Perhaps counter-intuitively, it turns out all the slices have the same amount of crust (*). Even more surprising to me was finding out that proving this geometric fact was one of Archimedes most prized discoveries. So much so, he had a diagram of a sphere enclosed in a cylinder inscribed on his tomb. Which got me wondering - how is it as a relatively well educated person, I'm only just now finding out about this in my late forties? And how did I miss out the fun of exploring this unexpected relationship?
|André Karwath CC-BY-SA-2.5 |
via Wikimedia Commons
I'm thinking now there are many lessons that could be built exploring the Archimedean relationships between spheres and the cyclinder - from very hands on practical explorations in Year 9, to ways to approach calculus at both introductory (limits, intervals and slopes) and more advanced levels (using integration of circles of revolution). Teach solid geometry this way and it will be both good fun and permanently etched in young minds.
While I might have to rework the lesson hook into something about chocolate shells, for me this problem will always be about the fight for equality of crunchy crusts.
(*) An excellent and not too technical explanation of the bread crust problem is presented at Math Central