Everybody thinks Manhattan’s avenues run north-south and the streets east-west, but it isn’t true. The whole grid is actually titled 29 degrees to the northeast. 29 degrees! This is why the Empire State Building looks so nice when seen from the “south” on a sunny afternoon—you’re actually viewing it from the southwest. In fact Broadway, which we think of as running diagonally, is the road that approximates north-south; and even Broadway actually tilts slightly to the east.
It fascinates me how few people are aware of this situation. Most NYU mathematics professors don’t know about it (I’ve asked them), nor does the man on the street (just one of the three I sampled). Maps depict avenues vertically, often without an arrow to indicate north. The NYU maps posted all around Greenwich Village compound the deception with an outright lie, including an arrow for North that points straight up.
Does it matter? What is truth? My feeling is that we get through life with all kinds of useful approximations, and until better data comes along, I’m going to take the view that their alignment with the truth, on average, is around 29 degrees.
[21 September 2013]
At the bike shop this morning, the technician was talking to the guy ahead of me about his bike. “I see you always ride in the top gears”, he said. “You might want to try to use your gears more evenly.”
Later I asked him about this and he said yes, it’s obvious when people are favoring a particular gear or two. “You can tell by the teeth,” he said. “They get worn down.”
I remarked that this is just how archaeologists figure out which prehistoric cultures had rough, grainy diets, requiring a lot of chewing. You can tell by the teeth.
This being Oxford, my bike technician understood and appreciated the analogy.
[15 March 2013]
I’ve long been puzzled by this. Our interest in love — sex— gossip — who pairs with whom — is inexhaustible. This is what animates almost any novel or movie or song. Yet food is equally indispensible to our fitness. Where are the movies and songs about food?
My latest theory puts it down to elasticity of demand. Food is essential, but once you have enough, you pretty much have enough. From starvation to obesity, the difference in intake is just a factor of ten, so we’ve evolved a kind of on-off interest in the subject. Love is different. Depending on how you play the game, you may have zero or two or ten or (if you are a man) 100 children. With such great elasticity in our potential, it is evolution’s business to make sure we pay attention to the subject. Our interest never turns off.
[26 March 2013]
Yogi Berra is famous for lines like “It ain’t over till it’s over,” “A nickel ain’t worth a dime anymore,” “When you come to a fork in the road, take it.” The trick is that the statements are literally false or empty or meaningless, yet suggest something true.
The challenge we more commonly face is statements of the reverse structure: literally true, yet suggesting something false. Here are two examples that blighted for a century their respective fields of fluid mechanics and numerical analysis.
(1) To test for instability of a 3D channel flow, it’s enough to look at 2D flow
perturbations. (Squire 1933)
(2) Given any system of interpolation points in an interval, there’s a continuous function f for which the polynomial interpolants diverge as n→∞. (Faber 1914)
Formulated precisely, (1) and (2) are as true as true can be: they are mathematical theorems. And yet, how misleading! What makes high-speed flows unstable in practice is not the theorist’s instabilities, but different 3D effects. Interpolants in Chebyshev points converge beautifully, so long as f is ever so slightly smooth.
You will find the wisdom of any field supported by confident assertions. It’s one challenge to spot the false ones, another to spot those that are true but misleading.
[8 May 2013]
The Economist has an article this week about how mouse brains work. If you apply an electric shock to the feet of a mouse when it’s in a cage, then for a long time after, it will freeze in fear if you put it in that cage again. That traumatic event becomes a lifelong memory for the mouse.
This effect casts a light on our perceptions of the meaning of life. Certain places, memories, and people have outsized importance for us. As long as I live, 23 Barberry Road, Lexington, Mass. will seem the truest home; memories of Emma and Jacob as toddlers holding out their arms to be picked up will seem the deepest family experiences. What powerful emotions these special things of ours evoke! I know in my head that my home and children are no more important to the universe than another man’s, but my head is nowhere near my heart. The mouse freezing in fear in the cage that hurt him is the same effect, only simpler. The timorous beasties with their tiny lives and worries lay bare the mechanistic nature of it all.
[29 July 2013]
As we drove away from the car park in Agrigento the day before yesterday, something seemed odd. I realized, I’m not wearing my glasses! But this never happens! For twenty years I’ve been unable to drive without them. What was going on?
In the bright Sicilian sunshine, my pupils had contracted more than they ever do in England.
[5 April 2013]
For many years, since 1970 in fact, I have been writing notes on social, mathematical, and other subjects, which I store on index cards. A collection of several hundred of these notes was published two years ago as Trefethen’s Index Cards: Forty Years of Notes about People, Words and Mathematics (World Scientific, 2011).
My son Jacob has persuaded me to experiment with posting new cards as I write them in a blog. So here goes. I’ll begin with a few notes from recent months. On average over the years, I have written these at the rate of one or two a month.
As my day job, I am Professor of Numerical Analysis at Oxford University (people.maths.ox.ac.uk/trefethen).
– Nick Trefethen, Oxford, 29 September 2013