Sunday, March 29, 2009

Hyperbolic Space

I recently ran across, via this post on Division by Zero, a way to make hyperbolic space from paper, a project that I couldn't resist. In fact, it claims to be a hyperbolic soccer ball. Digging through Reader to find the link again, I found this other recent post about shapes relating to soccer balls, so thought I'd share it as well.

Anyway, the idea is that to typically make a soccer ball, you place a ring of hexagons around a pentagon, and iterate. The pentagons introduce some positive curvature in the process (hexagons alone have 0 curvature - they tile the plane), and you end up with something fairly spherical. If you place hexagons around a central heptagon though, you get negative curvature.

The directions to actually make one yourself are at The Institute for Figuring, and are available here (pdf). Following the basic instructions, I ended up with the following:

Which my cats only took fleeting interest in:

The directions suggested that you could continue adding more rings of hexagons, with heptagons in appropriate locations, and extend the model. I was curious to see what would happen when I did, so I printed out six more of the base sheets (3 of which gave the starting model, above). It's a bit of a monster:

Tuesday, March 17, 2009

Semi-Sierpinski St. Paddy's Sugar Cookies

Today I made some sugar cookies, with hopes of them coming out something close to the awesome Sierpinski Cookies I found here. This was my second time trying. The first time I used a package of sugar cookie mix, and it didn't work out for me at all. This time, I followed the recipe linked to from the above page and made my dough from scratch. They just about came out reasonable:

In that picture I have a few levels of iteration of the general idea. You'll notice, though, that at the highest iteration, the cookies aren't quite right. There is supposed to be one big green square in the middle, and 8 little green squares around the outside. Let's have another look at how one actually came out:

Where's the missing green square? I honestly have no idea. But all of my cookies are missing at least one, and sometimes two, of the green squares.

Ah well, they still taste good. And they came out well enough for me to maybe try again sometime.

Thursday, March 12, 2009

Math Blogroll in OPML

I don't spend a lot of time visiting the actual pages for many of the blogs I follow, since I get all (or at least, most, and the main portion) of the content from their rss/atom feeds. Recently though, one of the feeds I had indicated that the author was quitting. For whatever reason (perhaps because it mentioned having the world's best math blogroll), I was inspired to visit the actual page, instead of just removing the feed from my list (or doing nothing).

The feed was from Vlorbik on Math Ed. Upon visiting the page, I found that Vlorbik kept a pretty substantial blogroll of math blogs. Liking to be in the know, I figured I might subscribe to some. Of course, I already probably do subscribe to some (and author some :)), but there are surely plenty there that I don't subscribe to. And perhaps some of them are ones I would like to follow. But I didn't want to click each link, load each page, find it's feed, and add it to Google Reader. I'm pretty lazy, and my computer would slow down a bit and frustrate me.

This evening, though, I decided to see if I could write a script to grab the rss/atom feeds for any or all of the linked pages in the blogroll. I had a great time doing so. Remembering some fun pattern matching variables in perl(like \$' and \$& (the \ there only because of how I'm doing LaTeX in Blogger)), and using curl to grab the pages... good times. Then some reformatting of appropriate strings, and out pops an OPML file. Handy, because that's what Google Reader expects if you want to import a bunch of feeds. I've played with similar things before.

Anyway, the long and short of it is, I thought perhaps other people might find this OPML file helpful. Blogger won't let me upload anything besides pictures (where's my damn GDrive?), so the file is currently (as of this writing) on my UVA personal page, here. If you'd like to blindly add these feeds to your feed reader, and then trim them down individually based on content or whatever, I encourage you to do so. The only reader I've used is Google's, so I'll give some instructions for that.

The first step is to download my OPML file, and save it somewhere convenient (you only need it temporarily on your computer). In Reader, at the bottom of the left-hand pane is the 'Manage Subscriptions' link. Click on that, and then the 'Import/Export' link at the top of the settings page that pops up. In the file upload form where it says 'Select an OPML file to upload', pick the file out from wherever you saved it, and then click 'Upload'. Wait patiently as Google imports the new feeds (it really doesn't take that long, though it might take longer for news items to start flowing in). It'll send you back to the main settings page, so click 'Back to Google Reader' to start reading. You'll notice that the feeds all show up in a folder in your subscriptions panel, called 'vlorbik' (if you already have such a folder, you might modify my OPML file before upload... I should have told you that earlier). If you already subscribed to one of the feeds, it won't mess anything up, and they won't show up as duplicates in your news stream. Of course, when making this file I grabbed the atom files, where available, so if you are subscribed to the rss feed (as I am, in many cases), then you will have duplicates. But whatever, I'll let you sort out your own subscription list.

So, with this success, I feel like perhaps I should visit actual pages (instead of just watching the news stream go by in reader) more often. Perhaps find some other blogrolls?

Anyway, enjoy. Sorry, Vlorbik, that I only started getting to know you on your way out.

Monday, March 9, 2009

Finding Mistakes

Of all the questions I get in the office hours for my calculus classes, the most frequent are probably from students who have worked through a problem and gotten the wrong answer, but can't find their mistake. I sit down with these students and go through each line of their work, ideally getting them to explain each of their steps to me. Sometimes, students are able to spot their own errors when we do this. Frequently, though, they can't.

While it's generally not terribly difficult for me to find errors, it's a skill I believe I've developed after several years in my math classes. It's a skill I'd like for my students to develop. Recently, I struck on an idea for how to run class that might help students find mistakes.

Our calculus classes are accompanied by an additional class period, called the fourth hour or discussion section. Mostly what happens during this time is that students ask questions from the homework, and the TA works them. Or, at least, gives some hints for how to work them. Sometimes the TA for the discussion section is just the instructor, sometimes it is another graduate student. While students certainly appreciate the chance to ask these questions and get answers to their homework, this setup has always frustrated me.

Part of the problem with this setup is the partition of the class into students who have started problems but gotten stuck or made a mistake, and students who have not started problem. The students who have not started are waiting for as many answers in the discussion as possible before doing whatever few remaining problems there are on their own. My hope is that these students do poorly on the exam, if I'm honest. The other students, the ones who ask the questions, because they have started their work, are also not gaining much from most of the time spent answering their question. This is because their mistake shows up, or they got stuck, mid-way through the problem, so all the time used in class getting to that point of the solution isn't much help. However, starting mid-way through the problem won't work, since most of the rest of the class will be lost.

I think a better plan would be to have students bring in their work, and spend most, if not all, of discussion sections working on finding mistakes. I'm trying to think about implementation details for this, and thought I'd see about getting some feedback here. I envision students writing each problem that they worked on but didn't get correct on a separate sheet of paper, and bringing those to discussion sections. Then, during class, the papers would all be gathered up and redistributed to all the students. Depending on how many papers there are, students might break into groups to tackle a paper, or perhaps collections of papers (or they could work alone). With all of those eyes, bugs are, famously, shallow. Groups would make notes on the paper about errors, or tips on how to proceed, and then papers would be returned to their owner. If, after this time, nobody can find a mistake on a paper, or everybody is stuck at some point in the same problem, the TA can talk through the problem with the class.

So, a few questions.
  1. Do I have students write their names on the papers, so that it's easy to get them returned? Or does this violate some sort of anonymity that should be preserved? Would writing some fixed number on the paper be better? Or perhaps initials? Maybe have students write something on their papers that they can identify, but other students wont, and then when passing papers back, just hand back all the papers at once, to be passed around so students can grab their own?
  2. What about students who haven't started the assignment? Or completed it successfully? Should I set things up so the assignment is due very shortly after the discussion section, encouraging students to have looked at it before-hand? Or will this lead to more students in office hours, avoiding postponing getting their problems fixed, and thus defeating the purpose of the group mistake-finding exercise? Should I have students who have already finished pick a problem and write up a fake solution, artificially introducing an error, to give somebody (whoever ends up with their paper) a chance to try to find the mistake?
  3. What could go wrong with this setup? What policies should be put in place? What do I need to be careful of, or think more about?
Thanks for any feedback.

Wednesday, March 4, 2009

Mm, Donuts

As a topologist (or, perhaps more accurately - somebody who has taken primarily topology courses recently), I see tori (donuts) all the time. And people always talk about a meridian circle on a torus, versus latitudes. I generally have a hard time remembering which is which. But recently I decided a fun way to remember it by thinking about donuts.

Your first bite into a donut, unless you're doing something wrong, is essentially along a meridian. That works out well for remembering things, because your first bite should also be accompanied by an 'mmmm', and 'mmmm' starts 'mmmmeridian'.