Squeezing bubbles into corners. (English)

Math. Teach. (Reston) 107, No. 6, 473-477 (2014).

From the text: I always seek activities that might stretch my students yet would be accessible to them; that might require logical thought yet would contain counterintuitive elements; that might provide the opportunity to venture into new mathematical realms yet would have a simple starting point. This article and the activity that inspired it did indeed arise by way of a relatively straightforward problem that I proposed to one of my classes. I asked students to draw a square $S$ of side length 2 units containing a circle $C_1$ of radius 1 unit. The challenge was to find the radius $r$ of the largest circle $C_2$ that could fit into one of the “corners" between $S$ and $C$. This problem can be solved in several ways. One approach is to imagine an infinite series of ever-decreasing circles. Later we deal with the extended problem of finding the total area of the infinite number of circles that cover $S$. Moving into three dimensions gives rise to the “bubbles" referred to in the article’s title. And this problem generalizes beautifully into $n$ dimensions.