Imagine a big round soft rubber mat, perhaps four feet wide and maybe maybe a couple of inches thick. On the underside are thousands of tiny ball bearings, probably less than quarter of an inch or so in diameter, but each made out of the same kind of high-density rubber that squash balls are fashioned
from, and each housed in a tiny casing which somehow - even though rubber is perhaps the most friction-prone substance on earth - ensures that they can move freely in any direction. Teflon, perhaps. Or maybe something a bit more realistic. Anyway, only a couple of millimetres of the rubber balls protrude from the underside of the soft mat - enough to gain purchase on whatever they're laid upon, but not enough to destroy the illusion that the mat is just sitting flush on the ground.
Around the circumference of this mat - say, every five degrees or so - there's a small wheel. Slightly larger than the ball bearings, but not by much. They're made of some substance with slightly less friction than the ball bearings (any mention of "custard" in the annos to this idea will be swiftly deleted), but each are connected to small, wafer thin motors buried within the (admittedly slim) bulk of the foamy mat. The motors have variable speeds - their top whack is perhaps two rotations per minute, but they're capable of slowing that down to anything up to half a rotation every twenty four hours.
Oh, and cover the whole thing with fluffy faux sheepskin, or something. Make sure it's more than comfortable to lie on.
Pointless, I know. And, as it stands, it makes little sense. But, the thing is, we've had a bit of a heatwave here in the UK, and my cat's been the first to take advantage of it. For once, the sun's been blazing through my dusty windows, and my cat's always been there, luxuriating effortlessly in those slanted squares of sun. In the morning he's sleepily supine on the right side of my room, and, come the evening, he's snoozing happily in that same diamond of burning light at the left-most corner of my flat. He follows the sun across the room, even though I never see him do it. He's always spreadeagled in the sunshine, no matter where the sun falls on my floor. Always in the middle of that shining skewed rectangle. Same is true in the wintertime - if I drop so much as a newspaper on the ground, I turn my back for five seconds and he'll be lying on it, insulating his ass against the cold, threadbare carpet. He's a heat seeking cat, and I want to steal some of his thunder.
So, aligned with the wheels on the outer rim of the rug, I'd also like to have some heat-sensors. The more heat the sensors feel, the more power their wheel gets. Remember, these are slow wheels. No throwing molotov cocktails in the direction you want to go and wearing asbestos pants - the wheels turn slowly, and the rug itself moves with still more deliberation. Obviously the wheels don't get their power from the heat, but with each wheel turning more or less in response to the heat it feels, the whole comfy rug should always naturally gravitate towards the most cosy toasty warm spot in the room. If it keeps nudging your cat out of the way, that's probably a good sign.
Imagine sprawling on a comfy rug in your house, being lost in the pages of a fascinating book, but still slowly crawling across the room to catch the sun without even having to lift a finger. Engrossed in your cocoon of comfort, you barely even notice as day turns into night: once the sun sets, your rug is swayed by the open fire across the room, and wafts you over there to enjoy the final pages of your novel in flickering firelight. Just don't leave your oven door open. Or live near a volcano.
I was originally going to use thousands of tiny suction-cups on the underside of the rug, and have it crawl towards the heat using tiny, millipede-leg type waves, but that was a bit too tricky to work out. And also a little creepy (in more ways than one).