Fun with numbers for November 22, 2019

How lucky can asteroid miners be?

So, I was speed-rereading Orson Scott Card's First Formic War books (as one does; the actual books, not the comics, BTW), and took issue with the luck involved in noticing the first formic attack ship.

Call it the "how lucky can you get?" issue.

Basically, the miner ship El Cavador (literally "The Digger" in Castilian) on the Kuiper belt had to be incredibly lucky to see the formic ship, since it wasn't in the plane of the ecliptic, and therefore could be anywhere in the space between 30 AU (4,487,936,130 km) and 55 AU (8,227,882,905 km) distance from the Sun.

The volume of space between $r_1$ and $r_2 $ for $r_2 < r_1$ is $4/3\, \pi (r_1 - r_2)^3$, so the volume between 30 and 55 AU is 219,121,440,383,835,000,000,000,000,000 cubic kilometers.

Let's say the formic ship is as big the area of Manhattan with 1 km height, i.e. 60 km$^3$. What the hay, let's add a few other boroughs and make it 200 km$^3$. Then, it occupies a fraction $9 \times 10^{-28}$ of that space.

To put that fraction into perspective, the odds of winning each of the various lotteries in the US are around 1 in 300 million or so; the probability of the formic ship being in a specific point of the volume is slightly lower than the probability of winning three lotteries and throwing a pair of dice and getting two sixes, all together.

What if the ship was as big as the Earth, or it could be detected within a ball of the radius of the Earth? Earth volume is close to 1 trillion cubic kilometers, so the fraction is 1/219,121,440,383,835,000, or $4.56 \times 10^{-18}$; much more likely: about as likely as winning two lotteries and drawing the king of hearts from a deck of cards, simultaneously.

Let us be a little more generous with the discoverability of the formic ship. Let's say it's discoverable within a light-minute; that is, all El Cavador has to do observe a ball with 1 light-minute radius that happens to contain the formic ship. In this case, the odds are significantly better: 1 in 8,969,717. Note that one light-minute is 1/3 the distance between the Sun and Mercury, so this is a very large ball.

If we make an even more generous assumption of discoverability within one light-hour, the odds are 1 in 42. But this is a huge ball: if centered on the Sun it would go past the orbit of Jupiter, with a radius about 1 1/3 times the distance between the Sun and Jupiter. And that's still just under a 2.5% chance of detecting the ship.

Okay, it's a suspension of disbelief thing. With most space opera there's a lot of things that need to happen so that the story isn't "alien ship detected, alien weapon deployed, human population terminated, aliens occupy the planet, the end." So, the miners on El Cavador got lucky and, consequently, a series of novels exploring sociology more than science or engineering can be written.

Still, the formic wars are pretty good space opera, so one forgives these things.

Using Tribonacci numbers to measure Rstats performance on the iPad

Fibonacci numbers are defined by $F(1) = F(2)= 1$ and $F(n) = F(n-1) + F(n-2)$ for $n>2$. A variation, "Tribonacci" numbers ("tri" for three) uses $T(1) = T(2) = T(3) = 1$ and $T(n) = T(n-1) + T(n-2) + T(n-3)$ for $n>3$. These are easy enough to compute with a cycle, or for that matter, a spreadsheet:

(Yes, the sequence gets very close to an exponential. There's a literature on it and everything.)

Because of the triple recursion, these numbers are also a simple way to test the speed of a given platform. (The triple recursion forces a large number of function calls and if-then-else decisions, which strains the interpreter; obviously an optimizing compiler might transcode the recursion into a for-loop.)

For example, to test the R front end on the iPad nano-reviewed in a previous FwN, we can use this code:

Since it runs remotely on a server, it wasn't quite as fast as on my programming rig, but at least it wasn't too bad.

Note that there's a combinatorial explosion of function calls, for example, these are the function calls for $T(7)$:

There's probably a smart mathematical formula for the total number of function calls in the full recursive formulation; being an engineer, I decided to let the computer do the counting for me, with this modified code:

And the results of this code (prettified on a spreadsheet, but computed by RStudio):

For $T(30)= 20,603,361$ there are 30,905,041 function calls. This program is a good test of function call execution speed.

Charlie's Angels and Rotten Tomatoes

Since the model is parameterized, all I need to compute one of these is to enter the audience and critic numbers and percentages. Interesting how the critics and the audience are in agreement in the 2019 remake, though the movie hasn't fared too well in the theaters. (I'll watch it when it comes to Netflix, Amazon Prime, or Apple TV+, so I can't comment on the movie itself; I liked the 2000 and 2003 movies, as comedies that they were.)

Late entry: more fun with Tesla

15-40 miles of range, using TSLA's 300 Wh/mile is 4.5 kWh to 12 kWh. Say 12 hours of sunlight, so we're talking 375 to 1000 W of solar panels. For typical solar panels mounted at appropriate angles (150 W/m2), that's 2.5 to 6.7 square meters of solar panels…

Yeah, right!

No numbers: some Twitterage from last week

Smog over San Francisco, like it's 1970s Hell-A

Misrepresenting nuclear with scary images

Snarky, who, me?

Alien-human war space opera – a comprehensive theory

Learning, MOOCs, and production values

Some observations from binge-watching a Nuclear Engineering 101 course online.

Yes, the first observation is that I am a science geek. Some people binge-watch Kim Cardassian, some people binge-watch Netflix, some people binge-watch sports; I binge-watch college lectures on subjects that excite me.

(This material has no applicability to my work. Learning this material is just a hobby, like hiking, but with expensive books instead of physical activity.)

To be fair, this course isn't a MOOC; these are lectures for a live audience, recorded for students who missed class or want to go over the material again.

The following is the first lecture of the course, and to complicate things, there are several different courses from UC-Stalingrad with the same exact name, which are different years of this course, taught by different people. So kudos for the laziness of not even using a playlist for each course. At least IHTFP does that.

(It starts with a bunch of class administrivia; skip to 7:20.)


Production values in 2013, University of California, Berkeley

To be fair: for this course. There are plenty of other UC-Leningrad courses online with pretty good production values. But they're usually on subjects I already know or have no interest in.

Powerpoint projections of scans of handwritten notes; maybe even acetate transparencies. In 2013, in a STEM department of a major research university. Because teaching is, er…, an annoyance?

The professor points out that there's an error in the slide, that the half-life of $^{232}\mathrm{Th}$ is actually $1.141 \times 10^{10}$ years, something that he could have corrected before the class (by editing the slide) but decided to say it in class instead, for reasons...?

The real problem with these slides isn't that handwriting is hard to read or that use of color can clarify things; it's the clear message to the students that preparing the class is a very low priority activity for the instructor.

A second irritating problem is that the video stream is a recording of the projection system, so when something is happening in the classroom there's no visual record.

For example, there was a class experiment measuring the half-life of excited $^{137}\mathrm{Ba}$, with students measuring radioactivity of a sample of $^{137}\mathrm{Cs}$ and doing the calculations needed to get the half-life (very close to the actual number).

For the duration of the experiment (several minutes), this is all the online audience sees:

Learning = 1% lecture, 9% individual study, 90% practice.

As a former and sometimes educator, I don't believe in the power of lectures without practice, so when the instructor says something like "check at home to make sure that X," I stop the video and check the X.

For example, production of a radioactive species at a production rate $R$ and with radioactive decay with constant $\lambda$ is described by the equation at the top of the highlighted area in the slide above and the instructor presents the solution on the bottom "to be checked at home." So, I did:

Simple calculus, but makes for a better learning experience. (On a side note, using that envelope for calculations is the best value I've received from the United frequent flyer program in years.)

This, doing the work, is the defining difference between being a passive recipient of entertainment and an active participant in an educational experience.


Two tidbits from the early lectures (using materials from the web):

 Binding energy per nucleon explains why heavy atoms can be fissioned and light atoms can be fused but not the opposite (because the move is towards higher binding energy per nucleon):

  The decay chains of Uranium $^{235}\mathrm{U}$ and Thorium $^{232}\mathrm{Th}$:

(Vertical arrows are $\alpha$ decay, diagonals are $\beta$ decay.)

Unfair comparison: The Brachistochrone video

It's an unfair comparison because the level of detail is much smaller and the audience is much larger; but the production values are very high.

Or maybe not so unfair: before his shameful (for MIT) retconning out of the MIT MOOC universe, Walter Lewin had entire courses on the basics of Physics with high production values:

(I had the foresight to download all Lewin's courses well before the shameful retconning. Others have posted them to YouTube.)

Speaking of production values in education (particularly in Participant-Centered Learning), the use of physical props and audience movement brings a physicality that most instruction lacks and creates both more immersive experience and longer term retention of the material. From Lewin's lecture above: