Number spiral

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. 

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 periodic table of the elements represents the classification that results on arranging the elements in order of increasing atomic number. This arrangement mirrors the distribution of primes within a natural-number spiral of period 24, well characterized in terms of a few simple concepts from number theory. 

The sum over all numbers from 0 to A is given by the triangular number k k + l)/2. The sum over numbers on each cycle of a periodic number spiral follows as ... 

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... 

Figure 2.7 Schematic diagram to represent natural numbers and their conjugates as two spirals that meet at infinity. The mirror image of each spiral represents negative numbers. Real and conjugate number spirals are chiral. In projection on orthogonal axes in the complex plane, together they create an achiral interface...

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. 

The Ulam spiral is a result of Ulam attending an uninteresting lecture, during which he started sketching his prime number spiral. This somewhat unusual format for writing sequences offers a two-dimensional pattern for a one-dimensional mathematical object, here, the sequence of prime numbers [124], In Figure 2.24, we have illustrated Ulam s spiral for prime numbers smaller than 100. 

Before leaving the Ulam spiral, we should mention that there have been a few modifications of Ulam s spiral. For example, R. Sacks constructed the Archimedean spiral by plotting integers uniformly on the spiral, and when composite numbers have been deleted, one obtains what is known as the Sacks prime number spiral [128]. On this spiral, prime numbers that are obtained from Euler s prime number generator x - x+41 are clearly seen on a line approaching the left horizontal axis. There are modifications of the... 

We thought that we might add a novelty to the topic of prime number spirals by considering only odd numbers in the construction of the spiral, hi Figure 2.26, we have illustrated a section of a so-modified Ulam s spiral on a 24 x 24 Cartesian grid. 

His claim was vindicated with the discovery of atomic number, but the theme remained undeveloped until it was conjectured by Plichta [6] that the electron configuration of atoms is mapped by the distribution of prime numbers. Based on the observation that all prime numbers >3 are of the type 6n 1, he defined a prime-number cross that intersects a display of natural numbers on a set of concentric circles with a period of 24. In Fig. 4, the construct is shown, rearranged as a number spiral. Noting that the numbers on each cycle add up to... 

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