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Insights into Fibonacci Constellations in 2D and 3D Dimensions

This text explores the role of Fibonacci numbers in non-spiral grids, forming Fibonacci matrices that influence deformation orientations. It delves into the structure of 3D "Fibonacci" constellations and their arrangement, along with Fibonacci points on a sphere and planetary-scale covariances. The concept of Gaussian-Fibonacci constellations for radial resolution profiles is also discussed.

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Insights into Fibonacci Constellations in 2D and 3D Dimensions

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  1. Fibonacci constellations in 2D and in 3D We have seen that, in two dimensions, it is not necessary that the grids formed from the Fibonacci constellations exhibit SPIRALS, although they always seem to in Nature. The most obvious manifestation of the FIBONACCI NUMBERS, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc., in the botanical context is in the tally of the spirals of each kind. So, a valid question is: What role do the Fibonacci numbers play in our non-spiral grids? Answer: They form the FIBONACCI MATRICES by which one valid Basis is expressed relative to another. [ ] [ ] 8, 5 5, 3 -8, 5 5, -3 is one example, but so is (The Fibonacci sequence can be followed in the negative direction.)

  2. It can be easily verified that the symmetric Fibonacci matrices composed of three consecutive entries from the two-way infinite Fibonacci sequence mutually commute (and their products are also Fibonacci matrices). It is the EIGENVECTORS of these matrices that determine the orientation of the principal components of the allowed deformation (i.e., the deformations that preserve the nice griddable properties of the constellation). What defines the 3D “Fibonacci” constellation? Again, the orientation of the allowed deformations’ principal components, relative to the “cubic” lattice of a “Fibonacci” grid are found as the eigenvectors of a family of commuting symmetric matrices, an example being: [ ] 1, 1, 1 1, 1, 0 1, 0, 0 What are the “Fibonacci” numbers in this 3D context?

  3. I.e., the “Fibonacci numbers” in the 3D context (which actually have nothing whatsoever to do with the TRUE Fibonacci numbers) have as their most natural arrangement a hexagonal “honeycomb” pattern. Each triangle, * * * * * * within this pattern defines one of the composition matrices, according to H_22 H_23 H_12 H_33 H_13 H_11, by which one valid 3D basis may be expressed relative to another.

  4. Fibonacci points on the sphere No obstacle prevents construction of Fibonacci constellations on a curved domain, such as the sphere. However, for covariance synthesis we need to ensure that singularities of the implied mapping are kept outside the used portion of the domain.

  5. There is no Fibonacci constellation free of at least one singularity on the surface comprising the entire sphere. There exists a perfectly uniform-resolution (in area) Fibonacci constellation extending to about 130 degrees of co-latitude (i.e., quite a bit more than a hemisphere) before singularities appear. But, if we relax the requirement for perfect uniformity of areal resolution, it is easy to construct a Fibonacci constellation whose single singularity is pushed to the antipodal 180 degree co-latitude! The areal resolution of this configuration varies no more than about 17 percent (plus or minus) around the equator.

  6. View from above the North pole Areal resolution remains fairly uniform over the hemisphere

  7. View from above the Equator:

  8. Even at 60 degrees south this “northern hemisphere” Fibonacci constellation retains sufficient regularity and smoothness to be useful as a framework for anisotropic filtering performed deep within the overlap margin. The deeper overlap of this configuration will allow covariances of a larger scale to be dealt with without significant distortion.

  9. What about planetary-scale covariances?

  10. What about planetary-scale covariances? This Fibonacci constellation exhibits an isotropic Gaussian density of grid points

  11. What about planetary-scale covariances? A superposition of Gaussian- Fibonacci constellations extends the range of radial resolution profiles. Such “nesting” is the natural generalization of the method of nested cubes shown by Sato and Purser (preprint, AMS 23rd WAP/19th NWP conference, Omaha, NE) to provide a way to use the polyhexad method to synthesize planetary- scaled covariances. But the Gaussian-Fibonacci constellation is almost certainly the more economical arrangement.

  12. MULTIGRID The multigrid structure allows the form, C= w1*C1 + w2*C2 + ……..wn*Cn, for very large n. This greatly enlarges the range a covariance shapes available to us, especially since there is no compelling reason in this formulation to require that the weights, w, be positive; the combination, B= C*C^T is automatically self-adjoint and non-negative. In contrast, by presently formulating B = w1*B1 + w2*B2 + … wn*Bn we MUST have all the w>0, which greatly restricts our present ability to fit covariance profiles realistically. The down-side of all this is that, for this step of the assimilation, we would sacrifice strict bit-reproducibility (as pointed out by John Derber).

  13. Illustration of covariance which requires a mixture of positive and negative weights:

  14. C: B:

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