can be predicated of 2ⁿ−1, for example, prime or not prime. E_s × E_l= (2ⁿ⁻¹)(2ⁿ−1) = the logical area or information dimension of an edge-sharing triangle = the total number of digits in the infinite expansion of the infinitely expanding triangle up to and including the longest edge of the edge-sharing triangle under consideration, and is a perfect number when a prime is the predicate of 2ⁿ−1.

As illustrated to the lower left of the infinitely expanding triangle as last depicted, in real trinary logic 0 (not nothing) can be taken to correspond to logical breadth,

(not not nothing) to logical depth, and the product of the former 0

= 1 (not not not nothing) to logical area.²¹ But, since 0 =

= 1, and

= 0, 000 =

, and 111 = 1,²² substantial unity of information additionally involves logical height 1 as that the product of which and logical breadth and depth = the logical volume of information 1 = 0

1, at once the product of the cubes of the discrete logical roots of the latter 1 =

000111. The identification of logical height 1, information 0

= 1, and volume of information 0

1 = 1 at once the integration of the cubes of the latter's roots

000111 = 1, constitutes as such the Unity predicated of Four or the categorical Fourthness in which the Third dimension is predicated of the Second, or substance is predicated of information, as illustrated here²³

Note that although for Peirce logical area or information is understood to be the product of logical breadth and logical depth,

the product of denotation and connotation, the latter are themselves not derived but have an absolute status vis-à-vis information. Missing from this derivation of information is very notation (height). In real trinary logic depth (connotation) is derived from height (notation) and breadth (denotation). Information is the substantial product of denotation (breadth) and connotation (depth) when the last is the identity of very notation (height) with denotation (breath), as here illustrated

where the (infinitely) linear structure that is no information (very breadth or denotation) is imaged in two dimensions and the (infinitely) flat structure of the real trinary logic series (1, 0,

, 1 . . ., height, breadth, depth, information . . ., analogue of the numeric geometric series) is imaged in three dimensions. Here what for Peirce would be no information (very breadth or denotation without depth)²⁴ is made information in the medium of intelligibility (depth) that is notation (height) identified with denotation (breadth). In the thinking occurring before now very comprehension underived equaled no information as did very extension underived. In the thinking now occurring for the first time the identification of notation and denotation is the comprehension that creates information where information was impossible when an underived denotation was the counterpart of an underived comprehension.

¹ Cf. D.G. Leahy, Foundation: Matter the Body Itself (Albany, 1996), Section IV.2, passim; cf. also, on this web, The v_maxwell^1/16 × 1 s/m Cube, Human Body Surface & BSA, & the Infinitely Flat Structure of the Universe.
² Cf. Leahy, Foundation, Section III.1.
³ Cf. Leahy, Foundation, Section IV.2, pp. 519ff.
⁴ The exponents of x here are actually the trinary logic arithmetic series beginning with 0 plus the successive addition of 1 and the logical sorts of 1, 0 and

, thusly, 0, (0)1, (01)0, (010)

, (010

)1, (010

1)0, (010

10)

, . . ., while the resulting trinary logic geometric series is actually the series beginning with 1 successively multiplied by the trinary logic mean proportional,

, and the logical sorts of

, 1 and 0, thusly, 1, (1)

, (1

)1, (1

1)0, (1

10)

, (1

)1, (1

1)0. . .. For

as the radical mean between 0 and 1 in the logical order, cf. Leahy, Foundation, Section III.1, Note to Rule 5 (cf. also Rule 6 and Note). See also, on this web, Measure Beyond Beyond Reach and Real Trinary Logic, the Imaginary Number i, & the Cyclic Series of Complex Numbers.
⁵ Cf. Leahy, Foundation, Sections III.1 and IV.2. If 0 and

are considered not in their equalizing arithmetic transcription but in their purely logical distinction from one another, then the sequence of numbers of digits in the pattern of sets of 1's and sets of 0's and

's, along the edge of the infinitely expanding major triangle, is the geometric series where x = 2, i.e., the series which governs the process of cell multiplication from which all organic life is built up, 1, 2, 4, 8, 16, 32, 64, . . .. So construed, the infinite iterative summing of the trinary logic geometric series is at once the foundation of the geometric series governing biological reproduction and growth. In this connection, see, on this web, The Rabbit Tree & The Logic of Life's Beginning and Beyond the Superego: The Pneumasomatic Human Person. Also, for the trinary logic as template of the life of the Trinity, see, on this web, Thinking Creation Ex Nihilo. For the comparison of real trinary logic and the alternative duality-based ternary logic, see, on this web, Quantum Gravitational vs. Quantum Logic: Virtually Left-handed Real Trinary Logic. In this connection see also D.G. Leahy, Faith and Philosophy: The Historical Impact (Aldershot, 2003), Chapter 7.
⁶ Cf. Leahy, Foundation, Section IV.2.
⁷ Cf. Leahy, Foundation, Section III.1, et passim.
⁸ Cf. Leahy, Foundation, Section IV.2.
⁹ Where α and δ are the Feigenbaum "chaos" constants, 2.502907875095892 and 4.669201609102990, then [(α + δ)/

]⁴

829.44^1.5, exact to the fourth place (cf., on this web, Linguistic Correspondence to . . . [for 82944 see Leahy, Foundation, Sections III and IV, passim; also, on this web, Thinking Creation Ex Nihilo, The Higgs Boson & the Absolute Dead Center Cube: Predictions Within the Probable Range, Report on a Dream & Related Matters & the God Particle, 82944 & the Four Fundamental Forces & the God Particle, and The Difference]).

also has the following relations (where α is the fine structure constant and φ is the extreme and mean ratio): 5769230.bmp (566 bytes)

^.125/

φ. LOG

⁻¹

8.2944^1.5/100. LN

⁻¹

(α × 10368^.5)². (φ

)⁸

82.944/9. 82.944α

[1.1]^½ ×

. (LOG

/LOG .644476259278)^.25

829.3184113/784 (where 829.3184113 is, with a difference less than 1 part per million, 1.110720735⁶⁴, the 64th power of the ratio of the circumference of the circle to the total length of the edges of its inscribed square). Finally, [LOG (.644476259278/

)]^−1/64

(37 ×

)

^/37

[1.1]^½ (the triple-cube body- face-diagonal ratio), and, where 10368^.5 is the edge of the 'dead center cube',

which is the value of the fine structure constant (0.007297 353 08) exact to the 8th place (for the 37 logical diagonals of the triple-cube, see, on this web, Logically Outfitted Cube(s), Maxwell's Constant & the Trinary Logic Triple-cube, and The v_maxwell^1/16 × 1 s/m Cube, Human Body Surface & BSA, & the Infinitely Flat Structure of the Universe; also Leahy, Foundation, Section IV.2, p. 524, n. 98). For the physical manifestation of the ratios

and

as foundational in astronomy and astronomically related areas of archaeology, see, on this web, Sorted Diagonals, Addenda. See also, on this web, The Langerhans-Epidermal Cell Ratio & the Structure of the Epidermis.
¹⁰ At infinity the ratio in such groups of the number of 1's to the total of 0's and

's is .75. Qua respective arithmetic values the ratio of the total of 0's and

's to 1's in such groups is

. For the counterintuitive relationship of these ratios in the prediction of body surface area, see, on this web, The Langerhans-Epidermal Cell Ratio & the Structure of the Epidermis. .75 ×

= .5, at infinity the ratio of the total value of the edge-sharing triangles to the total number of digits in the major triangle (see above in the text). See also, on this web, The Golden Bowl Structure: The Platonic Line, Fibonacci, and Feigenbaum . For

and the ratio of the values of the elementary charge and gravitational constant, see, on this web, Physical Variables, Placedness, & the Absolute Dead Center Cube. For

and the Beginning/Power Body ordered cornerstone, see, on this web, The Real Beyond the Void: the Beginning/the Power Body.
¹¹ Beginning at the point where the larger of the two is the first discernible triangle of 1's (shared edge = 4), 10E_s−1 - 3 is the series (where x = 4, y = 5 × 6, and z = 7), z, z + y, z + y + x⁰(yx), z + y + x⁰(yx) + x¹(yx), z + y + x⁰(yx) + x¹(yx) + x²(yx), z + y + x⁰(yx) + x¹(yx) + x²(yx) + x³(yx), . . ., or, 07, 37, 157, 637, 2557, 10237, 40957, . . .. Beginning with the edge-sharing triangle between the first two discernible edge-sharing triangles, note that the last two digits of the successive factors are alternately 37 and (37 + 20 =) 57. 57/37 = 1 + 20/37, where 20/37 is the ratio of the face diagonals to the total number of diagonals in the trinary logically outfitted triple-cube (cf. above, n. 9, including the reference to Leahy, Foundation).
¹² Cf. above.

¹⁴ Leahy, Foundation III.1. See also, on this web, Thinking Mathematics Sans Nothing.

¹⁵ Ibid., III.5, pp. 433ff.

¹⁶ See, on this web, The Simplicity & Syntax of the Concepts, Immediacy, Mediation, Omnipotence, & Beginning, where, effectively, singularity = placedness × particularity.

¹⁷ Leahy, Foundation IV.2, Excursus Circularis, pp. 535f., where the infinite straight line that is the circumference of Galileo's infinite circle is understood to turn "turning absolutely straight" qua the infinitesimal limit that is the fourfold minimum order, there initially a square inscribed within a circle.

¹⁸ Ibid., III.1, et passim.

¹⁹ C.S. Peirce, Writings of Charles S. Peirce: A Chronological Edition Vol. 2 (Bloomington, 1984), pp. 70ff. Cf., also, D.G. Leahy, Beyond Sovereignty: A New Global Ethics and Morality (Burlington and Aldershot, 2010), Section III.1-2.
²⁰ Peirce, loc. cit.
²¹ For 0 = not nothing, etc., cf., above, n. 13, also, on this web, Note to Faith and Philosophy Further to the Ontology of Real Trinary Logic, including further references.
²² Leahy, Foundation, Section III.1.
²³ For 3 predicated of 2, cf., on this web, Note to Faith and Philosophy Further to the Ontology of Real Trinary Logic. Also, for the minimum proportion 4 : 1 :: 2 : 3, cf. Leahy, Foundation, Section III.1
²⁴ Cf. Peirce, Writings, Vol. 2, p. 83: ". . . when there is no information, there is either no depth or no breadth, and conversely."