1238.00 Fourteen-illion Scheherazade Number

1238.20 Trigonometric Limit: First 14 Primes: The Fourteen-illion Scheherazade Number accommodates all the omnirational calculations of the trigonometric function tables whose largest prime number is 43 and whose highest common variable multiple is 45 degrees, which is one-eighth of unity in a Universe whose polyhedral systems consist always of a minimum of four positive and four negative quadranted hemispheres.

1238.21 45 degrees is the zero limit of covarying asymmetry because the right triangle’s 90-degree corner is always complemented by two corners always together totalling 90 degrees. The smallest of the covarying, 90-degree complementaries reaches its maximum limit when both complementaries are 45 degrees. Accepting the concept that one is not a prime number, we have 14 primes, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43which primacy will accommodate all the 14 unique structural faceting of all the crystallography, all the biological cell structuring, and all bubble agglomerating: the 14 facets being the polar facets of the seven and only seven axes of symmetry of Universe, which are the 3-, 4-, 6-, 12-great circles of the vector equilibrium and the 6-, 10-, 15-great circles of the icosahedron.

1238.22 Tetrahedral Complementations The sphere-to-space, space- to-sphere intertransformability is a conceptual generalization holding true independent of size, which therefore permits us to consider the generalized allspace-filling complementarity of the convex (sphere) and concave (space) octahedra with the convex (sphere) and concave (space) vector equilibria; this also permits us to indulge our concentrated attention upon local special-case events without fear of missing further opportunities of enjoying total synergetically conceptual advantage regarding nonsimultaneously considerable Scenario Universe. (See Secs. 970.20 and 1032.)

1238.23 We know the fundamental intercomplementations of the external convex macrotetra and the internal concave microtetra with all conceptual systems. Looking at the four successive plus, minus, plus, minus, XYZ coordination quadrants, we find that a single 90-degree quadrant of one hemisphere of the spherical octahedron contains all the trigonometric functioning covariations of the whole system. When the central angle is 90 degrees, then the two small comer angles of the isosceles triangle are each 45 degrees. After 45 degrees the sines become cosines, and vice versa. At 45 degrees they balance. Thereafter all the prime numbers that can ever enter into prime trigonometric computation (in contradistinction to complementary function computation) occur below the number 45. What occasions irrationality is the inability of dividends to be omni-equi-divisible, due to the presence of a prime number of which the dividend is not a whole product.

1238.24 This is why we factor completely or intermultiply all of the first 14 prime numbers existing between 1 and 45 degrees. Inclusive of these 14 numbers we multiply the first eight primes to many repowerings, which produces this Scheherazade Number, which, when used as the number of units in a circle, becomes a dividend permitting omnirational computation accommodation of all the variations of all the trigonometries of Universe.

1238.25 The four vertexial stars A, B, C, D defining the minimum structuralergo, triangulatedsystem of Universe have only four possible triangular arrangements. There are only four possible different topological vertex combinations of a minimum structural system: ABC, ABD, ACD, BCD. In multifrequenced, modular subdivisioning of the minimum structural system, the subdividing grid may develop eight positive and negative aspects:

ABC obverse (convex)

ABD obverse (convex)

ACD obverse (convex)

BCD obverse (convex)

ABC reverse (concave)

ABD reverse (concave) ACD reverse (concave) BCD reverse (concave)

1238.26

Three unopened edges AB, AD, BC. (Fig. 1238.26A.)

Four edge-bonded triangles of the tetrahedron. (Fig. 1238.26B.)

Three pairs of opened edges; three pairs of unopened edges. Each triangle has also both obverse and reverse surfaces; ergo, minimum closed system of Universe has four positive and four negative triangleswhich equals eight cases of the same.

The same four triangles vertex-bond to produce the octahedron. (Fig. 1238.26C.)

1238.27 In a spherically referenced symmetrical structural system one quadrant of one hemisphere contains all the trigonometric variables of the whole system. This is because each hemisphere constitutes a 360-degree encirclement of its pole and because a 90-degree quadrant is represented by three equi-right-angle surface-angle corners and three equi-90-degree central-angle-arc edges, half of which 90-degree surface and central angles is 45 degrees, which is the point where the sine of one angle becomes the cosine of the other and knowledge of the smallest is adequateergo, 45is the limit case of the smallest.

1238.28 Spherical Quadrant Phase: There is always a total of eight (four positive, four negative) unique

interpermutative,

intertransformative,

interequatable,

omniembracing

phases of all cyclically described symmetrical systems (see Sec. 610.20), within any one octave of which all the intercovariable ranging complementations of number occur. For instance, in a system such as spherical trigonometry, consisting of 360 degrees per circle or cycle, all the numerical intervariabilities occur within the first 45 degrees, 45 \times 8 = 360. Since the unit cyclic totality of the Fourteen-illion Scheherazade Number is the product of the first 15 primes, it contains all the prime numbers occurring within the 45-degree-limit numerical integer permutations of all cyclic systems together with an abundance of powers of the first eight primes, thus accommodating omnirational integrational expressibility to a 1 \times 10^{-42} fraction of cyclic unity, a dividend so comprehensive as to permit the rational description of a 22-billion-light-year-diameter Universe in whole increments of \sfrac{1}{10,000}ths of one atomic nucleus diameter.

image

image

Figs. 1238.26A, B, C: In this plane net of four hinged triangles the dotted line indicates the intersection of a great-circle plane passing through the assembled tetra.

Four edge-bonded triangles of the tetra with great-circle plane passing through.

The same four triangles may be vertex-bonded to describe an octahedron with alternate open and closed faces.

1238.29

(+)•(+)=(+) (+)•(-) = (-) (-)•(+) = (-) (-)•(-) = (+)

Multiply

Divide

(+)/(+) = (+)

(+)/(-) = (-)

(-)/(+) = (-)

(-)/(-) = (+)

Change: fix eqn

1238.41 Declining Powers of Factorial Primes: The recurrence of the prime number 2 is very frequent. The number of operational occasions in which we need the prime number 43 is very less frequent than the occasions in which the prime numbers 2, 3, 5, 7, and 11 occur. This Scheherazade Number provides an abundance of repowerings of the lesser prime numbers characterizing the topological and vectorial aspects of synergetics’ hierarchy of prime systems and their seven prime unique symmetrical aspects (see Sec. 1040) adequate to take care of all the topological and trigonometric computations and permutations governing all the associations and disassociations of the atoms.

1238.42 We find that we can get along without multirepowerings after the second repowering of the prime number 17. The prime number 17 is all that is needed to accommodate both the positive and negative octave systems and their additional zero-nineness. You have to have the zero-nine to accommodate the noninterfered passage between octave waves by waves of the same frequency. (See Secs. 1012 and 1223.)

1238.43 The prime number 17 accommodates all the positi-ve-negative, quanta-wave primes up to and including the number 18, which in turn accommodates the two nines of the invisible twoness of all systems. It is to be noted that the harmonics of the periodic table of the elements add up to 92:

2

8

8

18

36

8 .

18

18 !81

18]

_2

92 Change: fix eqn

There are five sets of 18, though the 36 is not always so recognized. Conventional analysis of the periodic table omits from its quanta accounting the always occurring invisible additive twoness of the poles of axial rotation of all systems. (See Sec. 223.11 and Table 223.64, Col. 7.)

1238.30 Cosmic Commensurability of Time and Size Magnitudes: Each degree of 360 degrees of circular arc is subdivided into 60 minutes and the minutes into 60 seconds each. There are 1,296,000 seconds of arc in a circle of 360 degrees.

1238.31 One minute of our 8,000-mile-diameter planet Earth’s great circle arc = one nautical mile = 6,076 feet approximately. A one-second arc of a great circle of Earth is 6,076/60= 101.26 feet, which means one second of great-circle arc around Earth is approximately 100 feet, or the length of one tennis court, or one-third of the distance between the opposing teams’ goal posts on a football field. We can say that each second of Earth’s great circle of arc equals approximately 1,200 inches (or 1,215.12 “exact”). There are 2\sfrac{1}{2} trillion atomic-nucleus diameters in one inch. A hundredth of an inch is the smallest interval clearly discernible by the human eye. There are 25 billion atomic-nucleus diameters in the smallest humanly visible “distance” or linear size increment. A hundredth of an inch equals 1/120,000th of a second of great-circle arc of our spherical planet Earth. This is expressed decimally as .0000083 of a second of great-circle arc = .01 inch; or it is expressed scientifically as .01 inch = 83 \times 10^{-7}. A hundredth of an inch equals the smallest humanly visible dust speck; therefore: minimum dust speck = 83 \times 10^{-7} seconds of arc, which equals 25 billion atomic-nucleus diametersor 2\sfrac{1}{2} million angstroms. This is to say that it requires seven places to the right of the decimal to express the fractional second of the great-circle arc of Earth that is minimally discernible by the human eye.

1238.40 Fourteen-illion Scheherazade Number: The Fourteen-illion Scheherazade Number includes the first 15 primes, which are:

1^{n}2^{12}3^{8}5^{6}7^{6}11^{6}13^{6}17^{2}19 23 29 31 37 41 43

It reads:

$3,128,581,583,194,999,609,732,086,426,156,130,368,000,000$

1238.50 Properties: The 3 fourteen-illion magnitude Scheherazade Number has 3 \times 10{43} whole-number places, which is 10^{37} more integer places than has the 1 \times 10^{6} number expressing the 1,296,0 00 seconds in 360 degrees of whole-circle arc, and can therefore accommodate rationally not only calculations to approximately \sfrac{1}{100}th of an inch (which is the finest increment resolvable by the human eye), but also the 10^{-7} power of that minimally visible magnitude, for this 3 \times 10^{43} SSRCD has enough decimal places to express rationally the 22-billion-light-years-diameter of the omnidirectional, celestial-sphere limits thus far observed by planet Earth’s humans expressed in linear units measuring only \sfrac{1}{1,000}ths of the diameter of one atomic nucleus.

1238.51 Scheherazade Numbers: 47: The first prime number beyond the trigonometric limit is 47. The number 47 may be a flying increment to fill allspace, to fill out the eight triangular facets of the non-allspace-filling vector equilibrium to form the allspace-filling first nuclear cube. If 47 as a factor produces a Scheherazade Number with mirrors, it may account not only for all the specks of dust in the Universe but for all the changes of cosmic restlessness, accounting the convergent-divergent next event, which unbalances the even and rational whole numbers. If 47 as a factor does not produce a Scheherazade Number with mirrors, it may explain that there can be no recurring limit symmetries. It may be that 47 is the cosmic random element, the agent of infinite change.

1238.52 Addendum Inspired by inferences of Secs. 1223.12, 1224.30-34 inclusive and 1238.51, just before going to press with Synergetics 2, we obtained the following 71 integer, multi-intermirrored, computer-calculated and proven, volumetric (third power) Scheherazade number which we have arranged in ten, “sublimely rememberable,” unique characteristic rows.

$2^\dots 3^\cdot 5^\cdot 7^\cdot 11^\cdot 13^\cdot 17^\cdot
19^\cdot 23^\cdot 29^\cdot 31^\cdot 37^\cdot 41^\cdot 43^\cdot
47^$

the product of which is

616,494,535,0,868

49,2,48,0

51,88

27,49,49

00,6996,185

494,27,898

13,35,17,0

25,22,

73,66,0

864,000,000.

If all the trigonometric functions are reworked using this 71 integer number, embracing all prime numbers to 50, to the third power, employed as volumetric, cyclic unity, all functions will prove to be whole rational numbers as with the whole atomic populations.

1238.60 Size Magnitudes

An Atomic Nucleus Diameter = A.N.D. =

Atomic Nucleus Diameters:

1238.70 Time Magnitudes: Heartbeats (Seconds of Time)

= 1 two-illion heartbeats ago

= 31 two-illion heartbeats ago

= 500 two-illion

heartbeats ago

= 1 three-illion heartbeats ago

= 2 three-illion heartbeats ago

= 2 weeks

=1 year

= 16 years (college)

= 32 years (prime life)

= Average lifetime

1238.80 Number Table: Significant Numbers (see Table 1238.80)