The Baker's Edge brownie pans, which are pans constructed in such a way that everybody gets an edge piece and nobody gets a piece from the middle, remind me of space-filling curves.
The isoperimetric inequality suggests that the only way to do the reverse -- to have pans where nearly everybody gets the middle and nearly nobody gets the edge -- is to have really big pans.
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You've got to think outside the pan, as it were. A brownie baked on the surface of a sphere would have no edges, and even less "bottom edge" per volume than a pan.
Of course, it would have to be fairly thick batter to adhere to the sphere during cooking, but that's a practical consideration, not a mathematical one.
I vote for injecting the brownie mix in between two concentric spheres. You lose the crackly, top crust, though.
Can anyone give an interpretation of variational characterisation of the isoperimetric constant in terms of brownies?
If you have a sufficiently large spherical pan with a solid centre, gravity would hold even quite runny brownie mixture.
Obviously you cook it by having it rotate sufficiently close to a star; the problem then is avoiding the undercooked polar pieces.
Eating the edgeless brownie is left as an exercise for the reader.
You can avoid undercooked polar pieces by having the pan completely surround the star.
I like the thin crispy cornbread parts so I bake it on cookie sheets maybe 1/4" thick, like crackers instead of the traditional deep (brownie) rectangular pan. ("Pie are round, cornbread are square", as the punchline to the old joke goes.)
The same principle should work with brownies, just use a cookie sheet. You'll get all edge AND the top.
Incidentally, if you want to know good, take your leftover cornbread crackers, pile them into a disheveled mess a few inches tall, in the center of a cookie sheet, shave lots of (real) butter on top, and place in oven on toast long enough to just turn some of the cornbread brown. Serve as desert.
"You can avoid undercooked polar pieces by having the pan completely surround the star."
Probably the most frivolous use of a Dyson Sphere yet.
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