Foo Fighters

 Interesting post from Robin Hanson about how the universe may be full of incompetent aliens who may have randomly been sending UFOs to us for centuries but never got anything done. I agree with most of Robin's points about how our power structures are enormously wasteful, sort of like the ancient Chinese empires, and we shouldn't really expect our civilization, or the aliens, to do any better on the average.

He uses the "Foo Fighters" seen by WWII aviators as an example of something that may have been alien craft:

... pilots flying over Germany by night reported seeing fast-moving round glowing objects following their aircraft. The objects were variously described as fiery, and glowing red, white, or orange. Some pilots described them as resembling Christmas tree lights and reported that they seemed to toy with the aircraft, making wild turns before simply vanishing. Pilots and aircrew reported that the objects flew formation with their aircraft and behaved as if under intelligent control, but never displayed hostile behavior. However, they could not be outmaneuvered or shot down. 

The main problem with this as an alien spacecraft, though, is that I have seen one of these myself.

And I know what it is.

It was a couple of decades ago. We were on a cruise ship, sailing up the Canadian/Alaskan Inner Passage, about a night out of Skagway. We had splurged for the trip, so we had a top-deck cabin with a balcony (and a butler). Just about bedtime, on the night in question, I stepped out onto the balcony for a breath of fresh air before turning in.

It was slightly overcast and very still. I could see shore lights in the distance, and the ship made steady progress through a sea that was almost like glass. There was no moon.

I looked forward. There was a glowing disc hovering maybe 200 feet over the bow. It dashed out in front, almost like a dog running in front of a car. It came back. It danced around. It made circles around the ship. It stopped overhead, never quite standing still, but I got a good look at it. Featureless, round, glowing softly.

It was completely silent.

I watched it for maybe 10 or 15 minutes. Then it disappeared. I went back in, mentioned to my wife that I had just seen the most amazing thing, and went to bed.

What could it have been? The only thing I have seen even vaguely like it was the spot from a searchlight dancing on the bottom of a cloud layer. But the ship didn't have the right kind of searchlight, it couldn't have done the fine dancing, it would have changed shape as it tilted low to go far out, and the beam would have raked the deck as the spot dashed from stem to stern and I would have seen it. Nothing like that happened.

But it was a clue. Something like the searchlight had happened, but not from below. The overcast layer was thin, of a kind I have seen many times flying. We were in northern waters, and as it turned out, some people in Skagway had seen aurora that night.

We were on an enormous iron object, moving, and that does interesting things to the Earth's magnetic field. In particular, it can focus a tendril out of an otherwise diffuse electron flow, as seen here:

So the ship had gone through what would otherwise been a nearly invisible aurora borealis, and concentrated it into visibility as it pierced the thin cloud layer. If you've played with a plasma ball like this, you will be very familiar with the character of its motion and why it seemed to be interested in the ship.

And it sounds a lot like the descriptions of the "Foo Fighters."

A Complex Treasure

 In the wake of this recent work, there has been a resumption of the sporadic debate over which imaginary numbers are real (pun intended) or not. Not so much by mathematicians, who tend to believe that they discover, rather than inventing, the structures of thought they employ, but physicists, who tend to think of the math they use as a different order of being than the actual physical world they describe.

From a pragmatic point of view it doesn't matter. If you have various mental tools, use the one that works best. The best meta-rule is Ockham's Razor.

In that spirit, I thought I would revisit a cute little puzzle that is often used to show how complex numbers are a bit more simple, nifty, or appropriate than just plain pairs of coordinates for solving a problem. The problem has nothing to do with the deeper properties of the quantum field; it's about a treasure map. 

It goes like this:

You have come into possession of a chart from a pirate long dead. It shows the location of a small island in the Spanish Main, and on the back are directions to find the treasure. "Pace from the cairn to the ash tree, turn right and pace the same, driving a stake. Again from the cairn to the bay tree, turning left and pacing the same, and a stake. Midway between your stakes is the treasure to be found."

You get to the island and find the ash and bay trees. Unfortunately in the meantime the island has been visited by a gang of kleptotaphophiles, who stole all the stones of the cairn, leaving it unmarked. How can you find the treasure?

Or, since this is really a math puzzle, how can you use complex numbers to prove your solution is correct?

The trick with a math problem is often to gain an intuition as to what the solution is, and then use the tools you have to show it is right. 

In my experience you are much more likely to be taught how to use the tools, and less likely to be taught how to gain an intuition. With that in mind, let's look at the map.

On our island it just so happens that the bay and ash trees are on an exact east-west line, and are exactly two furlongs apart. (A mathematician would phrase that, "Without loss of generality we may assume ..."). 

Now here's the essence of gaining an intuition. Take boundary cases, cases where something goes to 0 or 1, anything to simplify the problem without changing it overall. In the case of the treasure map, for example, start with cases where you can easily see the answer without doing any numerical geometry.

Let's call the point midway between the trees "Zero" and see what would happen if it were the cairn.

Pacing and staking, we get a simple diagram that shows the treasure would be exactly one furlong due south of Zero. Okay, what's another way to simplify?

Just pick the cairn as being one of the trees. Then the pacings for that tree are of zero length, and you drive the stake right there. When you pace the other tree you get:

Whaddya know, one furlong due south of Zero. And obviously it works the same starting from the ash tree.

What else can we eliminate? How about the string between the stakes? If they are in the same place, the centerpoint, and thus the treasure, will be right there. Put the cairn at one furlong due north of Zero:

Yep, it's at the same spot, one furlong south. And for a final flourish, what if we put the cairn right on the treasure?

Well, by now you will have gained the intuition that wherever you put the cairn, the treasure will be in the same place. And you even know where the place is. 

You now have your conjecture, and can prove it fairly straightforwardly using complex numbers.

But while you are doing that, I will have jumped to a conclusion, dug up the treasure, and escaped.

Bayesian Death Match

 How likely are you to die of, say, covid versus, say, heart attack next year? If you look at official figures you are met with a variety of bewildering metrics. Furthermore it can be confusing to interpret what they mean in the first place. Here is a website that uses CDC figures to tell you what your odds are of dying of various causes, looking kind of like this:

Cause of Death	Odds of Dying
Heart disease	1 in 6
Cancer	1 in 7
Suicide	1 in 88
Fall	1 in 106

Of course, this doesn't mean you have a 16% chance of dying of a heart attack next year; it means when all is said and done, given that you died, the chance it was by heart attack was 16%. This is beginning to sound pretty Bayesian, so let's see if we can turn it into something more intuitive.

Manipulating probabilities by Bayes' Rule is powerful but often less than straightforward. Luckily there is a way to do it that is quick and easy. The trick is to think in terms of the logarithm of the odds ratio. You are used to thinking of probabilities as a number between 0 and 1; just think of them this way instead:

• -30 -- a billion to one against; your chance of winning a rigged lottery
• -20 -- a million to one against; your chance of being struck by lightning in a year
• -10 -- a thousand to one against;  your chance of flipping ten heads in a row
• 0 -- even odds
• 10 -- a thousand to one for; the chance you won't flip ten heads in a row 

and so forth. What we have done is taken the base-2 log of the odds ratio. Why would we do this?

The reason for using this log-odds form is that we can apply Bayes' Rule simply by adding them. Here's an example: the logodds for some given random average American to die (of any cause) next year is about -6 (roughly 1%). The logodds of dying in a car accident given that you died is -6.7. So the logodds of dying in a car crash next year is -12.7.

But we can do better than that. You aren't a random American: you can improve your estimate of the prior by knowing, for example, your age. CDC says:

This translates to 

Age	Logodds
20	-10.5
30	-9.6
40	-9
50	-8
60	-6.8
70	-5.8
80	-4.5
90	-2.9

So rather than start with just -6, you'd start with the number associated with your age. Or you could have a table by sex, or whatever other division you thought made a difference.

Then add the number for the thing you're worried about. Dying from a fall is -6.7. So if you're 20, your total risk from falls is -17.2; you'll outlive Methuselah. But if you're 90, it's -9.6, well within the range of things to worry about.

The numbers for heart disease, cancer, and covid all stand pretty close to -2.6. Do the math.