## Wednesday, December 15, 2010

### Sharing out the bread crusts on a spherical loaf

Do you enjoy a crusty loaf of bread? Do you ever find yourself needing to share that loaf with family or friends to make sure everyone gets a fair share of the crust? An innocent enough problem posed in Professor Stewart's Cabinet of Mathematical Curiosities asks: If you had a spherical loaf of bread, and sliced it horizontally in slices of equal thickness, which slice would have the most crust on it?

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
Thinking back to my school days, solid 2D and 3D geometry was kind of - blah - not that interesting - lots of formulae to memorise - most of which I forgot later anyway. How much more exciting it was to discover that my favorite algebra could be used, courtesy of Descartes, to wrap around  those geometric lines and circles and turn them into something I could work with analytically.  But looking again now at Archimedes' methods of approaching circles, spheres and cylinders - through a sort of calculus lens without calculus tools, I'm left wondering why such wonders were never explored (or even just demonstrated) when I was a child at school?  Is it because by the time we teach and learn this topic (Year 9 & 10) - we've forgotten the joy of exploring these shapes - and are just on a roll of working through fairly meaningless formulae?

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

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

Sam

2. 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.

3. Bread and crusts + physics

Reminds me of:
http://uncyclopedia.wikia.com/wiki/Murphy's_law_application_for_antigravitatory_cats

Martin

4. In all seriousness, how about using an orange?

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

Sorry. Martin lurking on Pippa's facebook!

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

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