Golden mean

The system is regarded as commensurate if there is a rational value for the ratio a=al2iT=L/N, where a denotes the mean space of the particles, 27t is the potential period, and L indicates the number of potential periods within the calculation window. By using different combinations of L and N, one may be able to alter the commensurability of the system, which will become incommensurate when a approaches an irrational value. The golden mean, a=( /5- l)/2 144/233, for example, is usually chosen to characterize a typical incommensurate case. 

Unlike crystals that are packed with identical unit cells in 3D space, aperiodic crystals lack such units. So far, aperiodic crystals include not only quasiperiodic crystals, but also crystals in which incommensurable modulations or intergrowth structures (or composites) occur [14], That is to say, quasiperiodicity is only one of the aperiodicities. So what is quasiperiodicity Simply speaking, a structure is classified to be quasiperiodic if it is aperiodic and exhibits self-similarity upon inflation and deflation by tau (x = 1.618, the golden mean). By this, one recognizes the fact that objects with perfect fivefold symmetry can exist in the 3D space however, no 3D space groups are available to build or to interpret such structures. 

One way to solve the problem of unphysically short atomic distances is to project onto the Rpm subspace only those grid points included in a selected strip (gray area), with width of a (cos a + sin a) in the A per subspace. The slope of RPai shown in Fig. 1 is 0.618..., an irrational number related to the golden mean [( /5 + l)/2 = 1.618...]. As a result, the projected ID structure contains two segments (denoted as L and S), and their distribution follows a ID quasiperiodic Fibonacci sequence [2] (c.f. Table 1). From another viewpoint, the ID quasiperiodic structure on the par subspace can be conversely decomposed into periodic components (square lattice) in a (higher) 2D space. The same strip/projection scheme holds for icosahedral QCs, which are truly 3D objects but apparently need a more complex and abstract 6D... 

Table 1 The Fibonacci sequence and its relationship to the golden mean, 1.618...

The Fibonacci sequence can be generated by transformations of L LS and S—>L in each cycle. L/S represents the sequence of ACs that can exist for any QC system. With increasing order, the L/S ratio converges to the golden mean value... 

Further, larger prolate and oblate rhombohedra are also found to exist in 2/1 ACs, as shown in Fig. 23. The centers of the triacontahedra divide each edge by a value (1.615) close to the golden mean (x = 1.618), and thus they can be called... 

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. 

Covalent interaction in diatomic molecules depends on the golden mean t, the interatomic distance d and the radius ratio x r /r2 of the constituent atoms, as summarized in Figure 5.6. The golden mean is a universal constant that matches the geometry and topology of space-time, the radius ratio is a known function of atomic number and dl relates to the optimal wave-mechanical distribution of valence-electron density in the diatomic system. 

The equivalence between Sk, the infinite Farey tree structure and the nuclide mapping is shown graphically in Figure 8.4. The stability of a nuclide depends on its neutron imbalance which is defined, either by the ratio Z/N or the relative neutron excess, (N — Z) jZ. When these factors are in balance, Z2 + NZ — N2 = 0, with the solution Z = N(—1 /5)/2 = tN. The minimum (Z/N) = r and hence all stable nuclides are mapped by fractions larger than the golden mean. 

Because the range of nuclidic stability is bounded by fractions that derive from Fibonacci numbers, it probably means that nuclear stability relates directly to the golden mean. To demonstrate this relationship it is noted that the plot of A vs Z, shown in figure 13 for the A(mod4) = 0 series of nuclides, separates into linear sections of constant neutron excess (A — 2Z) and slope 2. Each section terminates at both ends in a radioactive nuclide. The range of stability for each section follows directly from... 

This space-time model is a conjecture that has been described in detail [28] and will be reconsidered in chapter 7. A new aspect thereof, which derives from number theory, is that the general curvature of this space-time manifold [26, 29] relates to the golden mean. This postulate is required to rationalize the self-similar growth pattern that occurs at many levels throughout the observable universe. 

Earlier speculations about the effect of the curvature of space on elemental synthesis and the stability of nuclides (2.4.1) are consistent with the interface model. The absolute curvature of the closed double cover of projective space, and the Hubble radius of the universe, together define the golden mean as a universal shape factor [233], characteristic of intergalactic space. This factor regulates the proton neutron ratio of stable nuclides and the detail of elemental periodicity. The self-similarity between material structures at different levels of size, such as elementary particles, atomic nuclei, chemical... 

This series of resonance conditions converges to 2.618..., implying that the most important bottleneck to intramolecular energy transfer is determined by a golden mean cantorus, that is,... 

Davis and Gray also demonstrated the existence of a series of intramolecular energy transfer bottlenecks, each corresponding to the breakup of a KAM torus. For example, for I2 in the vibrational state v = 20 they found intramolecular bottlenecks associated with frequency ratios equal to (3 + g) and up. However, Davis and Gray found that the last golden mean torus to be broken up is the most effective bottleneck to intramolecular energy transfer and is therefore... 

Another example of an algebraic number is the golden mean... 

The golden-section search, sometimes called the golden-mean search, is as simple to implement as the regular search, hut is more computationally efficient if < 0.29. In the golden-section search, only one new point is added at each cycle. 

The number (1 — ancient times. Livio (2002) gives a very entertaining account of its history and occurrence in art, architecture, music, and nature. 

The frequency of occurrence of letters L and S in this sequence is represented in Table 1.21, and it is easy to recognize that they are identical to the consecutive members (F +i and F ) of the Fibonacci series. The corresponding limit when the number of words, n, approaches infinity is the golden mean, t... 

The golden mean can also be represented as a continuous fraction, which contains only one number, 1, and therefore, it is sometimes referred to as the most irrational number. 

Throughout history, mathematicians have studied number patterns. Research Pascal s Triangle, Fibonacci and the Fibonacci Sequence, and the Golden Mean that was used by the Greeks in building the Parthenon. 

Using your favorite Internet search engine, type in "Escher Web Sketch." Many sites have this special (and free) online "sketching" interface that lets you create your own tessellations While online, research M.C. Escher, Penrose tiling, and the Golden Mean. 

The colors of these stars are blended between the dark (black, gray) and the golden, meaning that there is a blending of these two opposites in the seven stars. 

The solid curve between regions I and II is the boundary bottleneck between these two regions. Davis (1985) has suggested that the trajectory which defines this boundary has a frequency ratio with the worst irrational number. Such numbers are well known (Berry, 1978) and the golden mean. 

Figure 4.20 Classical surface of section associated with the CS stretch normal mode of OCS showing the following regions (A) total energy contour, (B) 3 1 resonance islands, (C) 5 2 resonance islands, (D) outer boundary of quasi-periodic region, (E) golden mean torus between chaotic regions labeled I and II, (T) turnstile (in golden mean torus) between regions I and II (located by arrow) (Gibson et al., 1987).

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