on little cat's feet

The cat is treading on dangerous ground these days. Granted, she's endured a huge amount of upheaval over the last three months, culminating in the indignity of being shoved into a kitty carrier and loaded into the cargo hold of an airplane, where she was imprisoned for several hours (plus, there was turbulence!). By the time we retrieved her from LAX Baggage Claim, she was a battered shell of a feline, meekly resigned to whatever fate was in store for her -- until she was released into her new home environment. Then the meowing, skulking and general outrage began.  Still, she bounced back remarkably fast, once she'd scouted out useful perches and hiding places and discovered the catnip toys, spiffy new automatic feeder and fountain water dish that Future Spouse had purchased in anticipation of her arrival. (Future Spouse clearly knows the way to a cat's heart.)

I was lulled into thinking the worst was over, when in fact Clio was merely plotting her revenge. Before going to bed, I placed my engagement ring -- yes, the one that gave me all that trouble -- on the living room coffee table. In the morning, it was nowhere to be found. I have some vague recollection of waking in the middle of the night and drowsily registering that Clio was delightedly batting something around the living room floor. I assumed it was a Diet Coke bottle cap, since everyone knows bottle caps are cats' natural enemy in the wild (tassels are a close second). But now it seems as if she knocked the ring off the coffee table and, entranced by the clatter as it hit the concrete floor, proceeded to bat it around until it disappeared into some unreachable space, at which point she became bored with the game and wandered off to find some new form of pseudo-prey.

Once I discovered the ring was missing, the frantic searching began. (Clio chose her vengeance well: she hid the one thing it would distress me no end to lose, apart from Future Spouse himself.) All the obvious places yielded no sign of the bauble, so I quickly turned to math and physics, specifically, a 2002 research paper that appeared in Physical Review Letters on optimizing encounters between biological organisms. My friend Buzz Skyline blogged about this a few months ago at The Physics of Sex, adapting it to the search for a mate (or the search for a one-night stand). But it applies just as well to predator/prey searches and bees seeking flowers to pollinate (which, arguably, is a form of "mating").

Basically, scientists from the University of Barcelona in Spain (plus a couple of other Spanish and Brazilian collaborators) studied two forms of search strategies to evaluate their success rates in various circumstances. The first strategy is known as a Brownian random walk, based in part on the physics concept of Brownian motion. In 1827, a scientist named Robert Brown was studying pollen particles floating in water under a microscope, and noted a strange "jittery motion." (Here's a nifty Applet demonstrating Brownian motion.) He thought perhaps the pollen were living creatures -- not really a wacky notion, considering that between 1673 and 1683, Anthony van Leeuwenhoek had observed protozoans in pond water and living bacteria in the mouths of two old men who had never once brushed their teeth. (Jen-Luc Piquant opines it was likely they had no remaining teeth; and yet they still had all that bacteria.)  But Brown wasn't one to make easy assumptions without further evidence. He repeated the experiment using particles of dust (which he knew to be inanimate), and saw the same odd jittery motion. He concluded that the pollen wasn't necessarily "alive," but could find no explanation for the origin of the motion. 

That task fell to Albert Einstein; one of his three seminal breakthroughs in his "Miracle Year" of 1905 was a paper on Brownian motion, drawing upon the work five years earlier of a French mathematician named Louis Bachelier, who wrote his PhD thesis on "the theory of speculation" -- basically studying stochastic processes in the fluctuations of the stock and options market. Similar analysis methods are still used today, which is why there are so many mathematicians and theoretical physicists traipsing the hallowed halls of Wall Street these days.

Anyway, Einstein used Bachelier's work as a means for indirectly confirming the existence of atoms and molecules, thereby putting to rest a debate that had raged for some two thousand years (since Democritus). His insight: the molecules in the fluid (e.g., water) would randomly move about and collide with other small particles, like pollen or dust suspended in the fluid, causing the "jittering" Brown had observed in 1827. Fans of Douglas Adam may recall that the Heart of Gold spaceship in The Hitchhiker's Guide to the Galaxy was powered by an "Infinity Improbability Drive," whose Brownian motion was generated by a nice hot cup of tea.

How could this possibly have anything to do with search strategies? Remember, we're talking about random collisions between particles, a model that can easily be adapted to lots of different systems, biological, physical, or otherwise. It's been adapted for stock market fluctuations, population genetics (specifically modeling genetic drift), neuron firing in the brain, and during World War II, Brownian random walks were used to model the distance that an escaped prisoner would travel in a given time. Think Tommy Lee Jones tracking Harrison Ford in the woods The Fugitive after that initial train/bus collision. It basically denotes a random search over a small area.

Applied to the search for my ring, using a Brownian random walk, I would look here and there in a very small focused area -- namely, the rug around the coffee table where I'd last left the ring. Which is exactly what I did, to no avail. However, the Barcelona researchers discovered that this is not the best search strategy to employ in this particular circumstance. Brownian random walks are most effective when that small search area is densely populated, thereby increasing the likelihood of colliding with something else, whether it be an object of interest or a potential partner. In contrast, I am looking for one particular thing, and since it is clearly not in the immediate vicinity of said coffee table, I must broaden my search. About all I can say for sure is that it must lie somewhere within the confines of the apartment.

That's a significantly bigger area to cover for search purposes, and since I'm looking for a single object much smaller than myself, the chances of success using a Brownian random walk are, frankly, extremely low. If only there were a similar search strategy designed for just such a circumstance. Well, there is! It's called a Levy Flight, in honor of another French mathematician, Paul Pierre Levy. It's a different kind of random walk, and the Barcelona scientists concluded that such a search is the best strategy for optimizing my chances of finding the ring since it has the advantage in circumstances in which "the searcher is larger or moves rapidly relative to the target, and when the target density is low."

A Levy flight basically involves searching a bit in one place, then moving to another likely spot and searching there, and so on. That was my second search strategy. I checked under the coffee table, then under the surrounding sofas, including under the cushions. Then I checked the bedstand and dresser in the bedroom, on the off-chance I'd moved it and had forgotten. I checked the office next, then the sidetable by the front door, then the bathroom, and finally the coin purse in my wallet where I sometimes temporarily store the ring if I'm worried about it getting damaged. Nada. Clearly Levy flights aren't the way to go either.

There's one last resort: the grid. That's the search strategy employed by the fictional CSI team in one episode where a random human femur was found by a dog playing fetch. On the assumption that where there's a femur, there must be other bones as well, the search team forms a grid and painstakingly searches through every minute patch of desert, placing little flags wherever they found human remains. They did find them, which was more than I can say, even though I crawled around on my belly with a flashlight checking every conceivable nook and cranny during my Levy flight search. There will be a painstaking grid search going on this weekend, and hopefully Clio's dastardly plot will be foiled.

Of course, there's one more possible explanation: extra dimensions of spacetime. Perhaps we've discovered proof of this seminal aspect of string theory right here in our Los Angeles loft. It would certainly explain the mystery of the missing socks in the dryer, too, not to mention where Clio's cat toys disappear to, only to magically re-emerge months later. These "lost" objects could just be popping in and out of hidden dimensions of spacetime at whim.

While I'm waiting for the probabilities to shift to my favor, I have this to say to my pernicious feline: that photo earlier in the post? That's not Clio, although it looks an awful lot like her. Jen-Luc found it on the Internets, on a Website devoted to pets available for adoption. Just her way of reminding Clio there are lots of cute tabby cats out there looking for a home. She can be replaced.

Comments

[Kristin]

Good luck with your search for your ring. I'm sure it'll turn up--and, of course, it'll be in the last place you look!

[Wilson]

I'm sure you thought of that, but... have you considered checking inside the cat?

[Jon H]

Hm. Next ring, try a stone that fluoresces under UV. That'd be easy to find, given a UV source.

Otherwise, I'd suggest putting a bottlecap (or the little plastic ring that's left when you remove the cap from a 2 liter bottle) on the table and, when the cat plays with it, seeing where it ends up. Then look in that place and in similar places in the area.

If you weren't in California, I'd suggest looking under any baseboard heaters.

[Lab Lemming]

Re the last comment, most diamonds DO fluoresce.

-LL

[Lab Lemming]

Oops. Clarification:
Diamonds fluoresce visably under Xray, not UV. However, about a third of them will fluoresce under both.

I suspect that the rest would fluoresce under vacuum UV as well as Xray, but pumping down your home may have a deleterious effect on your cat.

[Dr. Ellen]

My sweetie and I live with two cats. They're the default explanation whenever something goes missing. (This is also a face-saving measure on the part of the humans involved.) Nonetheless, we were delighted to find further proof of the essential evil of the Cat, in the form of http://www.catsthatlooklikehitler.com/cgi-bin/seigmiaow.pl

[TBB]

I found 12 items under my oven's bottom drawer once: pens, lighters, water bottle caps, magnets, plastic ring toys, etc. Some of these items were carried to the fun and slippery kitchen floor from other areas of the apartment. Basically anything that has a crack such as the couch, coffee table, closet doors, is amusing for cats, as you well know. They bat things under them and then enjoy the challenge of trying to get them out. One of my cats dropped toys into the toilet in her youth. Another did swallow something plastic, which I discovered in the litter box. Yuck, but it does happen.

I hope by now you've found it!

[Lee]

One last suggestion: forget the flashlight, run a yardstick beneath your appliances. Sometimes you can't actually see what's under there very well, even with a flashlight (and were you wearing your glasses?), but you can sweep it out. Stuff like this, I've found will obey some perverse statistical rule about rolling as far away from easy reach and behind the legs of the most inconvenient to move and largest object in the vicinity.

[coturnix]

Check the cat's water dish and the toilet. That's where I look for (amd always find) whatever Biscuit wants to hide.

[andy.s]

This is why cats need nine lives.

[Blake Stacey]

Nits:

1. In HHGG, it was the **finite** improbability generator which was built from a Brambleweeny-57 sub-meson brain connected to the circuits of an atomic vector plotter suspended in a strong Brownian motion generator (the nice hot cup of tea). The **Infinite** improbability generator was actually the Golden Bail of the Krikkit Gate, blasted into the whirling eddies of the space-time continuum countless epochs ago and reconstituted with the aid of the finite improbability machine. Its means of operation are not specified, although given Arthur Dent's troubles finding tea upon the **Heart of Gold**, one may plausibly conjecture that tea is not involved.

2. In **The Fugitive**, Tommy Lee Jones assumes that Harrison Ford is moving at an average speed of 4 mph (injured person over uneven ground) and has been traveling for ninety minutes, giving a radius of six miles for the search area. This doesn't require the mathematical model of Brownian motion (the 4 mph figure might have been derived that way, but even then the Brownian-motion part is "encapsulated"). It's just a rate-times-time problem.

Yes, there's a reason I have no friends.

(Our cat doesn't steal things. She just finds adult chipmunks outside, thinks they're her babies and tries to adopt them.)

[Bonita]

I was a fine arts major in school and was taught the Gestalt Theory. If you look at a picture and you look at each dot in the picture - you see a page of dots - but if you focus on the page as a whole you see a black and white illustration of a Dalmatian under the shade of a tree. This theory can apply to finding a small object on a large surface. Rather than look for the small object - look at the whole surface as a pattern and find the piece that is 'ring-shaped' within that pattern. I know it sounds crazy - but it works!

BTW - heard your podcast on the Baltimore Sun - good stuff!

[JoAnne]

OK, we're all dying to know. Have you found the ring yet???

[Jennifer Ouellette]

Sadly, the ring has not yet emerged... although we found a good number of paper clips and pens tucked away in out-of-the-way spaces. We'll see what we find when the movers drop off my stuff and the major reorganization kicks in.... I'm not giving up yet!

[joel hanes]

Here is a description of several well-researched methods for searching and finding

http://www.gksoft.com/a/fun/catch-lion.html