The Number Spiral

The sums cp = a, 3a, 5a, 7a. (a = 300) have coefficients that match the numbers of s,p,d,f electron pairs required to fill atomic sub-shells. The constant a = 300 corresponds to the maximum possible number of stable nuclides and a/3 to the maximum number of stable elements. 

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The conclusions of the previous section are summarized well in terms of the number spiral with a period of 24, shown in Figure 4.3. The arrows mark eight radial directions where all prime numbers, except for 2 and 3, and... 

To emphasize the periodicity of 8, suggested by the number spiral, the periodic table is rearranged as shown in Figure 4.5. Closure of the eleven periods coincides with the completion of electronic sub-levels, except for atomic numbers 62 and 94 that split the /-levels into sub-sets of 6 and 8. Additional structure, in complete agreement with the experimentally known sub-level order of the elements, is revealed by the zig-zag profiles that define the field of nuclide stability in Figure 4.4. 

The alternative derivation of atomic periodicity, based on the distribution of prime numbers and elementary number theory, makes firm statements on all of these unresolved issues. The number spiral predicts periodicities of 8 and 24 for all elements and nuclides respectively limits their maximum numbers, in terms of triangular numbers, to 100 and 300 respectively characterizes electronic angular-momentum sub-levels by the difference between successive square numbers (21 +1) and electron pairs per energy level by the square numbers themselves. In this way the transition series fit in naturally with the periodicity of 8. The multiplicity of 2, which is associated with electron spin, is implicit in these periodicity numbers. 

Recognition of space-time curvature as the decisive parameter that regulates nuclear stability as a function of the ratio, Z/N, with unity and the golden mean, r, as its upper and lower limits, leads to a consistent model for nucleogenesis, based on the addition of -par tides in an equilibrium chain reaction. This model is also consistent with the limitations imposed by the number spiral. 

The four periodic laws defined by figure 4 are related in the sense that each of them fits the compact periodic table (figure 3) such that all energy shells close in either period 2 or 8 [21]. From the spacing of points at the ratio 1.04 it is inferred that two extra groups of 24 nuclides become stable against /0-type decay. The total number of nuclides is thereby increased to 12 x 25 = 300, as required by the number spiral. The total number of elements increases to 102-2=100 as required. 

The energy spectrum of the nucleus according to the semi-empirical shell model [23] appears not at zero ratio, but in two disjoint parts at ratios 0.22 and 0.18. This shift relates to the appearance of the symmetric arrangement at ratio 1.04 rather than 1. An Aufbau procedure based on this result fits the 8-period table derived from the number spiral, but like the observed periodic table, at ratio r, the shell-model result also has hidden symmetry. At ratio zero, the inferred energy spectrum not only fits the 8-period table but also... 

Relative to the symmetrical situation of figure 15, the number of stable elements is reduced to 81 and the energy-level sequence that dictates the Aufbau procedure is inverted. The structure that positions closed-shell configurations in periods 2 and 8 is maintained by the appearance of three gaps along the number spiral. 

In the case of TiCl2 the number of propagation centers do not exceed 0.5% of the number of surface titanium ions this shows that the formation of the propagation centers proceeds at specific points on the surface of the crystalline catalyst (e.g. lateral faces, outlets of the spiral dislocations). 

Alternatively, each loop of the APH design may be constructed with variable radius to connect continuously (with no filling space) into an ascending Guggenheim-staircase pattern. In this construction the APH arcs upward from H (Z = 1) in ever-increasing energetic and atomic-number spirals, to the as-yet undiscovered realm at the head of the staircase. 

PHI is the ratio of any two sequential numbers in the Fibonacci (V sequence. If you take the numbers 0 and 1, then create each subsequent number in the sequence by adding the previous two numbers, you get the Fibonacci sequence. For example, 0, 1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. Ifyou sum the squares of any series of Fibonacci numbers, they will equal the last Fibonacci number used in the series times (10) the next Fibonacci number. This property results in the Fibonacci spiral seen in everything from seashells to galaxies, and is written mathematically as l2 + l2 + 22 + 32 + 52 = 5 x 8. 

Because only one repeat is present in the structure and the number of amino acids between repeats is variable, we cannot yet draw conclusions about whether the monomers spiral around each other or whether this will be a left-handed or right-handed spiral. Future studies (Section IV) will hopefully lead to structural information on the repeat-containing N-terminal half of the bacteriophage T4 short tail fiber and the long tail fibers. 

This experiment established the nuclear model of the atom. A key point derived from this is that the electrons circling the nucleus are in fixed stable orbits, just like the planets around the sun. Furthermore, each orbital or shell contains a fixed number of electrons additional electrons are added to the next stable orbital above that which is full. This stable orbital model is a departure from classical electromagnetic theory (which predicts unstable orbitals, in which the electrons spiral into the nucleus and are destroyed), and can only be explained by quantum theory. The fixed numbers for each orbital were determined to be two in the first level, eight in the second level, eight in the third level (but extendible to 18) and so on. Using this simple model, chemists derived the systematic structure of the Periodic Table (see Appendix 5), and began to... 

Each turn of the P-oxidation spiral splits off a molecule of acetyl-CoA. The process involves four enzymes catalysing, in turn, an oxidation (to form a double bond), a hydration, another oxidation (forming a ketone from a secondary alcohol) and the transfer of an acetyl group to coenzyme A (Figure 7.12). The process of P-oxidation operates as a multienzyme complex in which the intermediates are passed from one enzyme to the next, i.e. there are no free intermediates. The number of molecules of ATP generated from the oxidation of one molecule of the long-chain fatty acid pal-mitate (C18) is given in Table 7.4. Unsaturated fatty acids are also oxidised by the P-oxidation process but require modification before they enter the process (Appendix 7.3). 

Are you ready for the answer Here is the list of terms sorted in order of beauty as determined by my little survey of scientists and colleagues. The numbers in parentheses indicate the number of times the term was an individual s first choice dancing flames (31), snow crystal (20), mist-covered swamp (17), spiral nautilus shell (10), mossy cavern (5), kaleidoscope image (5), avalanche (4), computer chip (3), seagull s cry (3), tears on a little girl (3), trilobite fossil (2), glimmer of mercury (2), wine (2), asphalt (1). 

In the case of growth spirals originating from dislocations with large b, hollow cores with diameters of micrometer order are observed at the spiral center however, when a number of dislocations with small b concentrate in a narrow area, a basin-like depression appears at the central area of the composite spirals, since the curvature of advancement of the spiral steps is reversed near the center. A straight step may appear near the spiral center as an intermediate state in the reversal of step curvature. Several examples are shown in Fig. 5.11. 

If a new dislocation is introduced while a spiral step is advancing, the new step from the new dislocation will be affected by the advancement of the earlier spiral step, and can have only one-half or a couple of turns. This results in the coexistence of spirals with a small number of turns with those covering a wide area Fig. 5.14 is such an example. 

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