Golden logarithmic spiral

Fig. 5 The approximate golden logarithmic spiral, tana = x. Note that tp — 2ti/5 radians, emphasizing the relationship of T to a regular pentagon cos(27t/5) = t/2...

Fig. 4 Simulation of integer bond orders on a golden logarithmic spiral...

The findings reported here provide new evidence for the unity of micro- and macrophysics and refute the perception of separate quantum and classical domains. The known universe exists as a four-dimensional space-time manifold but is observed in local projection as three-dimensional Euclidean tangent space that evolves in universal time. The observable world, at either micro- or macroscale, can be described in either four-dimensional (nonclassical) or in classical three-dimensional detail. The descriptive model may change, but the reality stays the same. This realization is at the root of self-similarity between large and small. The symmetry operator, which reflects the topology of space-time, is the golden logarithmic spiral. 

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...

It has been argued [8] that the transformation from curved space-time to Euclidean tangent space is described by the golden ratio. This is not an entirely unexpected conclusion, in view of the prominence of r in the operation of selfsimilar symmetries, related to equiangular logarithmic spirals. 

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