Some Pi-day silliness

Yes, I realize this is a week late. Ilyen az élet - I'm trying to finish up my PhD here and punctuality seems to be a thing that happens to other people. Besides, the proper day to celebrate is going to be June 28^th anyway.

One thing I saw getting passed around for Pi-day last year (did I mention punctuality was not my thing right now?) was this image below. While it is kind of cute, it's also wrong. But it's wrong in some rather interesting ways, so I thought it might be worth expanding on why.

From all over the interwebs

While the image is correct that the decimal representation of pi is infinite and non-repeating, this does not imply* that all possible number combinations are contained within it. In fact, it's pretty trivial to come up with a number that doesn't to that: take the number 0.12112111211112111112.... for example. Or take pi and cross out every digit that's a 7. Either way, we know you're never going to see the number “79” somewhere in the pattern. So that's out.

But does pi contain all possible number combinations? Well, that's a profound “maaayyyybbbeeee?”, and here's where the story gets interesting.

A normal** number is a number where all the digits and combinations thereof appear with equal frequency if you look long enough. You should be able to find a Feynman point somewhere in there, as well as your pixel-perfect image of your death, the script to Monty Python's Silly Walks sketch, and a number that gives you a gzipped version of Corel WordPerfect 8 along with other illegal numbers. The great answers to everything is in there, as well as a far greater multiplicity of wrong answers. And wrong questions.

Though, of course, we shouldn't just talk about base-10. A number may be normal in one base b (being called b-normal) and not in another. So we have an extra definition; if a number is normal in all bases, then it is called absolutely normal.

So, is pi 10-normal? 16-normal? Is it absolutely normal? Efforts to calculate the firstseveral trillion digits of pi indicate that its digits are uniformly distributed, which is a good sign. But that's not enough to really prove anything. The problem with infinity is that's there's a lot of it, and so it's difficult to really say that you won't start to run out of certain digit sequences. Especially once you get to the sorts of digits positions where regular numbers are insufficient to write out how far along you are. We do not, in fact, know if pi is normal, or if it's absolutely normal. Perhaps we won't find our Ministry of Silly Walks script in there, nor some jpeg file of a Doge meme.

Normal numbers seem like they'd be magically rare beasts but, as a matter of fact, almost all real numbers are absolutely normal. This makes sense if you stop and think about it—how would you go about picking a truly random real number? Well, you might think about its decimal representation (and let's just think about numbers between 0 and 1 for now). First, randomly pick its first digits as one of 0 to 9. Then do the same for the second digit, then the third, and so on and so on. But this same way of trying to construct a real number would mean that all digits and digit sequences ARE uniformly distributed throughout its decimal representation, and would thus at least be 10-normal.

But just because almost all numbers are absolutely normal this does not mean that it's easy to prove that a specific number is. Outside of some numbers that seem to “cheat” (like Champernowe's constant 0.12345678910111213...) and a few other examples, we just don't know.

And this is, really, one of my favorite parts of math. There are so many proofs that say “I know this thing exists and is overwhelmingly common. But damned if I can give you more than a handful of nontrivial examples.” We can see this with ultrafilters, with Hamel bases, or with trying to see if some numbers are transcendental. I think it's so wonderfully curious that out of the uncountable infinity of numbers we have, in a sense we can only grasp at thin fraction of them. Heck, we can't even DESCRIBEmost numbers in a non-infinite way.

I think pi*** would be MORE special if they weren't normal in some way. Normal numbers are, truly, normal****. Contra Carl Sagan's novel Contact, it would be a far greater sight to find something that wasn't in pi than something that is. 

Perhaps something like this.

*Mathematically, it's necessary, but not sufficient.

**”Normal” is, IMHO, one of the most overused adjectives in math, followed by “regular”. Come on mathematicians, come up with some new phrases!

***Or tau, I'm not conceding this fight.

****Still no excuse for overusing the adjective.

200 beers!

For much of my time at Caltech, a bunch of friends and I had a Friday-afternoon drinking group*. I have many fond memories of "Beerhour", from getting a bottle-cap laurel wreath as I was elected Beer-lemagne to when we brought out the Moveable Feast to ride around campus and pester the attendants of TEDxCaltech.

Heady times at Beerhour.

And, of course, trying a wide variety of beers.

Sadly, the only recording I had ever done of my beer-drinking habits was a double-blind taste test with some of my department peers on a variety of cheap beers. While this was, in fact, a properly conducted experiment and I can now without shame embrace inner hipster when drinking Pabst Blue Ribbon, for the most part I could hardly remember what I'd tried and liked.

Around August, I realized this simply would not do, and I started keeping a small notebook with me to record all the beers I tried. I subsequently purchased a Raspberry Pi and had some rather grand ideas of turning it into a beer database until I realized it was far easier just to stick things into a spreadsheet on my computer and go from there.**

All beers were rated from 0 to 10, with the hope that these rather subjective ratings would generally average out to something useful. Notes were kept on the taste and mouthfeel, and other pertinent information. But collecting reviews was just the start; how does this let me know what other beers I might like? What sorts of beer-wisdom really holds true for me? And so I started to play around and see what sort of greater knowledge I could glean.

For the most part, I set up a system that just gives me averages, standard deviations, and the minimums and maximums for various conditions. I even took some t-tests to compare different classes of beer, which seems entirely appropriate even though I'm hardly getting a random sample: Outside of Switzerland (and whatever I picked up in the US) for this data I really didn't have access to the FULL range of beers, just whatever various companies think is worth importing across borders (viz, none of the really bad supermarket beer, nor anything really small-batch). Mostly I just went to Drinks of the World and would pick up things I hadn't rated before, hoping this would give a roughly representative sample. And unlike the cheap-beer taste test, this is also not remotely double-blind. Nor single-blind. Nor blind drunk. Nor methanol blind (thank goodness).

Also, do p-values really mean all that much, anyway?

Still, I tried to have the beers in a fairly consistent fashion; drinking them when they're fairly cool (not cold!) and straight from the bottle or can (because I like to make my German friends cringe). I usually drank with little taste in my mouth (but not always) and tried to keep myself sober. I did record things like temperature and drinking situation with the eventual hope that it would let me "adjust" the ratings at some point, though that's a project for later.

So, here's the data so far:

Cannae drive - I should update this thing or something

I set up this blog to practice writing, and look what happens. Oh well. In the meantime I post things on Quora and Facebook that would be perfect candidates for reposting here. Maybe I should cross-post? Let's do this thing.

A few days ago, some news of some new space thruster (the Cannae drive) that seemed to drive itself without shooting out some sort of reaction mass (apparently violating the conservation of momentum). A conference proceeding was posted and at the very end, it details being able to make some quite impressive trips around the solar system.

So, it was asked (on Quora): Does this new device disprove conservation of momentum?

Short guess: The drive doesn't work, and this is merely an experimental error. Conservation of momentum is not in question.

Pithy quote: "Extraordinary claims require extraordinary evidence." - Carl Sagan.

Another pithy quote: "Science is hard" - The most trusted name in news.

Longer answer:
Conservation of momentum is a fairly robust idea, and it would take a lot of evidence to show that it's been broken. The basic idea is a consequence of Noether's theorem--any symmetry of the universe leads to some conserved quantity. If we assume things are the same from one moment to the next, then energy is conserved. If things are the same if you move a few inches to the side, momentum is conserved. If things look the same when you rotate yourself, then angular momentum is conserved.

Doing accounting of these conserved quantities is a fundamental task of many working physicists, and very small discrepencies in the ledgers either means (generally most likely) that you made some experimental error/are dealing with too much noise or there is some new physics there. Momentum would not appear to be conserved with two magnets moving past each other--UNTIL you account for the momentum inherent in electromagnetic fields (or photons) themselves. Missing momentum and energy in particle tracks helped point the way towards things like neutrons and neutrinos, which generally otherwise would appear invisible but we've managed to find ways to detect and sometimes even use them.

And when we realized that parity (basically, left/right-handedness of things) is NOT conserved in certain particle interactions--after much methodical testing by Chien-Shiung Wu and others--that told us some very important things about the Weak interaction and the fundamental symmetries of the universe. Import enough to lead to the 1957 Nobel prize in physics. Rest assured there are physicists looking carefully for very tiny deviations in other conserved quantities, trying to tease out if there are any other interesting subtle effects to the universe.

Now, on to the Cannae drive--could it be generating thrust through some novel means, sticking the momentum into some unknown form? Yes. But I am EXTREMELY skeptical that they've measured anything real. And their comments about interaction with some quantum vacuum virtual plasma is pretty much just word salad. Nobody outside of NASA Eagleworks (who did the testing) uses that term, which should raise some red flags right away.

First, they are trying to do a torsion balance test, which are known for being extremely tricky and finicky (see Impossible Thruster Probably Impossible for some details on that front). This was tested over only a few days and only a few times, which is not much time for trying to iron out all of the issues. I would want to see months of testing to confirm any sort of results. Did I mention that Science is Hard?

Second, there are a number of real effects that could play a role. The tests were carried out at ambient pressure (confusingly, the original paper, which is mirrored here, talks about having a vacuum system to pump down the system--but then in their "Future Work" section they mention needing to change their RF circuitry to properly work under vacuum. Their official NASA abstract says the testing was done at ambient pressure), which means that there was plenty of air to interact with. If one side got warmer than the other, this would easily provide some thrust in a manner similar to a Crookes radiometer.

The report speaks of the difficulty of providing full shielding of the microwaves, and so any leaky waves (which would go about the distance of 1 wavelength, which for ~ 1GHz is about 1 foot) could interact with water vapor or other parts of the aparatus. They also mention several sources of error (they talk about waves from the Gulf Coast affecting measurements) but don't quantify how big they are. Major, major alarms here.

Third, their null test gave about as much thrust as their proper test. While one could argue that the null test was not "really" a null test as, since they don't really know how it works, maybe how they made it null (removing the slots) wasn't affecting it much, but it definitely kills their explanation of how it works (since their theory relied on the slots).

Fourth, this is a conference proceeding, not a paper. This means that nobody in the greater scientific community has formally vetted it (though now many are making comments in response to the news) and let me tell you, many many proceedings turn out to be wrong (I've caught mistakes in my own proceedings). These are more "interim" reports than anything final. So I'd give it fairly little scientific weight.

And, while this may sound a bit petty, the report looks sloppy, which does not inspire me with confidence. The fact that for their own proceedings they did things like take photos of graphs on their computer screens instead of downloading the data and putting it in nice graphs (which takes 20 minutes of effort, mostly to make the graph look nice): sloppy*. They talk about irrelevant things in the experimental set-up (like what sort of solder they used) and ignore important things like what the Cannae cavities are made out of (Copper? Or some kind of superconductor?). They have some inconclusive results and then end the report with some grand scheme of all the things that could happen if they scaled this up.

Just no.

This is the kind of thing I put out if I'm rushing to write a lab report for a class and am not putting too much thought into it. I will later accept the C grade and then never look at the report again. While you could have a shoddy paper for interesting results, this really does not instill any confidence in me that this test was undertaken with enough care and precaution to counteract all the sources of experimental error.

So I cannae vouch for the Cannae drive. Like the neutrinos we thought were faster than light (a measurement that was later found to be due to a loose wire) it's all too easy for these sorts of things to be some experimental hitch.

Science is Hard.

For more reading, I recommend John Baez's post: https://plus.google.com/11766301...

*Alternately, you could just press "Print Screen" on your keyboard, open up MS Paint, and crop the image to get something that looks nicer. That takes 30 seconds. IT TOOK MORE WORK TO TAKE THE IMAGES OFF OF A CAMERA LIKE THEY DID.

Netherlands to Belgium bike ride, day 4-5

Part 3 of the series. Part 1 Part 2

Robb riding through the Belgian countryside.

Saturday morning we awoke to a bright, shining sun—a wonderful sight, though after the previous day I had to wonder how long it would last. The air was still cold, made worse by the fact that the dampness still had not left. But at least we wouldn’t have to endure too much - today was the day to complete our trip, 40 miles to Brussels.

Netherlands to Belgium bike ride, day 3

Part 2 of the series. Part 1

Very, very wet. Photo by Robb.

Netherlands to Belgium bike ride, day 1-2

(First in a three part series, because this got long)

Spend some time reading the literature of cycling and alternative transportation devotees, and you will quickly gain the impression that the Netherlands are some sort of Shangri-La for two wheels. Time after time, bicycling advocates in the US will say “if only we could be like Amsterdam” - when we’re not trying to be Copenhagen or Paris or Bogota instead. Not for nothing is this a well-worn trope—I remember visiting Amsterdam a few years ago and being greeted with a practical sea of bicycles parked just outside the train station. The Dutch do like their velocipedes, quite a lot.

Amsterdam train station. The upper ledge in the middle of the picture was completely full of bikes.

And then when the Dutch (or the tourists) get stoned, they throw the bikes into the canals. Thus completing the circle of life.

So, given my status as “the bike guy” at Caltech (that’s what running a bike co-op does for you), when one of my fellow let’s-move-our-labs-to-Switzerland-expats, Robb, asked me if I’d be interested in doing a bike tour from Amsterdam to Brussels over a long weekend, I knew the answer had to be yes. I had never actually ridden around there, and now I too could properly experience this utopia of cycling.

The trip was to go as follows: on Wednesday night (October 9^th) we would fly up to the Netherlands, get ourselves settled, and get some rest. The next morning, we’d buy ourselves some nice bikes for the ride, load up on provisions, and do a pleasant 40-mile warm up trip down to Delft. Friday would be our push day, 100 miles down the Dutch coast, until we reached the Belgian border and made our way over to Ghent. This was to again be followed by another 40-mile light day down to Brussels, at which point we would get some well-deserved beer, rest, and further beer. Finally, Sunday we could take a high-speed train back to Zurich, by way of Paris.

A bit of Google Maps for you.

One syllable synposis

Trying to write an account of my bike trip last month has ended up taking longer than I expected. Oh well, something else in the meantime.

Kurt Gödel, a man operating in a different dimension than most of us. Image from Wikipedia

A few weeks ago, I stumbled across a fun paper, Gödel's Second Incompleteness Theorem Described in Words of One Syllable. The paper's behind a paywall (sadly) but here's a taste of the sort of writing we get:

Now: two plus two is not five. And it can be proved that two plus two is not five. And it can be proved that it can be proved that two plus two is not five, and  so on.

Thus: it can be proved that two plus two is not five. Can it be proved as well that two plus two is five? It would be a real blow to math, to say the least, if it could. If it could be proved that two plus two is five, then it could be proved that  five is not five, and then there would be no claim that could not be proved, and math would be a lot of bunk.

So, we now want to ask, can it be proved that it can't be proved that two plus two is five? Here's the shock: no, it can't.

He manages for nearly a page of this. Impressive! It's actually one of the better descriptions I've found, and certainly much shorter (if less fun) than working your way through Godel, Escher, Bach.

So, the natural exercise is to try and do the same with my own research. So, here goes:

********

There are a lot of times you might want to aim sound at some spot. It can help you try and find a crack in a plane's wing, say, or to look in you to see if your health is good. You can build a few sorts of things to do this: for one, you can make a lens. For two, have a bunch of spots that make sound that are timed just right, so that the right place all the sound adds up and gets loud, while far from that the sound waves don’t mix well and it all ends up soft and still. The last one is a bit hard to do: you have to know just when the sounds will play, and since the best way to make the sound will use a kind of clay that does not move that much, it’s hard to make it loud. But there’s a quite good thing with this way—you can choose where to aim the sound, while a lens will be fixed and thus can just aim at one spot.

My work is to build a new kind of way to aim sound, one that is based on how sound moves through a bunch of steel balls placed in a row. In most stuff, the key kind of sound is a tone: the stuff moves back and forth, and the sound goes through at one speed. All other sound can be made if you take these key sounds and mix them up. But, for the balls, the sound goes through in a new way; as a short pulse, one that can be quite strong as well. What’s more, you can change how fast the pulse goes if you squeeze the balls first*. The more you squeeze, the more swift the pulse.

This lets us make the new type of tool to aim sound. First, get a bunch of rows of balls and put them on a thing you want to aim sound in. Next, squeeze the rows in a way so that when the sounds hit the thing, they hit at just the right time so the sound they make will add up at the right place. Last, give the balls a good whack** to make the pulse. And that’s all there is to it.

I have two big goals with my work. One, I need to show that all these words are true, and that you can make a tool that will aim sound in the way I just said. And two, there are some ends to how well this can work, and I need to learn where these bounds are.

********

An "-ing", an "-ing", my kingdom for an "-ing".

EDITED 12/5 as someone pointed out I had a few 2-syllable words (basic, away, about, uses) in there. It's fixed now. Thanks Crowder!
12/6 Eve pointed out "inside" and "pulses". Thanks!

*My research also has pretty limitless potential for double entendre.
** See above.