So last night I remembered that thing that Tom sent out a while back, that was supposedly a part of a short proof of the four-color theorem. For those who don't remember, what it was actually a proof of was that, for any n, given any 2 parenthesizations of the expression
x1×x2×...×xn
where × is vector cross-product, there is some way of substituting in values for the xi from the set {i,j,k,-i,-j,-k} such that the results are equal and nonzero. How does this relate to the four-color theorem? I have no idea, and neither did Tom. I'm guessing it didn't actually result in a short proof of the four-color theorem... but the four-color theorem isn't the point of this entry. The point is the statement above, or it starts with it, anyway.
So last night in bed I was remembering this and I was reconstructing the proof, and I got most of it but I was missing one important lemma (that if one of the parenthesization-level-strings was less than its successor, then it's of the form 123...nn). I remember someone - Tom or Dr. Nevard - saying the proof actually had a mistake, but that it didn't matter. Because, unless I had the order he was using wrong, this lemma was pretty clearly false. I tried several variations on the order, but it didn't help. This is the mistake, then? But it doesn't matter? He definitely needed that... well, OK, I suppose there's some way around it... but I certainly was convinced of it when I read the proof; what was it? Was he using a different order, or was there actually a hole? Looks like I need to go look it up.
So, I go to where I've backed up everything from the desktop (I've since formattted the hard drive, though I've yet to get around to installing a new OS on it), and I look for it. It was a PDF, I recall. Hm... can't seem to find it. Well, it was an email attachment, presumably I never saved it to disk. I go and look in my mail folder - not that I'll find it in readable form, it'll be in base64 or something, but maybe I can at least find the email, and I notice - all the email files: 0 bytes. Empty. What the hell? How did that happen? Everything else copied over just fine; why not the email? Unless the mail wasn't actually being stored in those files, which is what I'm betting on. Otherwise I've just lost all the email from the old computer... most of it I have no use for anymore, sure, but I certainly had no intention of throwing it away so soon (especially because of things like this!). Anyway, it makes no sense that just the mail files would fail to copy properly, so I'm assuming that the mail is still their in those other files in the mail folder... I'll find out when I actually get the other computer working, I suppose...
-Harry
x1×x2×...×xn
where × is vector cross-product, there is some way of substituting in values for the xi from the set {i,j,k,-i,-j,-k} such that the results are equal and nonzero. How does this relate to the four-color theorem? I have no idea, and neither did Tom. I'm guessing it didn't actually result in a short proof of the four-color theorem... but the four-color theorem isn't the point of this entry. The point is the statement above, or it starts with it, anyway.
So last night in bed I was remembering this and I was reconstructing the proof, and I got most of it but I was missing one important lemma (that if one of the parenthesization-level-strings was less than its successor, then it's of the form 123...nn). I remember someone - Tom or Dr. Nevard - saying the proof actually had a mistake, but that it didn't matter. Because, unless I had the order he was using wrong, this lemma was pretty clearly false. I tried several variations on the order, but it didn't help. This is the mistake, then? But it doesn't matter? He definitely needed that... well, OK, I suppose there's some way around it... but I certainly was convinced of it when I read the proof; what was it? Was he using a different order, or was there actually a hole? Looks like I need to go look it up.
So, I go to where I've backed up everything from the desktop (I've since formattted the hard drive, though I've yet to get around to installing a new OS on it), and I look for it. It was a PDF, I recall. Hm... can't seem to find it. Well, it was an email attachment, presumably I never saved it to disk. I go and look in my mail folder - not that I'll find it in readable form, it'll be in base64 or something, but maybe I can at least find the email, and I notice - all the email files: 0 bytes. Empty. What the hell? How did that happen? Everything else copied over just fine; why not the email? Unless the mail wasn't actually being stored in those files, which is what I'm betting on. Otherwise I've just lost all the email from the old computer... most of it I have no use for anymore, sure, but I certainly had no intention of throwing it away so soon (especially because of things like this!). Anyway, it makes no sense that just the mail files would fail to copy properly, so I'm assuming that the mail is still their in those other files in the mail folder... I'll find out when I actually get the other computer working, I suppose...
-Harry
