"All right, let's hear it."

It was hot in small claims court that day, and I had better things to do. My mind had been wandering back to the last time I had to defend myself in a case like this. Then, the plaintiff was that infernal horn manufacturer--but that's a long story, in more ways than one. The gruff voice of the judge rattled me back to the present and I groped in vain to recall what had just been said. Absent-mindedness can be quite a liability in a situation like this, but luckily the judge repeated herself.

"You claim you have some reason that you refused the plaintiff your services. If you don't mind, the court would like to hear it now." The people in the courtroom responded with a rippling of laughter at the judge's barb, but were rewarded with a stern glance of their own. Apparently even laughing at the judge's sarcasm was unacceptable here. It's always reassuring to have a judge with a sense of humor.

Taking a deep breath, I began to sketch the argument for the court. Having a valid proof is good enough in a thesis defense, but legal defense is something else entirely, and not always directly connected to logic. I knew I had my work cut out for me, and there was nothing to do but plow forward.

"Your honor, if it please the court, I would like to enter into evidence my business card."

"It would please the court if you didn't pretend you had a legal degree. But if you want me to take a look at the card, hand it to the bailiff." This was going to be harder than I thought.

"As you can see on my card, my business has a motto--'Never a spot missed'. And it's not just a motto--it's a guarantee we put on every job. That's what the problem with this job was--there was no way that we could offer the guarantee on this one, and we don't do work we can't guarantee."

The judge's face hadn't moved a muscle, having the same contemptuous sneer left there by her last comment. Amazingly, she was able to preserve that look through her next comment as well. Thank goodness it was short, or she'd have caught me daydreaming again, this time wondering if was really true that your face could get stuck in a certain position.

"Yes, yes, the plaintiff has told us all of this. You were supposed to be adding information here. The only thing in dispute so far is your claim that you couldn't offer your guarantee on the job the plaintiff asked you to do. Why don't you enlighten us on that point?"

I could feel the smugness emanating from the other side of the courtroom grow thicker with each tongue-lashing the judge handed out. I tried to focus on my argument, but it was agonizing not to just peek over at the look of enormous self-satisfaction my opponent must have been wearing right then. Fighting back a growing discomfort with how things were going, I continued.

"Yes, your honor, I was about to get to that. The fact of the matter is that no one could paint a room like the one in question and not miss a spot. It's not a matter of taking adequate care, or anything like that. The problem is with this wall [I indicated the problematic wall on the blueprint displayed in the courtroom]. As the plaintiff has already stated, this wall was custom-designed and built by IRC [IRC, which is just an acronym for "Irc Recursive Construction", has been involved in cases like this before[. The method of construction is a trade secret, but the resulting structure is, of course, required by law to be completely and accurately described in documents filed in city hall." Now I was starting to get warmed up--these documents were at the center of my case, and I knew my case was a good one. I made sure that the judge had a copy of the description of the room and continued.

"Your honor, as you can see in this record, the room is laid out in a very ordinary fashion, 9 feet wide by 13 feet long, except for this one wall. This wall, where the other 9 foot wide wall would have been, has been replaced with a rather unusual structure. Allow me to quote from the building contractor's description:

'The wall can be thought of as having been created in the following fashion--first, an ordinary 9 foot wide wall is marked off into three even divisions. The middle section is removed, and we build outward to enclose a three foot square. If we stopped at this point, the wall would look something like this:

Note that we now have five three-foot sections of wall. Our next step is to take each of those wall sections and do the same thing to them that we did to the original wall--that is, we mark them off into three equal sections, remove the middle section, and build out to enclose a square. The next step would look like this:

Note that there are now twenty-five one-foot sections of wall. Our next step is to section each of these walls off, remove the middle section, and build out a square in each of them.

This time, there are 125 4-inch wall sections. The final product which we

create is exactly the same as what would result if you carried on this process (divide into three, remove middle third, build out a square) an infinite number of times. In light of the fact that this is a protected trade secret, we can not reveal in public record how this is accomplished, but in sealed documents we have shown that the result is equivalent to what would result from an infinite iteration of this process.'"

After quoting the documents and showing the diagram, I paused to let the judge study them. Everything else I would say would depend on her understanding of the structure of the room, so I wanted her to have plenty of time to look things over.

After a while she raised her head. With a slightly less unfavorable tone (I figured she had just let her guard down a little, distracted by the unusual description of the wall), she indicated that I should continue.

"Well, it's clear that with a wall like that, I can't guarantee that I haven't missed a spot." I knew that it _wasn't_ clear at this point, but I wanted dramatic emphasis. "After all, the area of the wall is infinite."

The last statement had exactly the effect I wanted it to. The courtroom was silent, and all eyes were turned toward me. I proceeded to show that that the wall's area was, in fact, infinite. This was easy--all I had to do was follow the contractor's description of how the wall was built, and show how the total surface area of the wall grew with each step of the process. The room had ten foot ceilings, so the area of the wall was simply ten times the length of the line on the blueprint that represented the wall. For example, the wall at first was 9 feet long, so the total area was 90 square feet. At the next step there were 5 three foot long sections of wall, giving 15 feet of "baseboard length"--or, 150 square feet of wall. With a little work, I showed how this pattern continued, and each step made the wall five-thirds as long as the previous wall length, and so the area kept increasing by a factor of five-thirds as well.

I pointed out that if you start with 90 square feet (or any number of square feet) and multiply it by 5/3 five times, you have already increased the area by a factor of more than ten (about 12.86, to be more exact). And each five times thereafter increases the previous area by a factor of more than ten. That means that, in effect, you add another zero to the end of the area each five steps. And since there are infinitely many steps, the area ends up being infinite.

"And so you see, what he is asking me to do is impossible. How can I guarantee that I haven't missed a spot on the wall when I know that it would take an infinite amount of paint to cover it?" I had been head to head with this judge before, and this time I knew I had her. I could barely contain my glee.

"So that's it? You're basing your whole case on the belief that it takes an infinite amount of paint to cover an infinite area?" My elation abated just a bit. Didn't she get it? Wasn't I clear enough? How could she be missing something so simple?

"Yes, exactly. I mean, if you multiply the thickness of the covering of paint, say, ten microns, by the amount of surface area you want the paint to cover, you get the volume of paint required to cover it. And if the area is infinite, that gives an infinite volume..." And I trailed off. That nagging doubt had just exploded into the realization that everything I had been arguing was barely relevant, if at all. But it was not a problem that I didn't finish the sentence, as the judge saw fit to do that for me.

"...as long as you assume that the thickness of the paint is uniform over the surface."

Oops. Maybe I should have thought that one out a little harder...

[editor's note--in case you don't understand what the judge was pointing out here--you can (mathematically speaking--see the paragraph below about how we're "cheating") cover an infinite surface area with a finite volume of stuff, despite the fact that this seems impossible. If you are familiar with infinite series, you can see how. Suppose you have an infinitely long wall that is ten feet tall. Open a can of paint, and use the first half- gallon to cover the first one-foot-wide strip (yeah, you'll have to put it on pretty think). Then cover the next one-foot-wide strip with a thinner coat--do this with only a quarter gallon of paint. And now, on the third strip, cover it with only one-eighth gallon of paint. Continue this way. You'll be able to cover the whole wall with a single gallon of paint!

Yes, we're cheating a little. If you follow this process, you will have to paint the tenth one-foot-wide strip with a little less than one one-thousandth of a gallon of paint. That's going to be problematic at best. But for the purposes of the story, it's not cheating--in the story, one construction company was able to create a wall with infinite surface area. If we allow that possibility, we have to allow for arbitrarily thin coats of paint. Fair is fair!]

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