Sierpenski's Gasket

This page describes Sierpenski's gasket, a fractal geometry which is created by starting with an equilateral triangle and dividing it up into 4 equal equilateral triangles, each with half the side length the orginal had, and then "removing" the middle one. This operation is then repeated on the remaining (surviving) triangles. Each iteration consists of applying this same rule and we have modeled 7 reiterations below. What is of interest is how both the number of elements and the log of the perimeter changes with each reiteration. Clearly, with each operation, each gets consistently larger.

.

Above is the first and second reiterations of the gasket.

.

Above are the third and fourth reiteration.

.

Above are the 6th and 7th reiteration.

Above is the 7th reiteration, and as far as we will go, but you can see how with more and more reiterations the black area is becoming a mere wisp of its former self.

A log-log plot of unit size (and in a manner equivalent to the ruler size as we have discussed before) versus the number of elements and the perimeter length (which in this case is a direct function of the number of elements and their side length) shows the hallmark linear relationship of a fractal relationship.

Copy of portion of Excel sheet that models the gasket.

Log-log plot of element size versus the number of elements. With a given ruler size one wouldn't 'see' the smaller elements.

Similar plot but now for the length of the perimeter of all the triangular elements.

 

How does this apply to anything? As a start consider if porosity had this general type of geometry in 2-D. Then the surface area is where the water clings to the grain, retarding water movement, and reducing permeability. The fractal geometry provides a model for that porosity. This might apply to the porosity/permeability of clays. You can build similar fractal geometries and relationships for porosity/permeability in 3-D. For tetrahedral shapes it would be similar, except the surface area would be a power law function of the unit side length (you would simply multiply the number of tetrahedral elements by 4 * (s^2*(3^.5/4)), where s is the side length (thesecond part is the formula is for the area of a equilateral triangle and each tetrhedra would have 4 of these). If you want more practice you can create an Excel sheet that models this. What application might this have? The size of the smallest clay particles could be a good lower limit for the scale over which the fractal property is applicable. Remember also that surface area is a very important material property in general, and can influence a variety of types of "surface energy".

Diagenetic mineral growth that influences porosity and which could contribute to a fractal geometry. From USGS site: http://pubs.usgs.gov/dds/dds-033/USGS_3D/ssx_txt/diagenes.htm .

There are many examples in the literature, but a Geology article by Aharanov et al. (19970 is one introduction if you would like to explore further: geology.gsapubs.org/content/25/6/547.full.pdf.