Logarithm spiral

The trajectories are logarithmic spirals (Fig. 6-4). For a > 0, they wind on the singular point (i.e., the rotation of the radius vector is clockwise) for a < 0, they unwind (i.e., the rotation of the radius vector is counterclockwise). 

In Fig. 10.3 it is seen that, if both Vr and Vu vary inversely to the radius r, the angle a is constant. Hence for such a case the path line is the equiangular or logarithmic spiral. For a case where B varies, Vr will not vary in the same way as Vu, and hence the angle a is not a constant. 

Figure 8.2 Logarithmic spiral with superimposed mean planetary orbits. The circles in blue define the orbits of inner planets on a larger (self-similarj scale. The divergence angle of 108° causes those planets at angles of 5 x 108° apart to lie on opposite sides of the spiral origin. These pairs are Neptune-Mars, Uranus-Earth, Saturn-Venus and Jupiter-Mercury. The hypothetical antipode of the asteroid belt, a second, unobserved group of unagglomerated fragments, has been swallowed up by the sun...

Conformal symmetry is very common in nature e.g. we can find it in the nautilus shell and the sunflower. These structures are clearly ordered, even if they do not give sharp diffraction patterns. Here the repetition is non-Euclidean, on a logarithmic spiral (nautilus), or on a torus (sunflower). We are inclined to say that any kind of repetition, conformal or isometric, even in non-Euclidean space, is ordered. However, classification of these more chaotic structures, as for liquids, is less certain. It may be that a liquid can be described as a structure with some of the characteristics of conformal symmetry or perhaps by a representation even more exotic, like a manifold of constant negative curvature. 

The equations of the spiral curves are considerably simplified by the use of polar coordinates. For instance, the curve for the logarithmic spiral (Fig. 35), though somewhat complex in Cartesian coordinates, is represented in polar coordinates by the simple equation r = o, . (1)... 

The golden ratio is superimposed on a logarithmic spiral, r = fP, by setting fj, = to produce the golden spiral, r = that leads to the... 

Figure 7.3 A set of logarithmic spirals, such as the golden, planetary spiral with divergence angle 2t, may serve as a model of Godd s compass of inertia, going through an odd number of involutions...

Figure 8.3 Seperate logarithmic spirals account for most of the yellow stars with red halos (left) and the blue stars (right) in the cluster...

The likeness that floral structures bear to shells and curved horns has been noticed many times and the basis of the likeness has been known for millenia. The logarithmic spiral, which is the only smooth curve that is selfsimilar at all scales, features in the description of all these growth structures. Mathematical description of the spiral, shown in Figure 3.32, depends on the three fundamental constants, tt, e and t. 

Slip line fields are used to analyse metal plasticity under plane strain conditions (see Appendix C). The slip line field consists of two families of logarithmic spirals, with equations in polar coordinates r, d... 

Basha and Babu (2009) derived expressions for total horizontal and vertical inertial forces acting on the part of logarithmic spiral H GJ, which can be written as follows ... 

Figure 4 shows the details of the computation of subtended angle of logarithmic spiral () and the angle of the rupture plane with the ground surface (), t/T ratio and seismic passive earth pressure coefficient () for typical values of =30°, <5 = /2, E/X=0.3, E/ri= 0.16,... 

Table 1 shows the effect of soil fnction angle (4>) and soil-wall interface friction angle ( ) on critical failure surface which is governed by the critical angles of logarithmic spiral (6 ), the rupture plane with the groimd surface () and t / T ratio fortypical values of k = 0.l, k = 0.05, ff/A= 0.3, ii/77= 0.16 and /= 1.0. It can be found from Table 1 that for constant ratio of 6 / 4>... 

Figure 8.42 Monochromator schematic with a logarithmic spiral curved crystal. www.bruker.com. Used with permission.)...

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