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?

AndrĂ© 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

oooh, I love this framing of the question (spherical loaf). It's kind of amazing! Thank you for sharing.

ReplyDeleteSam

I'm thinking now it should be manageable to actually bake a hemispherical bread and bring it into class - might need a mold to keep it exact.

ReplyDeleteBread and crusts + physics

ReplyDeleteReminds me of:

http://uncyclopedia.wikia.com/wiki/Murphy's_law_application_for_antigravitatory_cats

Martin

In all seriousness, how about using an orange?

ReplyDeleteStudents could easily cut each ring of peel and flatten to measure the area.

Sorry. Martin lurking on Pippa's facebook!

What happens if it is not a sphere? For a chocolate example, what would happen if you sliced an Easter egg?

ReplyDeleteWhat a great question. And I think you just gave me a (potentially very messy!) holiday activity to give to my students!

ReplyDelete