Foucault’s Pendulum

Over on the Astronomy Stack Exchange site, (obviously I follow the “new questions” feed in my RSS reader,) someone asked if it was possible, without knowing the date, to determine one’s latitude only by observing the sun. These are the sorts of random questions that grab me by the lapels and shake me until an idea falls out.

So my first thought was: Well if you’re in the arctic or antarctic polar circles you could get a good idea… when you don’t see the sun for a few days. Also, COLD. But that feels like cheating and doesn’t give a specific value. Which left me with this vague feeling that it would take me several months of observations. I could measure the highest position of the sun over the passing days and months and figure out what season I was in…

…wait, actually, I should be able to use knowledge of the Coriolis Force—our old friend that makes water circle drains different in the northern and southern hemispheres, and is the reason that computers [people who compute] were first tasked with complex trigonometry problems when early artillery missed its targets because ballistics “appear” to curve to do this mysterious force because actually the ground rotates . . . where was I?

Coriolis Force, right. But wait! I don’t need the sun at all! All I need is a Foucault Pendulum and some trigonometry… Here I went to Wikipedia and looked it up—which saved me the I’m-afraid-to-actually-try-it hours of trying to derive it in spherical trig… anyway. A Foucault Pendulum exhibits rotation of the plane of the pendulum’s swing. Museums have these multi-story pendulums where the hanging weight knocks over little dominos as it rotates around. Cut to the chase: You only need to be able to estimate the sine function, and enough hours to measure the rotation rate of the swing-plane and you have it all; northern versus southern hemisphere and latitude.



Predicting the behaviour of a sigmoid-like process is not fitting the parameters of a logistic curve. Instead, it’s trying to estimate the strength of the dampening term – a term that might be actually invisible in the initial data.

~ Stuart Armstrong from,

Wait! Don’t flee!

It’s a great explanation of sigmoids—you know what those are, but you [probably] didn’t know they have a general name. People toss up sigmoid curves as explanations and evidence all. the. time.

Ever make that slightly squinting face? The one where you turn your head slightly to one side and look dubiously, literally askance at someone? …that face that says, “you keep using that word, but I do not think it means what you think it means.” After you read that little article about sigmoids, you’re going to make that face every time some talking-head tosses up a sigmoid as evidence for a prediction.


Second order effects

In short, stop optimizing for today or tomorrow and start playing the long game. That means being less efficient in the short term but more effective in the long term. [… I]f you play the long game you stop optimizing and start thinking ahead to the second-order consequences of your decisions.

~ Farnam Street from,

Fundamentally, we humans and our lives are not mathematically tidy.

Aside: I had a math course once—I can’t even remember the material—and the professor said, “it’s a very subtle point that mathematics should model and predict reality.” …or something to that effect. It was mind-bending; but math is part of reality so why wouldn’t reality model itself? *smoke-emits-from-my-ears* The scene, the room, the lighting, everything are burned into my brain.

Heuristics are always and in all cases true but sort of false, because they are imperfect. But the purpose of heuristics is to enable us to wrap our meager brains around the vastly complicated universe. Maths, as in compound interest, exponential growth, 1/r^2 forces, and Fourier transformations, provide models of reality. The comment about second order consequences challenges us to dig deeper into our heuristics, (which are otherwise known more generally as “models.”)

I’ve said this before, here on the blog and out loud: Have you intentionally created the models you have of the world?


The answer is, “2.”

In this situation, before committing to a three year PhD, you better make sure you spend three months trying out research in an internship. And before that, it seems a wise use of your time to allocate three days to try out research on your own. And you better spend three minutes beforehand thinking about whether you like research.

~ ‘jsevillamol’ from,

This one caught my eye because the vague heuristic of spending increasing amounts of effort at each attempt to solve a problem felt true. But I was thinking of it from the point of view of fixing some process— Like a broken software system that occasionally catches fire. Putting the fire out is trivial, but the second time I start trying to prevent that little fire. The third time I find I’m more curious as to why does it catch fire, and why didn’t my first fix make a difference. The fourth time I’m taking off the kid gloves and bringing in industrial lighting, and power tools. The fifth time I’m roping in mathematicians and textbooks and wondering if I’m trying to solve the Halting Problem.

Turns out the context of the problem doesn’t matter. The answer is, “2.” Every time you attempt to solve a problem—any sort of problem, any context, any challenge, any unknown—the most efficient application of your effort is to expend just a bit less than twice the effort of your last attempt.

Not, “it feels like twice would be good,” but rather: Doubling your efforts each time is literally the best course of action.

…and now that I’ve written this. My brain dredges up the Exponential Backoff algorithm. That’s been packed in the back of my brain for 30 years. I’ve always known that was the chosen solution to a very hard problem. (“Hard,” as in proven to be impossible to solve generally, so one needs a heuristic and some hope.) They didn’t just pick that algorithm; Turns out it’s the actual best solution.


The Honeybee Conjecture

More than 2,000 years ago, Marcus Terentius Varro, a roman citizen, proposed an answer, which ever since has been called “The Honeybee Conjecture.” He thought that if we better understood, there would be an elegant reason for what we see. “The Honeybee Conjecture” is an example of mathematics unlocking a mystery of nature.

~ From

Every once in a while, you will have the chance to be alive when a multi-thousand-year old mystery is solved. Humans are awesome. Mathematics for the win. *drops mic*


Most people are not yet born

[…] recognize that at least in terms of sheer numbers, the current population is easily outweighed by all those who will come after us. In a calculation made by writer Richard Fisher, around 100 billion people have lived and died in the past 50,000 years. But they, together with the 7.7 billion people currently alive, are far outweighed by the estimated 6.75 trillion people who will be born over the next 50,000 years, if this century’s birth rate is maintained (see graphic below). Even in just the next millennium, more than 135 billion people are likely to be born. 

~ Roman Krznaric from,

50,000 years is somewhat of course. But it’s a good estimate of the span so far of recognizably-like-current-us human history. It’s obvious that today, most people are already dead. It’s those trillion yet to come that warp the brain and create perspective.

This article from The Long Now Foundation has 6 good examples of explicit ways to think long-term, rather than short-term.


Est provocationem

Today, I’m drawn to considering refinement of an idea I’ve mentioned a few times.

It’s clear to me that it’s impossible to be happy if my mind is unable to focus. Years ago, I regained my ability to intentionally focus, by disempowering the world—disabling as many as possible of the pathways for everything and everyone to actively grab my attention. Then I set about railing against everyone who has not yet regained their own ability to focus, (or perhaps, has never learned to focus.)

Internally, I often use an idea which I believe I stole from mathematical analysis. When facing some question, the idea is to find the largest contributor, and get a handle on that first; that’s the first-order item. Then find the next largest contributor, and get that second-order item sorted. And so on.

A few decades ago, the largest impediment to my being able to intentionally focus was external distraction. Having now sorted that first-order item, I can turn to the second-order item: Everyone’s inability to focus is polluting my attention. I remain easily distracted by others’ inability to focus.

Est provocationem!


Heating with math

Something a little different today: I’ve been considering switching to heating with gas and I recently ran some numbers.

tl;dr: I will be continuing to heat with solid fuel.

Preamble: We have already deeply insulated our attic, upgraded insulation in the walls which were opened during some remodeling, and replaced all windows and doors with modern versions. (Our house was originally built in 1954.) This is the obvious first place to begin improving heating your home.

Electricity: My electricity costs $0.0758 per kWh. I can basically turn on my electric baseboard heaters and this is what I’d pay (per kWh) to heat our house.

Methane: This is the proper way to heat a home in northern climates. Unfortunately, “street gas” is not present in my neighborhood. One block over, yes, here, no. They would install it for me… if I’m willing to pay the entire cost to rip up the street and put in the gas main.

Propane: Chemistry geeks know that propane has about 12% less energy per molecule compared to methane. But generally speaking, appliances (my gas cooking stove, a gas heating appliance I would need to buy/install) can be adjusted to burn either fuel. Anyway. I already have a small propane tank that serves my cooking stove, so I would “just” need a larger tank — possibly MUCH larger, possibly so large that safety ordinances would require me to put it underground. Anyway. My propane costs me $5.999/gal — if you know about petroleum, this is an incomprehensibly high number. Meanwhile, 1 gal propane = 27kWhr of energy. And a gas heater (I’m imagining replacing my wood stove with an appliance that sits in the same space) is effectively 100% efficient at turning that gas into heat. So simple math shows that propane would cost me $0.222/KWh — about THREE times the cost of electricity.

Firewood: This is MUCH harder to compute. First off, I have to estimate how much energy is available in the wood I’m burning; that’s affected by species of wood, and how it’s seasoned and stored (because the MORE water in the wood, the more heat is “lost” to vaporize that water and send it away up the chimney.) Some factors to consider: Where I live, there are several readily available “fuel” species of trees that are sustainably available. I’ve found a reputable supplier who is not hauling it long distances and provides me the right sizes etc for what I want. I also have the absolute best imaginable way of storing the wood in “cribs” that expose it to air drying while having it under cover.

So I’m guessing 20 million BTU per cord. (A cord is a stacked, pile 4 feet tall, with a foot print of 4×8 feet. Technically, it’s a pile of 4-foot LONG logs, 4 feet high and 8 feet wide on the ground. A true wood heating system is a separate unit outside that is meant to take 4 foot long logs. I purchase ~16″ pieces split, which still makes the 4×8 foot print computable. I digress.) Good fuel species can be up to 30MBTU/cord. So I’m being conservative with 20.

20M BTU is 5,861 KWh. I pay $300 per cord (fellow Pennsylvanians just twitched because that is pretty expensive — 225 or 250 is typical — but this is excellent wood species, all cut and split to the correct sizes for stove fuel, delivered early in the season, and dumped exactly where I want it. As usual, I digress. So math happens leading to $0.0512 / KWh. Even if I figure-in that the wood stove is only 80% efficient (we have a great stove made in Scandinavia which really does exceed 80% efficiency when operated correctly), that only bumps the cost up to $0.0639 / KWh.

Update in 2019: My electricity costs $0.07039 per KWh. (That’s down about 1/2 cent.) I’ve a new firewood supplier, with the price down to $225 per cord. That’s $0.038 / KWh, and still only $0.048 / KWh at 80% stove efficiency.

And finally some references…