Toilet paper rolls can fool you. You think you have a good quarter of a roll left, and next thing you know, you’re sitting on the pot trying to remember how to wipe your bum with just one square of paper.
The problem is that the toilet roll’s depth (i.e., thickness of paper on the roll) doesn’t diminish in a linear fashion. With an unused roll, you can get about three sheets in one rotation. But getting the same three sheets toward the end of the roll can take almost three rotations. A constant rate of use leads to an increasing rate of shrinkage in the remaining depth.
Using basic geometry and a measuring tape, I found the following:
• The diameter of a typical roll of toilet paper is about 10 cm, of which about 4 cm is space inside the tube and 6 cm is paper.
• The paper’s starting depth is about 3 cm (on each side of the tube), with 100% of the paper remaining.
• When the paper appears half gone, with 1.5 cm of depth, the actual fraction of paper remaining is only 39%.
• When the paper appears three-quarters gone, with .75 cm of depth, the actual fraction of paper remaining is only 17%.
But there’s more! I’ve derived formulae for the relationship between apparent fraction remaining (as measured by paper depth) and actual fraction remaining. Let X = apparent fraction remaining and Y = actual fraction remaining. Then:
• Y = [(3X + 2)^2 – 4]/21
• X = [sqroot(21Y + 4) – 2]/3
The first formula is probably the more useful, since you can use it to calculate the actual fraction left when you know the apparent fraction. The formulae still work for larger-than-usual rolls, if the tube is enlarged by the same proportion as the whole roll. (Otherwise, the formulae are a bit more complex.)
Now there’s news you can use!
Wednesday, September 01, 2004
Toilet Training
Tuesday, August 31, 2004
The Cause of Oz
I recently finished watching Season 1 of “Oz” on DVD, which got me thinking about the rape penalty again. I’m still appalled that prison rape is not taken at least as seriously as the death penalty, given that (a) it’s imposed without regard to the severity of your offense, (b) no judge or jury officially approves of the sentence, (c) it’s systematically inflicted on the weakest and most vulnerable of prisoners, (d) the transmission of HIV can make it a de facto death penalty, and (e) it occurs at least an order of magnitude more often than the death penalty. Why isn’t allowing prisoners to be raped considered cruel and unusual punishment?
When I presented my position to a group of college students this summer, most of them libertarians or libertarian-leaning, I was surprised by their willingness to defend prison rape. They relied primarily on a loosely intent-based argument: that while prisoners may unfortunately get raped, that is not the state’s intent when it jails them. In a sense, they pointed to point (b), above, as an excuse: nobody officially sanctions rapes; they just happen.
That argument, in my opinion, fails. Consider an analogy. What if the state imprisons a man in the same cell with a rabid dog? Let’s give the state the benefit of a doubt by assuming the intent is merely to imprison, not to expose him to attack by a vicious and diseased animal. Nonetheless, it’s easily predictable that the prisoner will get bitten and possibly infected. I hope this would be considered cruel and unusual punishment by most reasonable people.
But, it might be objected, the dog attack verges on certainty, which is not true of prison rape. Okay, fine. Let’s suppose we have ten prisoners and one rabid dog. The prisoners draw straws, and the one with the short straw gets put in the dog’s cell. Does that free the state from blame?
I will grant that the state cannot, at any cost, guarantee 100% safety in the prisons. Take the above example and continue increasing the number of prisoners, while keeping only one rapid dog. Assume for some reason we cannot remove the dog at low cost (a reasonable assumption, given that the dog plays the role of a rapist prisoner). For some sufficiently high number of prisoners, I would concede that the likelihood of attack for any given prisoner becomes low enough not to constitute an unconscionable risk.
But based on the numbers that I’ve seen, the actual risk doesn’t come close to being that low. The survey I cited in my previous post indicates that one in ten prisoners gets raped; hence the ten prisoners in my analogy above. Surveys can be unreliable, but in this case, I’ll bet the number is conservative, since some prisoners would be inclined not to admit having been raped (because of shame or fear of reprisal).
Moreover, I think the risk is probably much greater than one in ten for some prisoners. For instance, what if I went to prison? For those readers who don’t know me, I’m a slight guy (5’11”, about 160 pounds) with what a hard-up convict might consider “a purty face.” I figure my chance of getting raped would approach certainty. If prosecutors put me behind bars without some kind of protective custody, they would pretty much know I was going to get raped, or at least badly beaten.
My point is that the notion of intent is not – or should not be – limited to the primary goal of the state’s policies. When the agents of the state can predict with relative certainty that a given prisoner, or some substantial fraction of all prisoners, will suffer a peculiarly brutal and demeaning form of attack, they cannot plausibly pretend innocence on grounds of not “intending” for it to happen.
Monday, August 30, 2004
Theory and Application
On Saturday, I met a married couple with a couple of kids. They were pondering whether to have a third child - or get a horse instead. In class today, I felt compelled to use this as an example of diminishing marginal utility. (The example will probably resurface later this semester, when I replace diminishing marginal utility with diminishing marginal rate of substitution.)
Speaking Truth to Punkass
Speaking to a kid wearing a hammer-and-sickle T-shirt, Radley Balko calls it like he sees it:
So I added, "I know Soviet chic is hip right now, particularly on college campuses. But you really ought to think about the message you send by wearing that shirt. It has all the charm of a swastika."Right on. I’ve written before on why people give a moral pass to communism that they don’t give to Nazism, despite both systems having been responsible for the death of tens of millions of people. My assessment is about the same as Radley’s: anyone who still advocates communism in the modern era is at best monstrously ignorant.
And yet I, like Radley and some of his commenters, am willing to tolerate ironic expressions of communism, like the very pricey Red Square vodka bar in Mandalay Bay. I wouldn’t feel the same about a Nazi-themed establishment. What makes the difference? Perhaps it’s my sense that I can trust the communist irony more, given the obvious capitalist context.