Extremal Properties of Logarithmic Spirals

Phyllotaxis or the properties of spiral lattices. - II. Packing of circles along logarithmic spirals

Hyperinvariant subspaces and quasinilpotent operators

For a bounded linear operator on Hilbert space we define a sequence of the so-called weakly extremal vectors‎. ‎We study the properties of weakly extremal vectors and show that the orthogonality equation is valid for weakly extremal vectors‎. ‎Also we show that any quasinilpotent operator $T$ has an hypernoncyclic vector‎, ‎and so $T$ has a nontrivial hyperinvariant subspace‎.

Loxodromic Spirals in M. C. Escher's Sphere Surface

Loxodromic spirals are the analogues in spherical geometry of logarithmic spirals on the plane. M.C. Escher’s 1958 woodcut Sphere Surface is an image of black and white fish arranged along eight spiral paths on the surface of a sphere. By connecting the plane and spherical models of the complex numbers, we show that Sphere Surface is the conformal image on the sphere of a tessellation of fish o...

Fitting an Origin-Displaced Logarithmic Spiral to Empirical Data by Differential Evolution Method of Global Optimization

Introduction: Nature produces amazingly varied geometrical patterns. In particular, logarithmic spirals are abundantly observed in nature. Gastropods/cephalopods (such as nautilus, cowie, grove snail, thatcher, etc.) in the mollusca phylum have spiral shells, mostly exhibiting logarithmic spirals vividly. Spider webs show a similar pattern. The low-pressure area over Iceland and the Whirlpool G...

Some statistical properties of spiral galaxies

This paper presents some statistical correlations of 72 northern spiral galaxies. The results show that early-type spirals that are brighter, and thicker, and the axis ratios (Hz/Hr) of the disk tend to be smaller along the Hubble sequence. We also find that Hz/Hr correlates strongly with the galaxy’s color, and early-type spirals have larger values of Hz/Hr. The inclinations obtained by fittin...