ISWG a day late, and horns with no volume

A day late for ISWG, so I'll give you two posts in one! The first one is my insecurities, and my second one is being too smart for my own good.


New years resolutions, amiright?  I stopped setting them a while ago, but I still try to improve myself. I stopped setting them because of that god awful trap that I tended to fall into where I miss the goal and I think "Oh, damn it I failed! No I'll never be able to do it!" and then I would give up until next year.

Now, I've got stuff I do and I feel about when I can't do it as well as I can. It's a journey. Write when I can, try to improve my organizational skill, get better faster harder stronger. You know the whole drill.

How are you guys doing on your resolutions?

This month the Minions of Math are going to start talking about infinity. Not to ruin the surprise, but we'll talk more about infinity next month too. This is a video month, and I watched this video by Numberphile this morning before running 2 miles in 3^o Fahrenheit weather. That combination has proven time and time again to kick my mind into overdrive, so if figured I would take the time to address one of the paradoxes here: Gabriel's Horn.

There is a soft spot in my heart for Numberphile and Vihart because those two along with Gödel,Esher, Bach and the great Lewis Carrol did more for my interest in mathematics than 13 years of schooling. With that said, I've noticed that the deeper and deeper I get into math the Numberphile videos are a bit harder to watch since a majority of them seem to be aimed at I Fucking Love Science crowd. Instead of complaining about this, here are some more thoughts about Gabriel's Horn.

He points out that what makes it a paradox is the fact that it's a horn that tapers into infinity, and as such it has infinite surface area, BUT it has a finite volume. It seems paradoxical, until calculus is involved.

WAIT! Come back! I promise it's simple calculus! It needs to be, since I have yet to take an honest to god calculus class. First things first, I don't know much about the average reader's childhood, but a few of you must of noticed something about thirds at some point in your life. Namely this: 1 divided by 3 is equal to 0.3333 where the three just goes off into the distance. Multiply that number by 2 and you get 0.66666 repeating. So far so good, but if you multiply that first number by 3, depending on your calculator you won't get 1, you'll get 0.99999 repeating. That's a paradox, honestly, because you already divided 1 by 3 to get 0.333333, but multiplying 0.333333 by 3 doesn't give an answer of 1. This is simply because god hates math.

OK not true. The truth is 0.999999 repeating is so close to 1 that it's 1. That answer always struck me as unsatisfying when I was younger, but if we except that axiom we can do something more amazing with Gabriel's horn. More amazing than what the guy in video says with the whole "You couldn't paint the surface, but you could fill it with a finite amount of paint."

If there is a number that is so close to 1 that it's 1, then a number like 0.000 . . . 001 can be so close to zero that it's 0. Why Gabriel's Horn works is it's an infinite series, which can be best explained by a joke:

There's a math conference in town, and all the mathematicians walk into a bar. The first mathematician orders a beer. The second orders half a beer. The third orders a quarter. The forth orders an eight. The bartender stops them at this point and hands them two beers.

 It works for 2: 1 + 1/2 + 1/4 + 1/8 + 1/16 . . . to infinity. It works for 1: 1/2 + 1/4 + 1/8 + 1/16 . . . to infinity. In fact, it works with zero if ou work with small enough numbers. In a sense, the volume kind of works the same way for the horn, it's an infinite series. Less cool when you know how it works. What I've been working up to this whole time is that IF this is the case, AND a the sum of an infinite series can equal zero, THEN a Gabriel's trumpet exists that not only has infinite surface area, but also has a volume equal to zero. I'll leave it as an exercise to the reader.