A Purely Geometric Proof that the Square Root of Two is Irrational—With a Couple of Bonus Side-Trips

Lots of proofs are based on simple ideas, but get bogged down in notation or exposition that doesn’t bring out the salient points, as if the soloists and all the members of the choir were singing at equal volume.

Sometimes some warm-ups can help people understand what is essential in a proof and what is extra. They can help the simplicity of an idea shine through.

So the first warm-up is a purely geometric proof that the golden ratio is irrational. A number of years ago, I saw such a proof… but the diagram that went with it was laid out on a single line, and it got bogged down in a bunch of notation, and I kind of got it but it certainly didn’t excite me.

Then one day I was looking at my business card and I realized that the proof was right there.  When I was designing my business card, I tried to figure out a good logo, and I eventually settled on a golden rectangle and golden spiral:

Continue reading “A Purely Geometric Proof that the Square Root of Two is Irrational—With a Couple of Bonus Side-Trips”

Making Really Good Bread is Really Easy!

OK,  I have been meaning to do this post for almost a year now. In April 2016, I organized a workshop, and had dinner with four out-of-town colleagues afterwards. They asked if I was happy with how it went; I said that I was very happy with the workshop, but I mentioned that I was also happy because the previous week I had finally managed to make a loaf of bread that I was truly pleased with. All four were quite interested, and I thought I should put together a post about it.

I decided to do a demonstration using only equipment that most people have in their kitchens. So  I made some videos, but I got distracted and then let it sit until now. Continue reading “Making Really Good Bread is Really Easy!”

Mac Browser Wars… and the winner is: Safari! [oops, not really]

Update not all that long after I wrote this: I discovered that if I don’t keep too many tabs open while I’m also running a Windows virtual machine on VMWare, Chrome does not get so slow, so I’m back to using it, which is nice because I like it a lot more than Safari. So please just ignore this post. Continue reading “Mac Browser Wars… and the winner is: Safari! [oops, not really]”

A (long) Comment on Scott Alexander’s Review of “Unlearn Your Pain”

My son Ben texted me the other day that his favorite blogger, Scott Alexander, had published a post reviewing Howard Shubiner’s book Unlearn Your Pain.  Despite being a psychiatry resident, Scott manages to be a prolific writer in areas of interest which appear to be TNTC.  Ben knows him through Effective Altruism, and sent me the review because it mentions ISTDP (Intensive Short-Term Dynamic Psychotherapy), the kind of therapy that I do. I had read some of Scott’s other posts with interest, always wishing I had more time to keep up.

I enjoyed the review, and sat down to email a response to Scott, since my thoughts seemed a little long to post as a comment. Scott asked me to post them as a comment anyway, at which point I discovered that his blogging platform agreed with my initial gut feeling.  So I am posting them here, and will post a link to this post in the comment section of his review.

Continue reading “A (long) Comment on Scott Alexander’s Review of “Unlearn Your Pain””

OK, how big is a tablespoon, REALLY?

One of the handiest cooking factoids in our crazy US measurement system is that there are 4 tablespoons in a quarter cup.  If you are doubling a recipe for a sauce that calls for 2 tablespoons of flour, don’t measure 4 tablespoons, just measure 1/4 cup and you’re all set.

So I was a little surprised when I was looking at a source I consider reliable (Samuel Fromartz’s In Search of the Perfect Loaf, p. 87) to see “Mix 3 tablespoons (30 grams) lukewarm water…”  Now, “everyone knows” that a tablespoon is 15 ml, and 1 ml of water weighs a gram, so 3 tablespoons of water should be 45 grams.  OK, precision is overrated in cooking, but it is more important in baking, and this is a 50% difference (45 being 50% more than 30), and that is actually significant.

1 tablespoon “=” 15 ml

Continue reading “OK, how big is a tablespoon, REALLY?”

Data Ain’t What They Used to Be. Or Is They?

“Hmm… that makes sense,” said Ben.

“Uh, you sound surprised,” was my response.

“Well, usually when people start out by saying ‘Here’s my take on that,’ it tends to not be that helpful.” Well Ben’s a smart guy, and if he hints that it might be helpful, that was enough to get me to write this post, which has been rattling around in my head probably about since blogs were invented.

My “take” had been prompted by another iteration of a typically tedious discussion that you may also have been involved in periodically: whether “data” is singular or plural. For those of you too young to know, or too smart to care, “data” is the Latin plural of “datum,” meaning “a piece of information.” So when a hapless person would say “There isn’t enough data,” grammar snoots would correct them and try to get them to say “There aren’t enough data.”  It’s been a losing battle.  (David Foster Wallace fans will know that he actually called these people SNOOTS, all caps; an executive summary is here.) Continue reading “Data Ain’t What They Used to Be. Or Is They?”

“The Controversial Therapy That Deliberately Enrages Patients”

A sharp-eyed friend (with the help of google) alerted me to a piece about ISTDP on a site I had not previously known, vice.com, with the above title.  You can read it here.

I made a few attempts to post a comment, but they didn’t stick, maybe because I included my website URL (www.natkuhn.com).  Maury Yoszef did manage to get an intelligent comment up there, thanks Maury!

Continue reading ““The Controversial Therapy That Deliberately Enrages Patients””

Mavericks Upgrade: Go For It

Added later: this is of purely historical interest, if that.  I added a category “outdated,” but maybe “obsolete” would be more accurate…

I tend to be a late adopter of operating systems.  I went from Windows 95 to XP to 7, skipping Vista, Me, and a number of other dogs whose names I can’t even remember.  I would still be running XP except that when my son switched from Linux to Windows 7 I sat up and took notice. Continue reading “Mavericks Upgrade: Go For It”

The Land of Trac

In the third grade, our teacher Mrs. Smithy read us The Lion, the Witch and the Wardrobe, the first of C.S. Lewis’s Chronicles of Narnia, in which a group of kids find a secret door in the back of a wardrobe and enter a fantastical, undreamed-of world.

In the fifth grade, in 1968, I became part of a group of kids that had found such a door.  The group was called the RESISTORS, and the door led to the world of computers and what is now called “information technology.”

The group met in the house and barn of Claude Kagan, an engineer at the Western Electric Research Center in Pennington, NJ, near Princeton, where I grew up.  Claude was a complicated guy, by turns fun-loving, cantankerous, generous, childish, and more—but unswerving in his commitment to value of letting young people learn things and above all do things.  The principles of the RESISTORS were “Hands On” and “Each One Teach One,” and those principles have stood me in good stead for the last 45 years.  (As I side note, last night I read a piece by Atul Gawande in the New Yorker in which he wrote that an essential realization in the dissemination of the medical miracle of oral rehydration was that when teachers fanned out to villages, the teaching was much more effective if the villagers made the solution under the teacher’s instruction than if the teacher “showed them how.”  This was one of the first things we learned as RESISTORS: if you are teaching people things, THEY should sit at the Teletype [a primitive 100 bps terminal, which no self-respecting Bangladeshi villager would tolerate today] and YOU should sit next to them, talking them through it.)

We used a number of computers and computer languages, but the computational beating heart of the group was Claude’s PDP-8 computer at Western Electric, which we would dial into from a Teletype in his house.  It ran Trac, a computer language designed by Calvin Mooers, an independent thinker based in Cambridge, Mass.  The PDP-8 had 4K of RAM.  Yes, 4K, i.e. one one-millionth of the amount of RAM in the two-pound MacBook I’m typing on right now.  RAM was insanely expensive because it was made of little magnetic “cores” which were hand-strung, reportedly by armies of Filipinos.  OK, actually it had 4K of 12-bit words, so technically you could say it had 6K bytes.  A “Trac processor” (interpreter) could fit on such a machine, with some room left over for user-written scripts (programs).  There is really no computer today so minuscule that Trac makes sense as a language, and even fewer people would ever have heard of Trac today if Ted Nelson hadn’t happened upon the RESISTORS and mentioned Trac in Computer Lib.

Continue reading “The Land of Trac”