No wonder I couldn’t find it in my copy… neither cake nor pie appears in the index.
And, of course, there’s the next sentence:
GOD how I love that book!
[Homer]
Mmmmmmm… apple pie… mmmmmm
MMmmmmm… hypothetical, mathematical cake… mmmmmm
[/Homer]
I think the cake has to be round. If I have a rectangular cake the size of, say, a football field I imagine I could chop it in half and chop a half in half (and so on) 64 times and still have a decent size piece left at the end.
If you have a circular cake it doesn’t matter how big it is. Porcupine and I aren’t talking about a piece near the center since every piece has a part near the center and near the edge.
If that is the case I suppose a mathematically perfect circle would have a wedge that thinned to one atom on the third cut (chop circle in half, chop that half in two for two pieces with a 90[sup]0[/sup] angle and chopping a 90[sup]0[/sup] piece into two 45[sup]0[/sup] pieces. (Assume a sharp enough ‘knife’ or other cutting instrument.)
I think I understand what you are saying, but the shape of the delicious confection in question is irrelevant. If you read the OP, it’s pretty clear that we are talking about an eventual piece of cake that is exacty one-atom-size, not which comes to an exquisitely lethal “point” of one atom.
But even if we are talking about just a corner of the cake, then three cuts would do for any shape, in this crazy mixed-up three-dimensional world in which we live. Think about it for a second. With an infinitely sharp knife, three is the maximum number of cuts it would take to get that one-atom-wide-and-high-and-deep corner.
Which would then cut your tongue like a razor.