Sierpinski carpet coasters
November 2019
As one of my first forays on the Nueva laser-cutter, I cut a set of coasters in the shape of a Sierpinski carpet fractal:
A true Sierpinski carpet has zero area, but, as cutting one on a laser cutter would take an infinite amount of time, I only cut them down to three iterations. Even so, their perimeter, as a a function of the outermost side length \(d\), is:
$$\begin{align*} \text{perimeter} &= 4d + 4\left(\frac{d}{3}\right) + 8\!\cdot\!4\left(\frac{d}{3\cdot 3}\right) + 8\!\cdot\!4\left(\frac{d}{3\cdot 3\cdot 3}\right) \\ \\ &= 4d\left(1+\frac13 + \frac89 + \frac{8}{27}\right) \\ \\ &= \frac{184}{27}d \\ \\ &\approx 6.81d \end{align*}$$Compared to the \(4d\) perimeter of an uncut square, this has 70% more surface area for liquid absorption! The “laser-ply” plywood also helps: a tougher veneer over a lighter interior. They’re lightweight, absorbant, and aesthetic!
Over the next few months I laser-cut many sets of seven coasters (deliberately a non-even prime) to give away as gifts.