…But No Simpler!
Recently, I was talking with a coworker about how I try to simplify everything I do. The example I gave was the flipping the mattress every 6 months to even out the wear. Except I don’t actually flip the mattress every 6 months (simplify!), so I couldn’t remember if it proved my point or not.
The problem is this: After 6 months, I can’t remember what I did last time so I don’t know what the next orientation is supposed to be. What would be nice is if I had an Atomic Move that cycled through all the orientations. For instance, if I could flip and rotate every time and if that would put the mattress in all possible positions, it wouldn’t matter if I couldn’t remember because I know I’d get there eventually. Is there such an Atomic Move?
But after reading that, I wondered about mattresses of other geometries. Like what about a really thick mattress that basically has 4 sleepable sides? That one changes the side flip from 180° to 90° while leaving the end flip and rotate alone. Or a cubical mattress, where everything is 90°? Or what about the completely general case of the N dimensional mattress that has a sleepable surface every M degrees?
I spent some time thinking about the 4-mattress and finally managed to prove that it also was impossible. My method basically extended the one from the article, but it was a little more complicated because it doesn’t reduce down as well. Then I tried applying that to the cube-mattress and could immediately see it would be a nightmare.
Instead, I jumped directly to the fully-general case, which turns out to be way simpler (especially if you know anything about graphics programming).
Let’s say I have my N dimensional mattress. And let’s say I have some set of rotation moves that I claim is the the Atomic Move that will let me visit all possible orientations of the mattress. That set of moves can be represented by a single rotation around a composite axis1. When I apply my set of moves again, I just rotate around that composite axis a little farther.
But rotating around a single (albeit composite) axis will not let me visit all mattress orientations2. Therefore the set of rotation moves doesn’t exist. Which means that all higher-dimensional beings must also remember how they flipped their ultramegahypermattresses 6 months ago.
Hahaha, awesome. The mattress thing drives me nuts as well. I always end up trying to theorize about my state of mind last time I flipped it.
Oh, I forgot to mention: I conjecture that two Atomic Moves will always suffice, provided you choose them well. But of course, that requires memory.
Or does it? I just now thought of maybe an ideal fix to the problem. First of all, the plan has always been to flip the mattress when we change the clocks which (used to be a) guarantee that it would happen every 6 months.
What if we did this: I always do a sideflip in the spring and my wife can always do a rotation in the fall. Sort of an emergent algorithm distributed across multiple agents.
I taped a piece of paper underneath my matress with what I did last (After many years of forgetting) – turns out that the take stays sticky enough to be moved to the other side when I flip it over – that has worked for a couple of years now though the paper is getting a bit tatty.
I do like the emergent multi-agent flipping thing