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Nature produces amazingly varied geometrical patterns (Figure 1). In
particular, logarithmic spirals are abundantly observed in nature (Mukhopadhyay,
2004). Gastropods/cephalopods (such as nautilus, cowie, grove snail, Thatcher shell,
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 Galaxy also resemble logarithmic spirals. Many materials
develop spiral cracks either due to imposed torsion (twist), as in the spiral fracture of
the tibia, or due to geometric constraints, as in the fracture of pipes. Spiral cracks
may, however, arise in situations where no obvious twisting is applied; the symmetry
is broken spontaneously (Néda et al., 2002). Fonseca (1989) found that
rank-size pattern of the cities of USA approximately follows a logarithmic spiral.
In fitting spiral or conical curves in empirical data some important studies
have been made. Among those, Kanatani (1994), Werman and Geyzel (1995), Ho
and Chen (1996) and Ferris (2000) may be relevant in the present context.
The usual procedure of curve-fitting fails miserably in fitting a spiral to
empirical data. The author tried with several algorithms available for nonlinear
regression and nonlinear optimization, but was unsuccessful. The main reason for the
failure of these algorithms is easily discernible. |
