Blog-o! Notes from

Sun, 29 Jan 2012

The other day I was reading a tweet from Neil deGrasse Tyson, and it got me thinking… If you were going to cut a circle into three equal pieces using only two cuts, what would the angle to cut them be? (Θ in the diagram over on the right side there.) So I asked on Twitter, and eventually got a reply. This is my attempt to reconstruct the reasoning behind the answer.

The easiest way to figure this out, I believe, is to cut the circle in half, and then figure out what line divides the semicircle into a 2/3rds to 1/3rd ratio. So let’s do that, and label the points A, B, C, and O (for the origin), as shown on the left. We’ll also label the angle AOB as ɣ, because we'll be using it a little more later.

We know from this page that the area of a segment is r² × (ɣ - sin ɣ) / 2, and we want the segment formed by the line AB to contain 1/3rd of the total circle (or 1/3 × π × r²). Putting those together gives us the equation:
r² × (ɣ - sin ɣ) / 2 = π × r² / 3
multiplying both sides by 2, and diving both sides by r², we get
(ɣ - sin ɣ) = 2 × π / 3
Now all we have to do is simply solve for ɣ to get:
ɣ = 2.60533

That means that the angle BOC is π - ɣ, or 0.53626 radians (or 30.7º). Now, the inscribed angle is half the central angle, so in theory, to get the angle BAD, I should divide that by two, but then I’ll just need to multiply it by two again to account for the other half of the circle, so let’s skip all that, and just call it 30.7º.

[Posted at 22:29 by Blake Winton] link
Sun, 01 Jan 2012
Year in Review: 2011

Well, wasn't 2011 a piece of work? Lots of things happened, some great and some lousy. Let's start at the beginning.

In January we were taking Thomas the cat to the vet for excessive skinniness, strange loss of hair and general geriatric decrepitude. It turned out he had fleas, some kind of allergy which was giving him red spots all over, and lots of weird growths here and there: under his tongue, on his head, and probably in his bowel too. We treated him for fleas and gave him antibiotics for a while to clear up his skin problems, and put him on a permanent course of prednisone to treat his various tumours. (He's eighteen years old, so the growths are just because of old age.) We prepared ourselves for his imminent death; however, a full year later he's not dead yet, and indeed seems healthier than ever.

At the end of January Blake took off for Hawaii for work. I wasn't invited, which was particularly galling given that, you know, Hawaii in January. Not to mention, we hadn't been on a proper not-visiting-mum trip since before Cordelia was born. So come March Break I decided we should go to New York for the week. That holiday was more successful and fun than I expected it to be—the girls are at the perfect age to travel with. (Which is a bit ironic, now that I think about it, considering we won't have the money to go anywhere in the forseeable future.)

In May Greg Wilson and I published the paperback edition of The Architecture of Open Source Applications (Volume 1). In the process of publishing the book I learned an immense amount about TeX, typesetting, book production and publishing on, as well as getting a lead on a copyediting job. I enjoyed the whole process so much that I made it into a
business. So far I've had a handful of jobs and made some good connections, and I'm looking forward to growing the business this year.

February through June we were in reno upheaval. Our 80-year-old plumbing started leaking and we somehow decided, with impeccable North Toronto logic, that the only possible solution was to put a powder room in the basement and completely redo the upstairs bathroom. I've been meaning to post about that... It took a long time because our contractor usually runs much bigger jobs, and her trades were sneaking our bathroom in between other jobs. But when it was all finally done it was very satisfying and lovely.

In the middle of June one of the kindergarten students at the girls' school was hit by a car and killed. It shook up the staff and a lot of the parents pretty badly, and it's been on my mind a lot ever since. The little girl who died was the same age as Cordelia, and she was out with her mother when she was killed—actually her mum was hit by the car too. Imagine the fodder for rumination and imagination and nightmares that provided... It's a bit ridiculous that it would take something like this, but it made me understand that the continuing existence of my children is a gift. Anyway, that's a whole post on its own, really.

At the end of June I took a plane to Tokyo (by myself) to visit Dave. I'd never been to Japan, and it was a grand adventure.

By the time I got back from Japan the girls' summer vacation was well underway. The girls were at camp for a couple of weeks, then we went to Saskatchewan for a couple of weeks, then shortly after we came back the girls went to the cottage for a long weekend with Baba and Zaida. I felt like the real mooching-around-Toronto part of summer didn't start until the middle of August, but then we did manage to do our usual round of the Toronto Islands, High Park, Harbourfront, and a few trips to the park. I love summer.

I have to say I don't think anything terrifically interesting has happened since September. Greg and I have been working on the second volume of the software architecture book. (This time I'm actually going to copyedit the book, which will both be fun and educational, and improve the book.) Cordelia likes Grade One, Delphine likes Grade Three, Cordelia likes gymnastics and Delphine likes ballet. We're all pretty content to carry on into the new year as we've been carrying on.

[Posted at 22:08 by Amy Brown] link