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 8.  Geometries - Golden section - Special arithmetical operations, e. g. π + √2, - ES-series and Cheops pyramid - √1/3 and √2/3 Hopefully some advanced mathematicians will find this file on the web and get provoked enough to search for explanations! Geometrical relations in the two 12-groups: 385 x √2 is ~ 544. Through this factor √2 and inversions (/\ ) the codon groups 385 and 367 in the ES-series become related, figure 8-1 below.. How to interpret such relations? They are very likely only one example among many similar dimensional relations on a deeper underlying level that remains to investigate. (√2 should probably not be regarded as a relation diagonal/side in a square, sooner in its serial development?) Fig 8-1: Geometrical relations through √2: Two notes: 1) A right-angled triangle with the sides 176 and 209 (the number division of 385 within mixed codon groups) gives the hypotenuse 273.23.; 273 the mean number of two ams R + B unbound. 2) Cf peraps the relations between Z-numbers of ATP and NADPH(+H) around 258 - 387, a 2/3-relation, 6 x  43 --- 9 x 43. see Biochemistry. Intervals in the ES-chain with exponents 3/2 and 4/3: Intervals 40-44-84 in the ES-chain with exponent 3/2 gives numbers 253 292, and 770, as a kind of feed back (?) or mutual references , re-establishing the ES-numbers 5' and 4' (+1):    403/2 = 252.98. = 253.    443/2 = 291.86. = 292. ........Sum 544 +1. (Third number 208?: 84 - 49 = 35, → 353/2 = 207.06. = 208 -1)    843/2 = [292 - 208]3/2 = 770. = Cross- plus Form-coded ams. Compare with 42, 1/2 times the interval 84, divided and whole number:             Fig. 8-1-b Intervals in the three middle steps of the ES-chain, sum 152:        443/2 = 291.9.     493/2 = 343     593/2 = 453.2.......sum 1088.15. = 2 x 544. | Exponent 4/3 ? Testing this exponent - as related to a higher d-degree step 4 - 3 (?) - shows up to be another way to re-establish the starting number 292:    404/3 + 444/3  →  292.13. (Cf. 52/3 = 2.924.) Note that 404/3 is about 136 +1 and 444/3 about 156 -1: 136 mass of Hypxanthine, 156 mass of Orotate, the parents of the bases in the codons    844/3 = 367.9. = 208 + 159, +1;     x 2 = 736. = RNA- + Pair-coded ams +2. The golden section: The golden section (gs) as Fibonacci series appears also in this ES-chain and mass relations within groups of ams. Following series, figure 8-2, leads from 2' in the ES-series to total sum R + B of the 24 bound ams through steps as times gs:     The ES-numbers  208 and 367 gives for instance the groups  G+C 544 and U+A 960 approximately. Fig 8-2: Golden section (gs) steps in the ES-series: Higher numbers abridged. Number 257. = 208 + middle interval 49. In 7th step we get 2848, x gs = 2 x 2304 = 2 x 482.(24 = number of codons, ams.) Forms of R-chains: Elementary structure of R-chains as basis for grouping of ams may give three elementary groups: vary roughly ring-formed, "straight" and "branched": Ring-formed ams 292 + 159 (a loop 5' - 2'.) = 451, three aromatic ams 328, + His + Pro 123. "Straight" ams = 2 x 159 (2') = 318, Ala, 2 Ser, Thr, Cys, Lys, Meth = 317, + Gly 1.    Ring-formed + "straight" = 770 -1. Branched = 734 +1 (as at middle of the ES-chain): CHx-groups = 271; O-OH + O-NHx = 262; 2 Arg NHx-NHx = 202. π and √2 , an arithmetical curiosity: Reading and adding 2 x 5 two-figure-numbers (31 + 41 + 59..., 14 + 15 + 92...) from first eleven figures in these transcendent celebrities gives both whole and partial sums of the codon grouped amino acids as well as vertically numbers from the exponent series, see the figure below.    No motivation for the operation is offered here. Should there exist any sense in the operation, it's left to more advanced mathematicians to find it. Fig 8-3: π and √2: *Cf. fig. 4-1. Sums of numbers from π = 431 and from √2 = 321; cf. triplet numbers 432-321. A note: If √2 is seen connected to d-degree step 2-1 and π to step 3-2, the number 321 from √2 could be thought of as displaced, here one 10-power step: Fig. 8-3-x. Sum 431 + 321 with displaced 321 number    [Cheops pyramid:    ½ x 292 = 146 ~ the height of this pyramid - in meters!    ½ x 460 = 230 ~ side of this pyramid - in meters! If the old Egyptians were acquainted with cubic roots and exponents 2/3 is probably uncertain. The Pharaoh measure "ell" is said to have been about half a meter.    That the circumference of the base is approximately 2π times its height is a well-known fact. 460/292, x 2 = 3.15068, differs from π 3.1415... with ~ 3 pro mille. π and √ 2 in figure 8-3: the 2-figure numbers summed in another way:  One example: π       31 14 41 15    59  92 26 65   53 35 √ 2    14 41 14 42    21  13 35 56   62 23             210+2              367            175-2                  |                                      |                                       385     [212 x 173 = 366,76 x 10^2. 3' + 2' = 366,74.) What kind of relation could exist, which closely couples π and √ 2? π represents 1/2 of a unity circle, but √ 2 only 1/4, if illustrated as in the figure below? Could we interpret it in terms of a step or displacement in dimension degree 3 to 2: (4-3-1 → 3-2-1 as sums of the 2-figure numbers above)? π related to a 3-dimensional world, √ 2 to a 2-dimensional one. Fig. 8-4 Bell's theorem a.       b.   Assuming the definition of a dimension as characterized by two complementary “end-poles” (0 - 00 in figure a), the shortest step between the poles in an 1-dimensional world would be 2 r and the shortest step in a 2-dimensisonal world could be only √2. In a 3-dimensional world, the shortest way could be the circular one (with reference to Einstein)? (There is an association here to Bell's theorem and Aspect's experiments with photons in quantum physics: measurements in two dimensions (directions) as φ a branched way for two possible outcomes. If there were no coupling between the photons, the maximum result of Bell’s formula should be +/- 2. But the experiments showed on a maximum of +/- 2√2  showing that the photons were entangled.     The question arises: what would be the result if such measurements were carried out in three dimensions?) √2 representing the tangent of an angle, a direction? Arc tan √2 = φ 54.736. Sin φ = √2/3, Cos φ = √1/3 Treating these numbers sin - cos in the same way as π and √2 above gives these results: Fig. 8-5. Sin - Cos for arc tan √2 Taking each other 2-figure number from the cosine series and each other from sine numbers, the upper ten first, gives sums of the ES-chain: Fig. 8-6. Sin-Cos-series with sum 1011 of the ES-series   *

Individual research
E-mail: a.wohlin@u5d.net

Table 24 amino acids (ams)
R-chsins, A, Z, N

Background model

Files here:
 I. The ES-chain Introduction 3. The exponent 2/3 number series Mass of codon grouped ams 4. Mass division on atom kinds, and on other bases 5. The two 12-groups - details 6. Backbone chains 7. N-Z-division and H-atoms 8. Geometries - Golden section Special arithmetical operations 9. Glycolysis and Citrate cycle: codon grouped amino acids on the basis of origin 10. The bases: - Some annotations 11. Transmitters and the ES-chain

 II. Simpler numeral series 12. Quotients. - Exponents 5-1 13. The 2x2-chain 16. An x3 series?

 III. Transformations between number-base systems 17. First observations 18. Totals and other notable things 20. Additions to files 17-18 21. Triplet series - An alternative series 22. Other substances

 Discussion References

All these files in 3 documents, pdf:
Section I, files 0-11

Section II, files 12-16
Section III, files 17-22
Discusssion, References in section III

The 17 files as one document,
pdf

Latest updating
2018-11-20

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