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2016a.00587 Griffiths, Martin Squeezing bubbles into corners. Math. Teach. (Reston) 107, No. 6, 473-477 (2014). 2014 National Council of Teachers of Mathematics (NCTM), Reston, VA EN G40 I30 filling a square with circles total area geometric series plane geometry solid geometry filling a cube with spheres total volume generalization multidimensional spaces hypercubes hyperspheres student activities discovery learning
• http://www.nctm.org/Publications/mathematics-teacher/2014/Vol107/Issue6/Delving-Deeper_-Squeezing-Bubbles-into-Corners/
• 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.