Overthinking: October 2008 Archives

Optimal sock purchasing strategies

Most of my running socks are Wigwam Ironman Ultralites, but I recently tried some of the Injinji Tetrasoks, which seem to be the preferred choice of a lot of the ultra guys. The Tetrasoks are more like gloves for your feet than socks: each toe is in an individual pocket, thus at least theoretically preventing blisters between the toes. Takes a little getting used to, but they're pretty comfortable, actually.

Anyway, this creates a new problem in terms of laundry: ordinarily, any pair of socks the same color make a pair, but tetrasoks are chiral, so if you just pick a random pair of socks you have a 50% chance of having two lefts or two rights, neither of which is very useful. So, with ordinary socks it's likely optimal to have all your socks the same because that minimizes your search cost (no matter what your laundry strategy is). However, the situation with tetrasoks is more complicated. Let's say that you have a pile of laundry and you pull items out one at a time. If a sock matches a sock you've already seen you pair it up. If not, you put it aside waiting for a match. If it takes time X to figure out what color each sock is, and tiem Y to tell whether it's a left or right sock, then if X > Y, then you want to get all socks the same color: you just have a working pile of socks (either left or right). When a sock comes in it's either the other side, in which case you pull a sock off the stack and put the pair away. If it's the same side, you put it on the stack.

On the other hand, if Y > X, it's a little trickier. You need to maintain a separate pile for each color. If you only have one of each color, then you only ever incur X, since the second of each color must be its match. On the other hand, if you have more than one of each color, then you incur Y + X because you need to know both which pile to look in and whether a sock is a match or just more of the same (this can be partially optimized when you've already paired up all but one pair of a given color, at which point you go back to incurring X for that color). So, the breakeven point here depends on how many different colors of socks the manufacturer makes. Injinji only makes three colors of this sock (white, black, tan) so it probably makes the most sense to buy all one color.

Extra Credit: Is the situation the same if you just throw all your socks in the drawer and then try to pick out socks later? Scanning for colors isn't the same as deciding what color something is...

UPDATE: Apparently this is a problem many others are concerned with as well.

Can Superman read books without opening them?

An acquaintance recently brought me a homework problem which asked roughly "True or false: If you had x-ray vision you could read a book without opening it." Phrased like this, I'm reasonably confident the answer is "False", but it's a bit complicated to analyze.

First, we have to ask what x-ray vision is. It's natural to think of it as being like ordinary vision, i.e., your eyes are sensitive to x-rays. This doesn't make any sense, though: ordinary vision works because there is ambient light in the visible spectrum (from the sun, indoor lights, etc.) and it reflects off ordinary objects. What you're seeing is that reflected light. But there's next to no ambient x-ray radiation in the ordinary environment, so even if you could detect x-rays, it would be like trying to see in the dark.

So, in order for some sort of x-ray vision to work, you'd presumably need to be emitting some sort of radiation to illuminate the subject. Assuming that the source is attached to you, then x-ray absorption/transmission measurements (like with medical or dental x-rays) won't work, but x-ray reflection or x-ray fluorescence (XRF) will. The other possibility is that you're emitting some other sort of radition that the subject absorbs and then fluoresces in the x-ray region (really, it could be in any region, but remember we're calling it x-ray vision, not hard radiation fluorescing in the visible vision).

There are actually at least two aspects to this question:

1. Can you use x-rays to distinguish ink from paper?
2. Can you resolve the letters on individual pages?

I suspect that the answer to (1) is "Yes". Older inks, at least, often contained metals (e.g., iron gall ink), and so should have fairly different spectra from paper. In fact, this article describes the use of just such a technique: XRF to find iron gall inks in the Archimedes Palimpsest. It's less clear how modern inks would respond. I've seen some articles suggesting that x-ray fluorescence spectra of modern inks also are distinguishable from the background paper spectrum (see this article on cyan ink and this paper that mentions the use of XRF on postmark inks).

This only answers half the question, because books aren't just one page: they're a bunch of pages on top of each other and so even if the ink does fluoresce differently from the paper, you still need to read the book. There are a bunch of possible problems here: first, if you're using XRF possible your fluorescence isn't in the x-ray, it may get blocked by the paper on top of it. But let's assume you're irradiating in the hard x-ray and your fluorescence is in the soft x-ray and makes it out of the paper; we still have to ask whether you can spatially resolve the letters. This partly depends on the characteristics of your x-ray source.

If it's an uncollimated point source like a lightbulb, then you'll irradiate the whole book and each letter will be an independent emissions source, each of which will also be radially symmetrical, plus all the background noise from each sheet of paper. It's not at all clear you can disambiguate these sources. If you have a collimated source, the situation may be a little better, since then you just get a bunch of point sources along the line the beam is taking through the book). That's probably still not enough though, since the fluorescence from multiple radially symmetrical point sources still overlap on your eyes, and you need to distinguish light which is coming from incredibly small angular displacements (the thickness of a piece of paper apart at a distance of 10s of centimeters away). If you were able to take measurements from multiple independent angles like in a CT scanner, then you would have enough information to resolve things, though with an extremely large amount of computation. There are also multiple beam techniques like four-wave mixing, but that also seems biologically unlikely.

Bottom line, then, I think the answer is "False". I.e., it probably is possible to spectroscopically read books without opening them, but I doubt you could do it with any plausible (or even implausible) biological process.

Extra Reading: Man of Steel, Woman of Kleenex

October 2012

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