“Principia Numerologica” the Mysteries of # Theory, BOOKS I & II. Originally © Dec. 11, 2008, by: Jon Gee, Second edition © July 26, 2022, by: Jon Gee.
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“Principia Numerologica” the Mysteries of # Theory, BOOKS I & II. Originally © Dec. 11, 2008, by: Jon Gee, Second edition © July 26, 2022, by: Jon Gee.
BOOK I: the First 10 Mysteries. CHAPTER I.1: the mysteries of One. THEOREM THEOREM THEOREM THEOREM THEOREM THEOREM
I.1.1: I.1.2: I.1.3: I.1.4: I.1.5: I.1.6:
√1 (square root of One) = 1 (One) / (divided by) 1 (One). √1 (square root of One) X (times) √1 (square root of One) = 1 (One). √1 (square root of One) X (times) 1^2 (One squared) = 1 (One). 1^2 (One squared) = 1 (One) X (multiplied by) 1 (One). 1^2 (One squared) + (plus) 1^2 (One squared) = 2 (Two) X (times) 1 (One). 1^2 (One squared) X (times) 1^2 (One squared) = 1 (One) / (over) 1 (One).
PART I.1.1: One as Singularity; One in 1-space. POSTULATE I.1.1.1: One contains both Zero (either at one end of a #-line or in the middle of a graph) and Infinity (either at the opposite end of a #-line or within each co-ordinate thereon as fractions or decimals).
AXIOM I.1.1.2: Let any given unit, > (greater than) Zero and < (less than) Infinity, be = (equal to) One.
LEMMA I.1.1.3: On the #-line from 0 to ∞, 1 Intersects any point as an equal #-line at 90°.
THEOREM THEOREM THEOREM THEOREM
I.1.1.4: I.1.1.5: I.1.1.6: I.1.1.7:
0^0 (Zero to the zeroth exponent) = 1 (One). 1 (One) / (divided by) ∞ (Infinity) = (equals) 0 (Zero). Either Indeterminate Form (1 or ∞) X (times) itself (AKA squared) = itself. Either Indeterminate form (0 or 1) / (over) itself = itself.
Part I.1.2: One as Arithmetical; One in 2-space. POSTULATE I.1.2.1: One is the basis for Binary Probability; (plus/minus, yes/no, none/all, etc.) AXIOM I.1.2.2: One is a single Unit between two polar opposites: Zero and Infinity. LEMMA I.1.2.3: One (diameter) : (is to) 2-space :: (as) Zero (point) : (is to) Infinity (circumference).
THEOREM I.1.2.4: In 2-space, One (radius) is the GCF and Union of Zero (origin) and Infinity (area).
PART I.1.3: One as Exponential; One in 3-space. POSTULATE I.1.3.1: One is the Monad of 3-space; (Trinary, Trinity, Trigonometry, Dialectics, etc). AXIOM I.1.3.2: One is a single Object of any scale, reducible to Zero, expandable to Infinity. LEMMA I.1.3.3: One (atom) : (is to) 3-space :: (as) Zero (quantum) : (is to) Infinity (cosmos). THEOREM I.1.3.4: In 3-space, One is the LCD and Intersection of all co-ordinates (Zero, Infinity).
CHAPTER I.2: the mysteries of Two. POSTULATE I.2.1: Two is the first Prime #. AXIOM I.2.2: Two is evenly divisible only by itself and 1 (One) THEOREM I.2.3: 2 (Two) / 2 (Two) = 1 (One). THEOREM I.2.4: 2 (Two) / 1 (One) = 2 (Two). POSTULATE I.2.5: Two is the first Even #. AXIOM I.2.6: Every second # following 2 (Two) is Evenly divisible by 2 (Two).
LEMMA I.2.7: Zero is Even, One is Odd, and Infinity is neither Even nor Odd.
LEMMA I.2.8: Half of all #’s can divide evenly into two halves. PART I.2.1: Two as Fractal; Two in Mathematics. POSTULATE I.2.1.1: 2 (Two) = exponential expansion.
AXIOM I.2.1.2: any Even # (divisible by Two) squared = an Even #.
LEMMA I.2.1.3: any Even # X (Times) another Even # = an Even #.
THEOREM I.2.1.4: 2 + 2 = 2 X 2 = 2^2. POSTULATE I.2.1.5: 3 (Three) = triangular expansion.
AXIOM I.2.1.6: any Odd # X (times) another Odd # = an Odd #.
LEMMA I.2.1.7: any Odd # X (times) any Even # = an Even #.
THEOREM I.2.1.8: 2 (2^1) : 3 (3^1) :: 6 (3X2) : 9 (3^2) :: 12 (3X2^2) : 18 (2X3^2) = 2/3.
PART I.2.2: Two as Gnomon; Two in Geometry. POSTULATE I.2.2.1: A = πr^2, where A = area & r = radius; of a circle.
THEOREM I.2.2.2: 4π = π2^2. THEOREM I.2.2.3: 16π = π2^3. POSTULATE I.2.2.4: for any Even # n, nπ = [2π X (1/2 n)]^2.
THEOREM I.2.2.5: 6π = (2π3)^2. POSTULATE I.2.2.6: for any Odd # n, nπ = [2π X (n-1 X 1/2)] + π.
THEOREM I.2.2.7: 5π = (2π2) + π. THEOREM I.2.2.8: 7π = (2π3) + π.
PART I.2.3: Two as π spiral, 2 at 90°.
POSTULATE I.2.3.1: correspondence between the Archimedean (circular) and Equiangular (90° right angle or square) spirals, due to the equivalence in Area between any given circle and its certain square, or vice versa, implies a relative pattern of overlapping defined by a ratio of √2 to √3.
AXIOM I.2.3.2: √2 = the diagonal of a base-1 Unit square. AXIOM I.2.3.3: √3 = the diagonal of a base-1 Unit cube. THEOREM I.2.3.6: √2 : √3 :: 1^2 : 1^3 :: 1/3 : 2/3 :: 1:3 :: 2:3 :: n^x : n^(x+1). PART I.2.4: Two as ⲫ spiral, 2 @ 45°. POSTULATE I.2.4.1: Twin π progression spirals with overlapping origin points, oriented at 90°, 60° or 30°, form harmonious scalar wavelengths.
POSTULATE I.2.4.2: Twin ⲫ progression spirals with overlapping origin points, oriented at 36°, 45° or 54°, form harmonious scalar wavelengths.
AXIOM I.2.4.2: Two spirals, one π and one ⲫ, form a “ⲫ/π” graphic spiral.
LEMMA I.2.4.3: the ⲫ/π spiral forms harmonious scalar wavelengths at 30°, 36°, 45°, 54°, 60°, 90° (and their co-linears in other quadrants).
CHAPTER I.3: the mysteries of Three. POSTULATE I.3.1: 1 X 1 = the Origin Point, Singularity or Monad.
AXIOM I.3.2: 1 / 0 = Zero; or, One divided by Infinity. POSTULATE I.3.3: 1^1 = the Unit Line, Light in Extension, Ray.
AXIOM I.3.4: √1 = One; or, One divided by One. POSTULATE I.3.5: 1^2 = the Unit Square, Love Under Will, πr^2.
AXIOM I.3.6: √2 = Unit Square’s diagonal & circle’s diameter.
POSTULATE I.3.7: 1^3 = the Unit Cube, Life In Liberty, 2πr^2.
AXIOM I.3.8: √3 = Unit Cube’s diagonal & sphere’s diameter.
PART I.3.1: Trigonometry; 3 in 2-space = 180°. fig I.3.3.1: four types of regular triangle. POSTULATE I.3.1.1: 3 is the minimum possible # of angles at all scales. AXIOM I.3.1.2: in 1-space, 3 acts as a measure of number. (@ 30°) AXIOM I.3.1.3: in 2-space, 3 acts as a measure of distance. (45° / 90°) AXIOM I.3.1.4: in 3-space, 3 acts as a measure of depth. (36° / 54°) AXIOM I.3.1.5: in 4-space, 3 acts as a measure of duration. (90° < +)
fig.s I.3.3.1.A - I.3.3.1.I: special cases of triangles in 2-space.
fig. I.3.3.1.J: trigonometric expansion @ 30° & 45°.
fig. I.3.3.1.L: 10^2 between 8^2 and 12^2.
fig. I.3.3.1.K: 7^2 between 6^2 & 8^2.
fig. I.3.3.1.M: 13^2 between 12^2 and 16^2.
PART I.3.2: Three as dimension; 3 in 3-space = 270°. POSTULATE I.3.2.1: 3-d^4 Cosmos is 3, Infinite, 2-d sheets or planes, intersecting at all 1-d points.
fig. I.3.2.2. B: 2-s^3-t = 2-d space. D: 4-s^5-t = 4-d hyperspace. A: 1-s^2-t = 1-d time. C: 3-s^4-t = 3-d spacetime. CHAPTER I.4: the mysteries of Four. POSTULATE I.4.1: 4^1 = most basic, Even Shape.
AXIOM I.4.2: 4^1 = 2^2. AXIOM I.4.3: 4^2 = 2^3. AXIOM I.4.4: 4^3 = 2^4. AXIOM I.4.5: 4^4 = 2^5. THEOREM I.4.6: 4^n = 2^(n+1).
POSTULATE I.4.7: 4 squared circles with overlapped centerpoint constitute a ray at 180°.
POSTULATE I.4.8: 4 levels of non-verbal communication arise when this ray is at 90°.
THEOREM I.4.9: Both angles depict the same 4 levels of the ray, lowest = least, top = most.
fig. I.4.10.A: trig. of circle^2 up to A = 256.
fig I.4.10.B: intersections of the ⲫ/π spiral.
CHAPTER I.5: the mysteries of Five.
POSTULATE I.5.1: re. the “stealth” tile.
AXIOM I.5.2: exponential expansion rate.
LEMMA I.5.3: legs count dots, hypotenuses, diagonals.
THEOREM I.5.4: 6 dots = 5^2.
fig I.5.A: base-5 “stealth” tile.
fig. I.5.B: base-5 “tree” tile. POSTULATE I.5.5: re. the “tree” tile. AXIOM I.5.6: arithmetical expansion rate. LEMMA I.5.7: dots counted horizontal, squares counted vertical.
PART I.5.1: the Pentacle. fig. I.5.1.C: the base-4 “Tetractys” of 10 + 5 Pents.
fig. I.5.1.D: the “skip-1” Pentagram.
POSTULATE I.5.1.1: re. the “skip-1” Pentagram Star.
AXIOM I.5.1.2: 5/5 (stellations divided by legs).
LEMMA I.5.1.3: “1:5” (in a Pentagram of 5 stellations, 1 is transected by the Pent’s legs).
fig. I.5.1.E: the “Pythagorean,” 5-in-5 Pentagram.
fig. I.5.1.F: the 5 “Platonic” Solids.
POSTULATE I.5.1.4: re. the “5 within 5” Pythagorean Pent.
AXIOM I.5.1.5: 5 “within” = smaller. 5 “w/out” = larger.
THEOREM I.5.1.7: 5^1 : 5^5 :: (1:5).
fig.I.5.1.G: the Pentagram & Hexagram.
fig. I.5.1.H: the “5 within 5” Pent, color-coded.
CHAPTER I.6: the mysteries of Six. POSTULATE I.6.1: if 1 -> 2 = 1/4 of A circumference, &, if A -> B = 1/6 of “I” circumference, when n-space = 3; # possible Vectors = 6…
AXIOM I.6.2: as position 1 -> 2, so space A -> B…
LEMMA I.6.3: Vector = n-space (#-dimensions) X 2…
LEMMA I.6.4: “Real” Vector = # of possible directions of movement…
THEOREM I.6.5: for 3-space, thus 6-Vectors… AXIOM I.6.6: as space A -> B, etc. “chord” I = time…
LEMMA I.6.7: “Actual” Vector = only spin or counter-spin per direction…
THEOREM I.6.8: for 4-space, thus 8-Vectors… LEMMA I.6.9: Conservation to 1 = effects all scales…
POSTULATE I.6.10: Conservation of manifestation = Probability collapsing dimensions.
fig I.6.A: base-6 from inertia. fig. I.6.B: Six, base-6 circles. POSTULATE I.6.11: due to scale, base-4 : time :: base-3 : space.
LEMMA I.6.12: Cosmos = 3-space + 2^2-time.
THEOREM I.6.13: (1/4) Volume = (1/3) n-space.
PART I.6.1: the Hexagram. POSTULATE I.6.1.1: 6 “within” 6 “counter-rotate,” thus: (fig. I.6.B).
LEMMA I.6.1.2: 6 “within” rotate clock-wise. LEMMA I.6.1.3: 6 “w/out” rotate counter-clockwise.
THEOREM I.6.1.4: 6 “in” + 6 “out” = time’s gears.
POSTULATE I.6.1.5: re. the Unicursal Hexagram.
AXIOM I.6.1.6: there are 6 combined, base-4 elements that do not repeat as doubles.
LEMMA I.6.1.7: not depicted: 4 repeated double elements.
THEOREM I.6.1.8: the Unicursal Hexagram expresses a pattern between combined elements.
POSTULATE I.6.1.9: re. the 6 Questions of Reasoning. AXIOM I.6.1.10: can be arranged on a hexagon / hexagram. LEMMA I.6.1.11: can be related by geometry thereon. THEOREM I.6.1.12: using √2 and √3 to determine thus, the best assignment for each vertex to one of the 6 Questions can be found.
POSTULATE I.6.1.13: re. 6 in 3-space
LEMMA I.6.1.14: hexagon : cube :: hexagram : stellated-octahedron.
LEMMA I.6.1.15: stellated-octahedron : cube :: hexagram : hexagon.
THEOREM I.6.1.16: 8 = 6 + 2, where 2 = √4 as time & 6 = directions in 3-space.
POSTULATE I.6.1.17: re. the “Meta-form” (3 ring gyroscope + 3-star intersection).
AXIOM I.6.1.18: any point on the gyroscope or the 3-star’s edges / corners is spun by 6.
LEMMA I.6.1.19: in this event (model) we see 3 directions are spin-Vector, 3 counter-spin.
THEOREM I.6.1.20: 3-space X 3 Vectors = Time. POSTULATE I.6.1.21: re. any object dropped through this gyroscope would enter a holographic wormhole for the time it would be inside the gyroscope.
AXIOM I.6.1.22: any point within the Meta-form is reducible to its central origin-point.
LEMMA I.6.1.23: within this event (model) we see 6 directions are spun simultaneously.
THEOREM I.6.1.24: 3-space X 6 Vectors = Time^2.
PART I.6.2: the Hyper-Cross. POSTULATE I.6.2.1: the hyper-cubic hyper-cross of hypercubic hyper-crosses [HCHCoHCHC] is formed from an alternating pattern of hyper-cubes and hyper-crosses, replicated throughout.
AXIOM I.6.2.2: a hyper-cube X a hyper-cross = a hypercubic hyper-cross. a hyper-cubic hyper-cross X a hypercubic hyper-cross = a [HCHCoHCHC].
LEMMA I.6.2.3: the purpose of the hyper-cubic hyper-cross of hyper-cubic hyper-crosses [HCHCoHCHC] is as a “memory castle.” THEOREM I.6.2.4: by focusing on certain forms & associating them with memories, our memories become clearer.
POSTULATE I.6.2.5: the hyper-cube and hyper-cross are such forms. AXIOM I.6.2.6: when applied to physics, this model prospers.
LEMMA I.6.2.7: it can be taken to stand for spin.
LEMMA I.6.2.9: Such explains the QBLH, as a hyper-cube @ perigee or a nested hyper-cross.
THEOREM I.6.2.10: ⲫ^2 + e^2 - π^2 = 0.13748568659; alpha = 1/137.03599913.
CHAPTER I.7: the mysteries of Seven.
fig. I.7.B: the Triskele of 6’s and Penrose triangle of 7’s. fig. I.7.A: the “skip-2” Heptagram. fig. I.7.C: 7 classic Planets. AXIOM I.7.1: 7 is the Prime, Odd Sum for combining trig. (3) and base-4 physics.
POSTULATE I.7.2: by changing base-unit, the first 7 “Kamea” represent 7 forms of direct, verbal communication.
AXIOM I.7.3: the heptagram has 2 stellations divided by one leg. (cf. Pentagram.)
AXIOM I.7.4: the 7 “Planetary” spheres can overlap as stations on a flattened torus.
THEOREM I.7.5: the mysteries of 7 combined: (3 + 4 = 7) + (3 X 4 = 12) + 3 = 22. where: 22 / 7 = π.
CHAPTER I.8: the mysteries of Eight.
POSTULATE I.8.1: re. the model of the “hidden” 8th cube within the model of the base-7 hypercross, and re. the model of the combined, double, base-4 Elements on the diagonal of a hypercube.
AXIOM I.8.2: the dual-outlined cube between the inner-cube and the surrounding 6 of the hypercross represents a “hidden,” 8th, hyper-cube central to and surrounded by the hyper-cross.
LEMMA I.8.3: the interior dual-Elements correspond to the exterior trigrams, and the exterior dual-Elements correspond to the interior hexagrams (from the I Ching). Lower-right shows the diagonal.
THEOREM I.8.4: 6 (combined doubles) + 2 (doubled doubles) = 8 (double trigrams), X 2 = 8 (double hexagrams).
PART I.8.1: the Tesseract.
POSTULATE I.8.A.1: @ 45° between apogee and perigee, the hypercube appears as an octagonal / octagram “Tesseract.”
POSTULATE I.8.B.1: the Tesseract is a 4-space shape, and can thus lattice the Probabilities of Vectors over Time = Manifestation.
AXIOM I.8.B.2: a lattice expressing a certain # of relationship types between a certain # of variables can depict equations graphically.
LEMMA I.8.B.3: by re-combining “Probability” and “Quanta,” it is possible to lattice the equations governing a temporal singularity.
LEMMA I.8.B.4: for Zero-time and 1-space, the equations collapse beyond “c,” the Variable Speed of Light in a vacuum.
THEOREM I.8.B.4: @ “c,” matter > energy; & @ c^2, matter < energy POSTULATE I.8.C.1: the interior and exterior cubes of a hypercube have the same volume, because they are the same cube over time.
AXIOM I.8.C.2: the interior and exterior square of the shadow of a hypercube, a Tesseract, have the same area.
LEMMA I.8.C.3: therefore, the 4-space hypercube and its shadow, the Tesseract, have the same Area^3 as the Tesseract^2.
THEOREM I.8.C.4: 1^1 = Origin Point; 1^2 = Unit Square; 1^3 = Unit Cube 1^4 = hypercube, depicted in 2-space as a Tesseract.
THEOREM I.8.C.5: hypercube : Tesseract :: cube : square :: hex : origin. THEOREM I.8.C.6: hypersphere : Torus :: sphere : circle :: Zero : origin. POSTULATE I.8.D.1: certain #-sets correspond 1:1 with infinite and transfinite measurements that transcend the physical Cosmos.
THEOREM I.8.D.2: the Tau-sub-Tau Tesseract > aleph-sub-omega outer hypersphere > aleph-sub-sigma inner-sphere > aleph-sub-n torus.
POSTULATE I.8.E.1: within Tau-sub-Tau are the “Planetary” spheres, the sefirot and the engine of creation of Cosmos from singularity.
AXIOM I.8.E.2: the engine of creation = aleph-sub-n; the “Planets” = aleph-sub-sigma; the sefirot = aleph-sub-omega; time = Tesseract.
THEOREM I.8.E.3: the measurements (origin point -> hyper-shape) are 1:1 = the Tau-sub-Tau lattice of the infinite & transfinite #-sets.
POSTULATE I.8.F.1: the pattern connecting these concepts across levels forms a spiral over time according to the equation: QBLH = 137.
AXIOM I.8.F.2: ⲫ and π are universally ubiquitous measurements because Geometry : dimension :: Gravity : manifestation.
AXIOM I.8.F.3: the equation, 1 / (ⲫ^2 + e^2 - π^2) = alpha, the “fine structure constant” occurring throughout most equations of physics.
THEOREM I.8.F.4: the ⲫ (negentropy) / π (entropy) spiral therefore transcends the Tau-sub-Tau tesseract of time.
THEOREM I.8.F.5: The relationship between ⲫ/π and Tau-sub-Tau is operative @ a 45° angle to apogee of spiral and perigee of tesseract.
CHAPTER I.9: the mysteries of Nine.
CHAPTER I.10: the mysteries of Ten.
fig. I.10A: the 10 sefirot as min. sphere “packing.” fig. I.10B: the Tetractys dates back to Genesis. fig. I.10C: the lengths of lines of a hyper-tetrahedron, a tesseract and combined lattice.
POSTULATE I.10.1: because 10 is the minimum # of points on a 2-cube lattice (the shadow of a hypercube at perigee, seen from 45°, or above one of its “vertexes”), it is seen as the maximum expansion of the singularity, point, or “event” in time, as the min. # of Vector-directions in 4space.
LEMMA I.10.2: 10 retains this essential corollary in any configuration. Therefore, regardless of the arrangement of 10 attributes, the same essential characteristics will describe them: 4-space Vectors.
AXIOM I.10.3: although there can be more (in accordance with the ⲫ/π spiral harmonic), there are a minimum of 10 Vectors in 4-space. Thus, 4-space mathematics is minimum base-10.
AXIOM I.10.4: the maximum regular 2-d shape to translate into the minimum regular 3-d solid is the base-5 Pentagon, as the base-12 Dodecahedron. Therefore, the min/max of 4-space is base-10.
BOOK II: the Mysteries of the Combined #’s. CHAPTER II.1: the mysteries of Expansion Rates. PART II.1.1: Basic Mathematical Expansion Rates; Squares & rectangles.
fig. II.1.A: the regular expansion rate of a square. POSTULATE II.1.1: re. the Even expansion rate of 4^2. AXIOM II.1.1.2: consider the Unit square as 1^2, and the square of the unit square as 2^2 in planar 2-space.
LEMMA II.1.1.3: the Unit square in 2-space is 1X1 with a diagonal “root” = √2.
THEOREM II.1.1.4: the Unit square squared in 2-space is = 2^2, or the Area-4 square.
AXIOM II.1.1.5: consider the square of the Unit square as = 2^2; thus the cube of the Unit square in 2-space is = to 2^3 = 8 in volume, & in 2-space = to 4 in Area.
LEMMA II.1.1.6: thus in 3-space, the Unit square cubed = the same as the 4-square in 2-space.
THEOREM II.1.1.7: the Unit square cubed in 3-space = the Area-4 square in 2-space.
AXIOM II.1.1.8: expansion in 2-space measures the shadow of expansion in 3-space.
fig. II.1.1.B: the irregular expansion rate of a 1:2 rectangle. POSTULATE II.1.1.2: re. the Odd expansion rate of 6+8.
AXIOM II.1.1.9: the “expansion-rate” of any regular, Unit-square #-base system is Area^2 + 8.
LEMMA II.1.1.10: this applies to all systems that use the Unit square as the base-unit, irregular, regular or both.
THEOREM II.1.1.11: “[n - (minus) 1] n” = “n^2 (minus) n.”
AXIOM II.1.1.12: the “expansion-rate” of regular, square base systems is Even.
LEMMA II.1.1.13: any Odd # raised by an Even # square will = a regular square expansion-rate, & vice versa.
THEOREM 1.14: any Even #-base square system will be = to the base-Unit + 8 per each squared Area.
AXIOM II.1.1.15: the “expansion-rate” of irregular, rectangular base systems is Odd.
LEMMA II.1.1.16: any Even # raised by an Odd # square will = an irregular, rectangular expansion-rate, etc.
THEOREM II.1.1.17: any Odd #-base square-system will be = to the base-Unit + 8 per each squared Area.
PART II.1.2: Basic Geometric Expansion Rates; Stellations divided by legs.
POSTULATE II.1.2.1: re. the skip-1 Pentagram = 1 stellation divided by 1 leg = base-5 Single-Unit.
AXIOM II.1.2.2: the base-5 Single-Unit is = 5 X 1 = 5, therefore the Pentagram contains 1 Pentagon.
LEMMA II.1.2.3: the base-5 single Unit is 1/5 X 5, therefore 1 stellation divides by 1 leg.
THEOREM II.1.2.4: 540° / 5 angles = 108°, & 180° - 108° = 72°; per pentagon & pentagram.
POSTULATE II.1.2.5: re. the skip-2 Heptagram = 2 stellations divided by 1 leg = base-7 Unit X 2.
AXIOM II.1.2.6: the base-7 Unit by 2 is = 7 X 2 = 14, therefore the Heptagram contains 2 Heptagons.
LEMMA II.1.2.7: the base-7 Unit is 1/2 X 14 = 7, therefore 2 stellations divide by 1 leg.
THEOREM II.1.2.8: (7 - 2) 180° = 900°; & 180° - (900° / 7) = 52°; per Heptagon & heptagram.
POSTULATE II.1.2.9: re. the skip-3 endecagram = 3 stellations divided by 1 leg = base-11 Unit X 3.
AXIOM II.1.2.10: the base-11 Unit by 3 is 11 X 3 = 33, therefore the Endecagram contains 3 other endecae.
LEMMA II.1.2.11: the base-11 Unit is 1/3 X 33 = 11, therefore 3 stellations divide by 1 leg.
THEOREM II.1.2.12: (11 - 2) 180° = 1620°; & 180° (1620° / 11) = 33°; per endecagon & endecagram.
POSTULATE II.1.2.13: re. the skip-4 tridecagram = 4 stellations divided by 1 leg = base 13 Unit X 4.
AXIOM II.1.2.14: the base-13 Unit by 4 is 13 X 4 = 52, therefore the tridecagram contains 4 other tridecae.
LEMMA II.1.2.15: the base-13 Unit is 1/4 X 52 = 13, therefore 4 stellations divide by 1 leg.
THEOREM II.1.2.16: (13 - 2) 180° = 1980°; & 180° (1980° / 13) = 27°; per tridecagon & tridecagram.
POSTULATE II.1.2.17: re. the Even and Odd expansion rates wherein the (+) series = Even & (X) series = Odd.
AXIOM II.1.2.18: #’s added to Even #s are 1/2 a spiral expansion rate, #’s multiplied by Odd #’s are the other 1/2.
LEMMA II.1.2.19: where n is a base-Unit, the expansion rate need not necessarily follow the base-n Unit’s squared#’s.
THEOREM II.1.2.20: 5 + 9 = 14; 14 + 13 = 27; 27 + 6 = 33; 33 + 19 = 52; 52 + 20 = 72; 14 + 5 = 19; 13 + 6 = 19; 9 X 3 = 27; 11 X 3 = 33; 24 X 3 = 72; 360 / 72 = 5; 180 / 36 = 5.
PART II.1.3: Advanced Mathematical Expansion Rates; Number ladders.
POSTULATE II.1.3.1: the “Sequential set of Multiplicatives” counts every other multiplicative factor in sequence.
AXIOM II.1.3.2: the sequence begins with the line 1 X 2 = 2, then skips one place ahead to the line 2 X 2 = 4, etc.
LEMMA II.1.3.3: here the sequence is depicted extending to limit-15, on the line 30 = 2 X 15, in the lower-right.
THEOREM II.1.3.4: the sequential set is 1/2 the sums on the opposite side, is base-2, and comprises the “Even set.”
POSTULATE II.1.3.5: the “Odd set of Multiplicatives” counts every other multiplicative factor as an Odd #.
AXIOM II.1.3.6: the sequence begins with the first line, 1 X 1 = 1, the skips one place to the line 3 X 1 = 3.
LEMMA II.1.3.7: here the sequence is depicted extending to limit-31, the last interval in the lowest-right chart.
THEOREM II.1.3.8: the Odd set is 1 X the same # opposite it in the Sequential set, is base-1, & compliments the “Evens.”
POSTULATE II.1.3.9: the “Sequential Set of Additives” are the #’s inner-most on the chart, counting in sequence.
AXIOM II.1.3.10: the sequence begins with 1, next 2, etc. LEMMA II.1.3.11: the sequence here ends with limit-30. THEOREM II.1.3.12: the products of the multiplicative factors are herein interpolated, both Even and Odd.
POSTULATE II.1.3.13: the “Factorial Set of Additives” is = to the Even set of Multiplicatives X (times) the Odd set.
AXIOM II.1.3.14: the sequence begins with 1 X 1 = 1, but is not differentiated until line 1 X 3 = 3, skip to 3 X 2 = 6.
LEMMA II.1.3.15: here the sequence ends with limit-465, the last sum and factor in the lower right of the chart.
THEOREM II.1.3.16: the Factorial Set is a ratio of the Even and Odd multiplicative sets, and thus combines both.
POSTULATE II.1.3.17: the complete chart given here graphs arithmetic and exponential expansion rates both.
AXIOM II.1.3.18: the Even Multiplicatives yield arithmetic expansion, & the Odd Multiplicatives yield exponential.
LEMMA II.1.3.19: the Even set is base-2, and the Odd-set is base-1; the Factorial Additive set is sums between them.
THEOREM II.1.3.20: for any # n; if n = Even, then n + (n -1) + (n - 2) + (n - 3), ... [n - (n - 1) = 1], etc. = n (2n + 1) = 2n^2 + n.
THEOREM II.1.3.21: for any # n; if n = Odd, then n + (n -1) + (n - 2) + (n - 3), ... [n - (n - 1) = 1], etc. = 2 (n + 1) + 2 (n 1) = 4n.
PART II.1.4: Advanced Geometric Expansion Rates; Ideal lattices.
POSTULATE II.1.4.1: a combined hyper-tetrahedron & hyper-cube lattice in 4-space.
AXIOM II.1.4.2: a hyper-tetrahedron & hypercube, both at 45° & perigee, form the combined lattice.
LEMMA II.1.4.3: the hyper-tetrahedron is @ perigee what a stellated-octahedron is at apogee or “nested.”
LEMMA II.1.4.4: the hypercube is @ 45° what a tesseract is at perigee, seen from above a vertex.
POSTULATE II.1.4.5: the 36 legs of the lattice formed by a tesseract over-top a stellated-octahedron.
LEMMA II.1.4.6: the 22 legs of a hypercube lattice @ 45° perigee: 3 horizontal, 7 vertical, 12 diagonal.
LEMMA II.1.4.7: the 12 legs of a hyper-tetrahedron @ 45° perigee: 2 horizontal, 3 vertical, 7 diagonal.
AXIOM II.1.4.8: the 36 legs of the combined lattices @ 45° perigee: 5 horizontal, 11 vertical, 20 diagonal.
POSTULATE II.1.4.9: re. the 17 vertexes of the lattice formed by a tesseract over-top a stellated-octahedron.
LEMMA II.1.4.10: 10 vertexes of hyper-cube lattice @ 45° perigee: 4 types: 2X bases-3,5&6, 4X base-4.
LEMMA II.1.4.11: 7 vertexes of a stellated-octahedron @ 45° perigee: 2 types: 6X base-3, & 1X base-6.
AXIOM II.1.4.12: 17 vertexes of the mixed lattices @ 45° perigee: 5 types: 2X base-3&6, 3X base-8, 4X base-4, & 6X base-5.
POSTULATE II.1.4.13: the central 7 vertexes of the lattice of a tesseract over a stellatedoctahedron.
AXIOM II.1.4.14: 2 stellated-octahedron legs over-lap the central vertical axis, and 3 hyper-cube legs over-lap it. AXIOM II.1.4.15: 5 legs within 5 vertexes besides + 1 leg each for the top- and bottom-most vertexes.
AXIOM II.1.4.16: there are 6 vertical legs between the 7 central vertexes, as 7 chakras and 64 I Ching hexagrams.
AXIOM II.1.4.17: the central 7 vertexes also correspond to the 7 methods of verbal communication.
POSTULATE II.1.4.18: re. the 72 traits of the lattice.
THEOREM II.1.4.19: 3 X 2 = 6; 6 X 2 = 12; 12 X 3 = 36 36 X 5 = 180; 360 / 5 = 72.
THEOREM II.1.4.20: 22 + 14 = 36; 36 + 14 = 50; 50 + 14 = 64; 64 + 8 = 72.
THEOREM II.1.4.21: 22 + 50 = 36 X 2.
PART II.1.5: Greater Mathematical Expansion Rates; “Square” & “compass.”
POSTULATE II.1.5.1: re. the base-13 “compass” is comparable to a candelabra or apron in motif, however it conveys symbolically the formation of a cone implied by the square-base expansion rate - as the square-base forms a step-Pyramid, a circle’s circumference folds up by 1/3 to form a cone.
POSTULATE II.1.5.2: re. the base-15 “square” trestle is comparable to a “tiled” or checkered board that has been divided along the diagonal to form a right triangle. It counts out to a diagonal hypotenuse of 15 squares and 16 dots, starting from 1 dot and 1 square at the right-angle apex.
PART II.1.6: Greater Geometric Expansion Rates; Multiplying the “double-slit” experiment.
fig. II.1.6.A: 17 double and triple “slits” in 6 walls. POSTULATE II.1.6.1: re. the base-2 expansion rate.
AXIOM II.1.6.2: Twin or binary states of “c” (particle & wave), indicate L&R Vectors.
LEMMA II.1.6.3: L&R Vectors implies spin & counter-spin for both L & R.
THEOREM II.1.6.4: L X spin & counter-spin + R X spin & counter-spin + L + R = 6.
POSTULATE II.1.6.5: re. the base-6 expansion rate.
AXIOM II.1.6.6: 6 directions (Up, Down, Left, Right, Front, Back) around a central 7th.
LEMMA II.1.6.7: the 7th direction is binary either inward or outward.
THEOREM II.1.6.8: 8 interpolate into 7; 7 is surrounded by 6; 6 walls with 9 “slits.”
POSTULATE II.1.6.9: re. the base-8 expansion rate.
AXIOM II.1.6.10: 6 walls with 9 “slits” divide 7 refraction-gradient types of light.
LEMMA II.1.6.11: 6 walls + light + film = 8; 9 “slits” - 1 = 8; 7 spectrum + light or film = 8.
AXIOM II.1.6.12: min. “slits”/wall = 2; max. = 4 (up) 3: 2: 3: 2: 4: 3 = 17 total “slits”/6 walls.
fig II.1.6B: the Minimum “Packing” Space of 7,10 & 12. POSTULATE II.1.6.13: Minimum “Packing” Space using 7, 10 and 12 spheres.
AXIOM II.1.6.14: the 7 red spheres represent the 7 “Planetary” kamea.
AXIOM II.1.6.15: the 12 black spheres labeled in green represent the 12 zodiac signs.
AXIOM II.1.6.16: the 10 #’ed blue spheres represent the base-10 “Sefirot” Emanations.
LEMMA II.1.6.17: 12 measures time (the Aeons), 7 measures space (the Archons). 10 is their combination / alignment.
LEMMA II.1.6.18: the alignment of 7 in 12 corresponds to an event in space and time.
LEMMA II.1.6.19: the “invisible” Sefirot will be “revealed” when this event occurs.
AXIOM II.1.6.20: on 5/5/2000 AD, 5 Planets aligned with earth, our moon and the sun.
THEOREM II.1.6.21: This event occurs cyclically.
CHAPTER II.2: the mysteries of Combined Sets. PART II.2.1: the Set of Twelve; first Even “combined” set.
POSTULATE II.2.1.1: the Minimum “Packing” Space of 12. AXIOM II.2.1.2: Only one configuration exists to relate a circle of base-Unit ratio to 12 self-similar circles.
LEMMA II.2.1.3: this configuration can be expanded & contracted upon; it cannot be distorted nor altered.
LEMMA II.2.1.4: 12 circles the same size as a central circle overlap each one’s mid-point with the circumferences of the two adjacent to it, each on either side.
THEOREM II.2.1.5: there is 30° between each circle’s midpoint and the intersection of its circumference with the midpoint of the next adjacent circle.
POSTULATE II.2.1.6: the 12 Aeons of 12 Archons each. THEOREM II.2.1.7: 1 Month = 1/12 of 1 Year; 2000 years = 1/12 or 1 Aeon. 1 “Archon” = 1/12 mo. X 2000 years.
THEOREM II.2.1.8: 1 “Archon” = 1/12 of 1 Aeon. 1 Aeon = 1/12 of a “Solar Day.” 1 Season = 3 Aeons = 1/4 a “Day.”
THEOREM II.2.1.9: 1 “Solar Day” X 2 = a “Great Age.” 1/24 a “Great Age” is an Aeon.
THEOREM II.2.1.10: 1 Archon X 12 = 1 Aeon; 1 Aeon X 12 = a “Solar Day;” 1 “Solar Day” X 2 = 1 “Great Age.”
THEOREM II.2.1.11: 1 Archon = 1 month X 2000 years; 1 Aeon = 12 months X 2000 years; 1 Season = 6000 years; 1 “Solar Day” = 24000 years; 1 “Great Age” = 48000 years.
THEOREM II.2.1.12: 1 “Archon” = 1 month or 1 zodiac sign per year every year for 2000 years.
THEOREM II.2.1.13: 1 Aeon = 2000 years. POSTULATE II.2.1.14: the Lemurian “cork-screw” calendar. cf. document “Ordo Historia” of the POD.
AXIOM II.2.1.15: the calendar is “calibrated” for Spring Equinox on the year Zero, between Gemini and Cancer.
LEMMA II.2.1.16: in “Pythagorean Year” Zero, the Archon of the Spring Equinox changed from Gemini to Cancer.
AXIOM II.2.1.17: all signs relate by quadrangles and trines. Signs apply to “Archons” and Aeons.
AXIOM II.2.1.18: by Precession, the signs “counterrotate,” by months / year -> clockwise; & Aeons / “Day” -> counter.
AXIOM II.2.1.19: the base-12 calendar includes the measure of 3+4+5, and thus encompasses a Pythagorean triangle.
THEOREM II.2.1.20: 11 X 12 + 12^2 = 12 X 23.
PART II.2.2: the Combined Set of Seven and Twelve; first Odd “combined” set.
THEOREM II.2.2.1: 3 + 4 = 7; 3 X 4 = 12. POSTULATE II.2.2.2: Additive Table of 3 & 4; expressed as 7 and 12 sums.
AXIOM II.2.2.3: on the left, recombinations of 1 -> 4; on the right, of sums plus 1 -> 4.
LEMMA II.2.2.4: 1 - 4 recombine in the left column to form 7 sums (2 -> 8 inclusive).
LEMMA II.2.2.5: 2 - 8 recombine on the right quadrants to form 10 sums (3 -> 12 inclusive).
AXIOM II.2.2.6: this accounts entirely for the use of #bases 3,4,7,10&12 in metaphysics.
LEMMA II.2.2.7: thus, each digit corresponds to 3 signs, 1 sign for each #-base system.
LEMMA II.2.2.8: such is the basis of all QBLH. POSTULATE II.2.2.9: the base-12 “Wheel” or “Mill,” basis of the Zodiac clock, is a circle divided @ 30°.
AXIOM II.2.2.10: the object is in getting the base 7 to “fit” into the base-12 with correspondent meaning.
LEMMA II.2.2.11: ultimately, all attempts at this are equally arbitrary, and none rightfully “mystical.”
LEMMA II.2.2.12: thus, none have yet finally solved the geometry for the combined base-7 & 12 system.
POSTULATE II.2.2.13: the “pseudo-Gnostic” arrangement.
AXIOM II.2.2.14: the base-7 heptagram, or “Apocalypse star,” aligns perfectly with the “Wheel” in 2 places.
LEMMA II.2.2.15: cf. the “Sun” glyph at the nadir and the division directly 180° opposite from this, at the apex.
LEMMA II.2.2.16: thus the base-7 system corresponds to the base-12 system only twice in this model.
POSTULATE II.2.2.17: the “Magus” lamen array. AXIOM II.2.2.18: the base-7 system is here laid out as 6 bars (1 divided into 2) across the base-12 “Mill.”
LEMMA II.2.2.19: however, this by no means is the final model of alignment between the bars and the Wheel.
THEOREM II.2.2.20: Properly calibrated to Year Zero, Gemini is the Moon and Cancer is the Sun.
POSTULATE II.2.2.21: the “Magus” permutations. substituting digits for the classical “signs.”
AXIOM II.2.2.22: assume the sums 1 -> 12 on the “Mill.”
AXIOM II.2.2.23: the #’s 1 -> 12 yield 10 sums (3,5,7,9,11,13,15,17,19,23).
THEOREM II.2.2.24: the Prime-Unit in this system is base 13, however the expansion rate is max 78.
LEMMA II.2.2.25: all remaining #-base systems of the ancient sages’ mysteries derive from this.
THEOREM II.2.2.26: for the expansion rate of Prime#’s cf. basic geometric expansion rates (above, 1.2)
PART II.3: the Greater Combined Sets of 12; The “Loom” on the “mill-wheel.”
POSTULATE II.3.1: where the basic angle-sums of all lesser stellations and polygons are in 12.
AXIOM II.3.2: quadrangles and trines are squares and triangles; hexes are dual trines, pent’s from ⲫ.
LEMMA II.3.3: the basis for the Pentagrams is the ⲫ ratio triangle contained within 12 = 3 + 4 + 5.
THEOREM II.3.4: 30°, 36°, 90° & 120° angles form a “Cosmic Loom” on the “Mill-Wheel” of base-12.
LEMMA II.3.5: as the Mill-Wheel counter-rotates over time, it renders the ⲫ/π spiral expansion rate.
LEMMA II.3.6: the “Loom’s” Precession = shapes.
PART II.4: Other Greater Combined Sets; Combined Sets using base-3 and base-4.
POSTULATE II.4.1: the set of 3 X 4. AXIOM II.4.2: without repeats, there are 64 sets where each base-unit is 3 “signs” from a 4-set.
LEMMA II.4.3: there are thus 64 X 3 = 192 base-units.
LEMMA II.4.4: thus these correlate to other #base’s.
THEOREM II.4.5: any #-base set “n” is = to any other use of the same #n, even when “Combined” or “fit.”
AXIOM II.4.6: “3 X 4” is a Combined (“Odd”) expansion rate.
LEMMA II.4.7: base-64 + expansion rate 8 = 72.
THEOREM II.4.8: 2^6 = 4^3 = 8^2. THEOREM II.4.9: 2 X 36 = 3 X 24 = 4 X 18 = 6 X 12 = 8 X 9.
POSTULATE II.4.10: the set of 4 X 4. AXIOM II.4.11: 4^2 = 16; the # of base-units / square.
AXIOM II.4.12: 16 X 12 = 192; the # of “Elemental” base-units / each “Elemental” set.
AXIOM II.4.13: 192 X 4 = 768; the # of base-units in total.
AXIOM II.4.14: 4 X 4 = “Even” / regular expansion rate.
LEMMA II.4.15: 16 + 8 = 24 base expansion rate.
THEOREM II.4.16: 2 x 32 = 64. THEOREM II.4.17: 2 x 36 = 72. THEOREM II.4.18: 4 X 16 = 64. THEOREM II.4.19: 4 X 18 = 72. THEOREM II.4.20: 8 X 8 = 64. THEOREM II.4.21: 8 X 9 = 72. AXIOM II.4.22: a base-72 set combines with Odd expansion rate 3; the base-64 set combines only Even #’s. Thus, they share Even factors, not Odd.
LEMMA II.4.23: Even expansion rate = √2 or π; Odd expansion rate = √3 or ⲫ.
THEOREM II.4.24: ⲫ^2 + e^2 + (i/ e)^2 - π^2 = 0.00215040335.
PART II.5: Other Greater Sets; Regular Sets.
fig. II.5A: Gnomonic expansion rate.
fig. II.5B: the circle^2. fig. II.5C: the ⲫ/π Vortex.
APPENDIX 1: Bibliography. APPENDIX 1.1: essential reference works: 1. 2. 3. 4. 5.
Huntley, H.E. “The Divine Proportion.” Dover Books, 1970. Michell, John. “The Dimensions of Paradise.” Adventures Unlimited Press, 2001. Cousto, Hans. “The Cosmic Octave.” Life Rhythm, 2000. Stirling, William. “The Canon.” Samuel Weiser, 1999. Lawlor, Robert. “Sacred Geometry.” Thames & Hudson, 1982.
APPENDIX 1.2: other reference works: 1. 2. 3. 4. 5. 6. 7.
Ghyka, Matila. “The Geometry of Art and Life.” Dover, 1977. Simon, William. “Mathematical Magic.” Dover, 1964. Ogilvy, C. Stanley & Anderson, John T. “Excursions in Number Theory.” Dover, 1966. Aczel, Amir D. “The Mystery of the Aleph.” Simon & Schuster, 2000. Maor, Eli. “e: the Story of a Number.” Princeton University Press, 1994. Beckman, Peter. “A History of Pi.” Dorset Press, 1971. Livio, Mario. “The Golden Ratio.” Random House, 2002.
APPENDIX 2.1: On Trans-finites and Number Sets. The maximum ordinal (1-d, radius) “natural number” (N) is ∞ (infinity). The maximum cardinal “natural number” is aleph sub-null (transfinite). “Natural” #s include both all “whole” #s (including zero) AND “counting” #s (excluding zero) because these have a 1:1 ratio to one another. (0, 1, 2, ..., ∞ = N sub-zero) is equal to the same cardinal sum as (1, 2, 3, ..., ∞ = N sub-one) because zero is the empty set and thus adds no measured value. The maximum (2-d, diameter and area) “integer” (Z) sum is aleph sub-one because it is both +∞ and -∞ and is therefore a quantitatively different sum of “infinity” than the “natural number” set, which is necessarily smaller by exactly 1/2 of (+∞) + (-∞); thus aleph-null is 1/2 the size of alephone. Because positive infinity plus negative infinity cancels out to zero, the entirety of the “integer” set is arithmetically reducible (topologically compactable) to zero. An integer maybe either: even (divisible by 2) OR odd, AND either: prime (factorially indivisible) OR composite (having multiple factors). The set of all positive integers is +∞ and that of all negative integers is -∞, and these combine to form the set of all integers; likewise, the infinite set of all “even” integers and that of all “odd” integers also sum the same quantity, as do the combinations of the infinite set of all “primes” and that of all “composites.” The maximum (3-d, volume) “rational number” (Q) sum is aleph sub-N (where N is the largest Natural number, AKA aleph sub-zero; thus another expression for aleph sub-N is aleph sub-aleph sub-zero. “Rational” numbers introduce fractions or decimals, making each integer infinitely divisible (linking one to infinity) and adding a “Y” axis to the “X” axis of all integers, increasing the dimensionality of the number set. If measured by decimals, rational numbers may either “terminate” (come to an end after a certain sequence of digits) OR “repeat” (repeating a finite sequence of digits ad infinitum, cf. 1/3 or 33.3-bar %). Thus the sum of all “terminating” rational numbers plus all “repeating” rational numbers equals the maximum rational number sum. The set of all “rational” numbers therefore has a maximum “cardinal” sum (aleph sub-N), however the set of all “irrational” numbers is uncountable and therefore (presumed) to comprise a larger sum than that of all “rational” numbers. Insofar as “rationals” add “depth” to the “number-line” by infinitely expanding every integer into fractional parts, “irrationals” have a maximum cardinal sum of “aleph sub-N+1,” where, no matter how “infinite” the rational number set maybe, the irrational set is more so by a minimum of one additional sum - in the context of irrationals, this “one sum” difference between “rational” and “irrational” number sum sets maybe reducible to the “absolute value of one” (or the circle with radius one surrounding the origin-point at zero). The maximum (4-d, time) “real” number (R) is thus a combination of both all “rational” and all “irrational” numbers. Therefore, if the maximum sum of all “rational” numbers is reckoned as “aleph sub-N” and that of all “irrational” numbers as “aleph sub-N plus one,” then the maximum “real” number is the sum of both these, expressed as “aleph sub-omega;” this expression is shorthand notation for “aleph sub-N” plus “aleph sub- N+1” and is synonymous with the concept of “(2 aleph sub-N)+1,” “Beth sub-1” and / or “c” (the “cardinality of the continuum). A “real” number is one whose division into fractions can continue on in an infinite (unending) sequence post decimalplace. Because of these reasons, the sum of all “real” numbers is considered “uncountable” as it combines multiple “uncountably infinite” sub-sets. The set of all “real” numbers and that of all “imaginary” numbers maybe 1:1. Thus, the (5-d) set of all “complex” numbers (C) that includes both “real” and “imaginary” numbers can be reckoned roughly as “aleph sub-omega squared” (or as the “cardinality of the continuum squared”).
APPENDIX 2.2: On Indeterminate Forms Because the other 5 usual “indeterminate forms” are transformable (under certain limiting conditions) to the 2 indeterminate forms 0/0 and ∞/∞, and because ∞/∞ maybe expressed as ∞ • ∞^-1, it maybe possible that the indeterminate form 0/0 can also be expressed as (0 • 0)^-1. Also, assuming “complex infinity” (~∞) maybe allowed as one of the variables along with 0, 1 and ∞, then there are (at least) 13 “indeterminate forms,” including complex “duals” for some of the “simple” indeterminate forms:
CENTER I = 0/0 (or: (0 • 0)^-1 ?).
LEFT II = ∞ - ∞ III = ~∞ - ~∞.
RIGHT IV = 0 • ∞ V = 0 • ~∞.
FRONT VI = ∞/∞ (or: ∞ • ∞^-1) VII = ~∞/~∞.
BACK VIII = ~∞/∞ IX = ∞/~∞.
DOWN X = ∞^0 XI = 0^0.
UP XII = 1^~∞ XIII = 1^∞. Because the 6 simple indeterminate forms are all commutable to zero/zero (under certain limiting conditions) it stands to reason the other 6 indeterminate forms (based on “complex infinity”) should be also. Thus, 0/0 maybe the “GCF & LCD” indeterminate.
APPENDIX 2.3: Some Brief Conclusions Consider this “hyper-cross” model’s “core” indeterminate form (0/0) alike the ordinal maximum sum of all “natural” numbers (∞); the “left” pair of indeterminate forms (∞ - ∞ and ~∞ - ~∞) alike the cardinal maximum sum for the “naturals” (aleph sub-zero); the “right” pair (0 • ∞ and 0 • ~∞) like the maximum sum of the “integers” (aleph sub-one); the “front” pair (∞/∞ and ~∞/~∞) like the max sum of the “rationals” (being aleph sub-N); the “back” pair (~∞/∞ and ∞/~∞) like that of the max sum of all “irrationals” (being aleph sub-N plus one); the pair “below” (∞^0 and 0^0) like the “cardinality of the continuum” of all “real” numbers (expressed as aleph sub-omega); and the pair “above” (1^∞ and 1^~∞) maybe alike the max sum of all “complex” numbers (aleph sub-omega^2). The reason for placing the “indeterminate forms” in this order and relating them to the maximum sums of each number-set in this sequence is due to the “order of operations”: the “subtractive” pair precedes the “multiplicative,” which precedes the (2 pair of) “divisions” which precede the (2 pair of) “exponential” functions; where, also, 0 < 1 < ∞.
APPENDIX 2.4: Bibliography Wikipedia articles: https://en.wikipedia.org/wiki/Number https://en.wikipedia.org/wiki/List_of_types_of_numbers https://en.wikipedia.org/wiki/Natural_number https://en.wikipedia.org/wiki/Integer https://en.wikipedia.org/wiki/Rational_number https://en.wikipedia.org/wiki/Real_number https://en.wikipedia.org/wiki/Complex_number https://en.wikipedia.org/wiki/Number https://en.wikipedia.org/wiki/0 https://en.wikipedia.org/wiki/Brāhmasphuṭasiddhānta https://en.wikipedia.org/wiki/Zero-dimensional_space https://en.wikipedia.org/wiki/Division_by_zero https://en.wikipedia.org/wiki/Zero-product_property https://en.wikipedia.org/wiki/Indeterminate_form https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero https://en.wikipedia.org/wiki/1 https://en.wikipedia.org/wiki/%E2%88%921 https://en.wikipedia.org/wiki/Infinity https://en.wikipedia.org/wiki/Calculus https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise https://en.wikipedia.org/wiki/De_analysi_per_aequationes_numero_terminorum_infinitas https://en.wikipedia.org/wiki/Line_at_infinity https://en.wikipedia.org/wiki/Plane_at_infinity https://en.wikipedia.org/wiki/Point_at_infinity https://en.wikipedia.org/wiki/Beth_number https://en.wikipedia.org/wiki/Transfinite_number https://en.wikipedia.org/wiki/Actual_infinity https://en.wikipedia.org/wiki/Absolute_Infinite https://en.wikipedia.org/wiki/Lemniscate
“Principia Numerologica” the Mysteries of # Theory, BOOKS I & II. Originally © Dec. 11, 2008, by: Jon Gee, Second edition © July 26, 2022, by: Jon Gee. Including a new appendix on indeterminate forms.