Perimeter-area Relation Research Articles

Fractal dimension measured with perimeter-area relation and toughness of materials.

This paper shows that the fractal dimension ${D}_{m}$ as determined by the perimeter-area relation is not the intrinsic fractal dimension ${D}_{0}$ of a fractured metal surface. The measured value of ${D}_{m}$ depends on the length of the yardstick used to measure the perimeter and has a quantitative relation with ${D}_{0}$. When the yardstick is small enough, ${D}_{m}$ approaches ${D}_{0}$. It is also shown that the origin of the negative correlation between ${D}_{m}$ and toughness of materials is that the yardstick used by many authors for measuring ${D}_{m}$ is too large. As we expected, a positive correlation is obtained when the length of the yardstick becomes smaller than a critical length of the side of the Koch generation. It seems hopeful to use fractals to characterize fractured surfaces with which material parameters can be correlated.

Area-Perimeter Relation for Rain and Cloud Areas

Following Mandelbrot's theory of fractals, the area-perimeter relation is used to investigate the geometry of satellite- and radar-determined cloud and rain areas between 1 and 1.2 x 10(6) square kilometers. The data are well fit by a formula in which the perimeter is given approximately by the square root of the area raised to the power D [See equation in the PDF], where D is interpreted as the fractal dimension of the perimeter. It is concluded that rain and cloud perimeters are fractals-they have no characteristic horizontal length scale between 1 and 1000 kilometers.