Fourier Transforms of Tubular Objects with Spiral Structures ()

G. B. Mitra

Indian Association for the Cultivation of Science, Jadavpur, West Bengal, India.

**DOI: **10.4236/jcpt.2012.24024
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Indian Association for the Cultivation of Science, Jadavpur, West Bengal, India.

Crystal structures of several naturally occurring minerals are known to contain various deformities such as cones, cylinders, and tapered hollow cylinders with different apex angles, which have been described as solid and hollow cones, “cups”, “lampshades” as well as rolled cylindrical planes. The present study was undertaken to determine how these different shapes within a crystal structure can be explained. Since the usual method of observing them is by either X-ray and electron diffraction or electron microscopy, we investigated Fourier transforms of these forms, which were considered in terms of spirals with varying radii. Three types of spirals were considered, namely: 1) Archimedean spiral; 2) Involute of a circle or power spiral and 3) Logarithmic spiral. Spiraling caused the radius *r* to be modified by a factor *f(θ)*, so that* r *becomes *rf(θ)*, where *f(θ)* = *θ* for Archimedean helix,* θ*^{n} for power helices like *θ*^{1/2} for Fermat’s helix, *θ*^{-1} for hyperbolic helix and *e ^{θ}* or

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G. Mitra, "Fourier Transforms of Tubular Objects with Spiral Structures," *Journal of Crystallization Process and Technology*, Vol. 2 No. 4, 2012, pp. 161-166. doi: 10.4236/jcpt.2012.24024.

Conflicts of Interest

The authors declare no conflicts of interest.

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