We finished our discussion of the Sierpinski triangle, noting that just like the “final” version of the Koch Curve is “nothing but angles,” this geometric oddity is “nothing but edges,” as the area of the triangle converges to zero as the iterations continue. This conclusion also presented an interesting contradiction. For the Koch curve, we argued that an infinite number of segments, each of length zero, resulted in an infinite perimeter (effectively, ∞ * 0 = ∞). Here, we have an infinity of triangles, each again with an area of zero, resulting in an area of zero (effectively, ∞ * 0 = 0).
What this reveals is that the expression “∞ * 0” is what is called an Indeterminate Form, an expression the defies definition. We can create a reasonable argument that defines it as infinity, and we can create a just-as-reasonable argument that defines it as zero. Therefore, it must be defintionless.
We finished the day by opening up the PC laptop mobile lab and downloading FractaSketch to each device. We’ll be using this software extensively over the next few weeks!