Friday, December 18, 2009
Wednesday, December 9, 2009
George Hart presents what he says is a calculus problem: What is the ratio of the surface area of [the double mobius] cut to the surface area of the usual planar bagel slice?
Calculus is not necessary to answer this question and I think there are some key insights if you kick it old-style.
Let's start with the easy part: The area of the regular "sandwich style" bagel cut.
π(ρ+r)2 - π(ρ-r)2 = 4πrρ
Now, on to the (Double) Mobius Bagel.
(This image is very confusing--refer to the above link for a better visualization of the bagel in 3D.) Note that the red and blue edges of the cut must be the same length because they are the same cut, just offset by π radians around the bagel. OK, now unroll the bagel into a cylinder.
The double mobius cut is now a slice through the center of a cylinder. If you imagine the knife traveling down the cylinder, it rotates 2π radians while traversing the length. (Aside: The solution to the question of how to make a true mobius cut in a bagel should now be easy to visualize.)
I'm going to argue that unrolling the bagel didn't change the cut area. If you imagine re-rolling this cylinder into a bagel, you'll see that one side stretches while the other side shrinks. The red and blue lines are symmetrically spaced around the cylinder, and the two edges are the same length in the rolled state, so the stretching and shrinking should be the same for both. But do the stretch and shrink cancel out?
We get a stretch of 2πr with a shrink of -2πr. Therefore unrolling the bagel doesn't change the cut area.
Another way to think of it is with a trapezoid. If you shorten the top by the same amount that you lengthen the bottom while the height is constant, the area of the trapezoid is the same.
So the area of the cut in cylindrical form is the same as the area of the cut in bagel form. But what is that area? We need to deal with the twist. For this, we unroll the cylinder.
The width of this rectangle is the original cylinder length: 2πρ.
The height of the rectangle is the original circumference of the cylinder: 2πr.
By Pythagoras: b = sqrt((2πρ)2 + (2πr)2)) = 2π*sqrt(ρ2 + r2).
b is the length of the edge of the cut. The "depth" of the cut is 2r (i.e. the diameter of the "tube" of the bagel). Therefore the total area of the cut is 4πr*sqrt(ρ2 + r2).
This area is larger than the sandwich-style cut by a factor of ρ/sqrt(ρ2 + r2).
Wednesday, September 23, 2009
Old Sk00l Unix Guru #1: Right.
Old Sk00l Unix Guru #2: Right.
Me: But I need to do it from a script, so I'm checking the exit code....
OSUG #1: Right.
OSUG #2: Right.
Me: And that works great on Linux. But on Solaris, it's 0 on both success and failure.
OSUG #1: Can't you just put it in backticks and grep the output?
Me: Well sure, I can. But it seems like such an old utility should have a way to return 'yes' or 'no'. What's the Old Sk00l UNIX Way to do this?
*all three characters pore over Solaris man page, which was last updated in 1992*
Me: I guess I'll do that. I just wanted to be sure that if I did the backtick and grep thing, someone isn't going to look at that later and say "what a n00b".
OSUG #1: You can write a comment that says "let me know if you have a better way".
OSUG #2: Or "to overcome braindead Solaris return code values".
Me: I guess leaving a comment about someone being braindead IS the Old Sk00l UNIX Way.
Monday, September 7, 2009
Sunday, August 30, 2009
Many cans are unlabeled, but some have a name scrawled on them, or some tape, or a business card stuck under the tab. However, these solutions have a couple problems. First, they only provide identification, not security. Second, some of these are kind of time-consuming. The tape, for instance, can't take less than 60-90 precious seconds during which you could be reading some hilarious emails forwarded from the secretary.
Thus my invention, which I call "Club Soda". You slip it on, snap it shut and club it with a tiny key. It would be personalized with your name via a tag or engraving or something. Security and identification in under 10 seconds. MSRP $4.99.
Saturday, August 29, 2009
Notes for next year:
- Plant only about 4 tomato plants, not 6.
- Same for the jalapeños.
- MORE green peppers. Or maybe they were just shaded by the tomatoes?
- Plant the tall things in the back. Which is to say, the tomatoes.
- Tomato cages have very little resistance to bending from being overloaded. Stake them.
- The Garden-Fresh Vegetable Cookbook has a trillion good ways to use all the results. (By the time I re-read this next year, I should know if they are actually yummy.)
- I will also know if making my own crushed tomatoes worked.
- Add snap peas, basil, cilantro and possibly watermelons.
Tuesday, July 21, 2009
My officemate and I discussed this one on-and-off throughout the day, drawing and redrawing many extra lines and triangles. We did finally solve it, but by a kind of questionable meta-method. Here's the basic idea:
Imagine blowing up the circle just a little bit. The line on the left can still be 12 units long. It will still be possible to have another line cut a chord 7 units long. And it will still be possible to have those two lines meet at a point. The angle they form will be different, though, and the value of x may be different. But the angle isn't a given and neither is the diameter of the circle. Therefore that problem is the same problem as this one. Since we know this puzzle has a unique solution, the value of x must actually be a constant and we can make that angle whatever we want. We will make it such that the chord is actually a diameter.
From that point the puzzle is easy and this method did yield the correct answer. The questionable part is assuming there was a unique solution. If we had come across this puzzle "in the wild" this wouldn't have been at all kosher. So how did the first person solve this puzzle? What's the real solution?
Friday, July 10, 2009
- Woooo! On 03/26/2007 I started at 259. Today I'm at 199.
- I've only recently started using multiple colors of clay together. I have an irrational fear of "using up" the clay when I mix it, even though a pound of clay is like $1. However, a clay model gathers impossible-to-remove dust and small nicks. So I've finally bitten the more expensive bullet of switching to sculpey.
A 4 year old of my acquaintance calls this "the treasure cat" and a 2 year old, also of my acquaintance, calls it "measure cat". (Also, any Cheshire Experts will notice right away the wrong color I didn't notice until too late.)
- I have a scrollsaw from way back but the blade broke and it turns out that that size is hard to find. Then last weekend I found a bandsaw at the flea market! After some tuning and tweaking, it's running great.
Sunday, June 28, 2009
Friday, June 12, 2009
- Never played before. Any halfway game is annoying the first few times as you learn The System and what is and isn't important, etc.
- Takes a lot of time to set up.
- I feel like a real dork saying things like "leathern helm" or "dwarf mage".
Which brings me to the nerdliness. Playing D&D. Around the dining room table. While eating pizza, hot dogs and nachos. At least none of us has a neckbeard or obesity problem. Although one of us runs Linux. Oh and Mom brought in a sugary snack, plus we had our first game-based injoke.
Anyway, he actually did really, really well at it. I mean, we're wandering around without a clue in a very haphazard dungeon and he's giving hints pretty often ("do you want to check behind you?")...but he has the right attitude. He's confidently telling us what's what and taking our unanticipated actions completely in stride. Kudos, proto-NERD!