Radian in not-a-unit shocker

October 6th, 2015

One of the nice things about scmutils is that it tracks units, so you can’t accidentally add 10 seconds to 5 metres.

 (& 10 &second)
 (& 5 &meter))
=> Units do not match: + (*with-units* 10 (*unit* SI ... 1)) (*with-units* 5 (*unit* SI ... 1))

When dealing with angles, it initially seems to do the right thing too:

 (& pi/2 &radian)
 (& 90 &degree))
=> 3.141... (ie. its converting everything to radians)

But this is less cool:

(/ (& pi &radian) (& 1 &second))
=> (& 3.141592653589793 &hertz)

Err, pi radians should be 0.5Hz. The trouble is, scmutils treats radians as a unit-less number.

To check whether this was a reasonable thing to do, I checked my old favourite Frink. In frink’s units.txt files, we have the following:

// Alan’s editorializing:
// Despite what other units programs might have you believe,
// radians ARE dimensionless units and making them their own
// unit leads to all sorts of arbitrary convolutions in
// calculations (at the possible expense of some inclarity if
// you don’t know what you’re doing.)
// If you really want radians to be a fundamental unit,
// replace the above with “angle =!= radian”
// (This will give you a bit of artificiality in calculations.)
// The radian was actually a fundamental base unit in the SI
// up until 1974, when they changed it, making it no longer
// be a special unit, but just a dimensionless number (which
// it is.) See the definition of the “Hz” below for a
// discussion of how this broke the SI’s definitions of
// basic circular / sinusoidal measures, though.

And down a bit, on the section about hertz, we have:

// Alan’s Editorializing: Here is YET ANOTHER place where the SI made a
// really stupid definition. Let’s follow their chain of definitions, shall
// we, and see how it leads to absolutely ridiculous results.

// The Hz is currently defined simply as inverse seconds. (1/s).
// See: http://physics.nist.gov/cuu/Units/units.html
// The base unit of frequency in the SI *used* to be “cycles per second”.
// This was fine and good. However, in 1960, the BIPM made the
// change to make the fundamental unit of frequency to
// be “Hz” which they defined as inverse seconds (without qualification.)
// Then, in 1974, they changed the radian from its own base unit in the SI
// to be a dimensionless number, which it indeed is (it’s a length divided by
// a length.) That change was correct and good in itself.
// However, the definition of the Hz was *not* corrected at the same
// time that the radian was changed. Thus, we have the conflicting SI
// definition of the radian as the dimensionless number 1 (without
// qualification) and Hz as 1/s. (Without qualification.)
// This means that, if you follow the rules of the SI,
// 1 Hz = 1/s = 1 radian/s which is simply inconsistent and violates basic
// ideas of sinusoidal motion, and is simply a stupid definition.
// The entire rest of the world, up until that point, knew that 1 Hz needs to
// be equal to *2 pi* radians/s or be changed to mean *cycles/second* for
// these to be reconcilable. If you use “Hz” to mean cycles/second, say,
// in sinusoidal motion, as the world has done for a century, know that the SI
// made all your calculations wrong. A couple of times, in different ways.
// This gives the wonderful situation that the SI’s Hz-vs-radian/s definitions
// have meant completely different things in the timeperiods:
// * pre-1960
// * 1960 to 1974
// * post-1974
// Thus, anyone trying to mix the SI definitions for Hz and angular
// frequencies (e.g. radians/s) will get utterly wrong answers that don’t
// match basic mathematical reality, nor match any way that Hz was ever used
// for describing, say, sinusoidal motion.
// Beware the SI’s broken definition
// of Hz. You should treat the radian as being correct, as a fundamental
// dimensionless property of the universe that falls out of pure math like
// the Taylor series for sin[x], and you should treat the Hz as being a
// fundamental property of incompetence by committee.
// One could consider the CGPM in 1960 to have made the original mistake,
// re-defining Hz in a way that did not reflect its meaning up to that point,
// or the CGPM in 1974 to have made the absolutely huge mistake that made
// the whole system inconsistent and wrong, and clearly broke the definition
// of Hz-vs-radian/s used everywhere in the world, turning it into a broken,
// self-contradictory mess that it is now.
// Either way, if I ever develop a time machine, I’m going to go back and
// knock both groups’ heads together. At a frequency of about 1 Hz. Or
// better yet, strap them to a wheel and tell them I’m going to spin one group
// at a frequency of 1 Hz, and the other at 1 radian/s and let them try to
// figure out which one of those stupid inconsistent definitions means what.
// Hint: It’ll depend on which time period I do it in, I guess, thanks to
// their useless inconsistent definition changes.
// It’s as if this bunch of geniuses took a well-understood term like “day”
// and redefined it to mean “60 minutes”. It simply breaks every historical
// use, and present use, and just causes confusion and a blatant source of
// error.
// In summary: Frink grudgingly follows the SI’s ridiculous, broken definition
// of “Hz”. You should not use “Hz”. The SI’s definition of Hz should be
// considered harmful and broken. Instead, if you’re talking about circular
// or sinusoidal motion, use terms like “cycles/sec” “revolutions/s”,
// “rpm”, “circle/min”, etc. and Frink will do the right thing because it
// doesn’t involve the stupid SI definition that doesn’t match what any
// human knows about sinusoidal motion. Use of “Hz” will cause communication
// problems, errors, and make one party or another look insane in the eyes
// of the other.

Quantum Scheme

September 30th, 2015

I’m doing the Stanford “Quantum Physics for Engineers” online course just now. Separately, a few months ago I was reading the Sussman “Structure And Interpretation of Classical Mechanics” book which is notable for using scheme as a mathematical notation, thereby avoiding a lot of the ambiguities of ‘normal’ maths notation (a big win in Lagrangian mechanics, which makes heavy use of partial derivatives).

Anyhow, the Stanford Quantum course requires you to do various exercises, such as the following:

An electron has a 1nm wavelength. Is it reasonable to treat this electron as an approximately non-relativistic particle (i.e. traveling much slower than the speed of light)?

As usual, this requires plugging the supplied numbers and a bunch of physics constants into the right equation. At school, I would’ve done this by hand – hopefully remembering constants like ‘c’ (3e8 m/s) and h (6.62e-34).

But I can also do this using scheme, as per the SICM book. The ‘scmutils’ library comes with a bunch of built-in constants, with the correct units:

=> (& 299792458. (* &meter (expt &second -1)))

=> (& 6.62606896e-34 (* (expt &meter 2) &kilogram (expt &second -1)))

In scmutils, the ampersand function attaches units to a number.

So now I can use de Broglie’s wavelength relation to find velocity as a function of mass and wavelength:

(define (velocity mass wavelength) (/ :h (* mass wavelength)))

then plug in the appropriate values to find the velocity:

(velocity :m_e (& 1e-9 &meter))
=> (& 727389.4676462485 (* &meter (expt &second -1)))

The question actually asked “can you treat it as non-relativistic” so we want to know if it’s close to the speed of light or not:

(/ (velocity :m_e (& 1e-9 &meter)) :c)
=> 2.43e-3

So it’s much slower than the speed of light, and the answer is “yes, it’s reasonable to treat this as a non-relativistic particle). But thanks to scheme/scmutils, I’m also pretty confident I haven’t made errors with units (because scheme tracked them for me) or constants (because I didn’t have to enter them).

Although not required for this exercise, the scmutils package also handles symbolic differentiation which is pretty nifty! For example:

(define (foo x) (log x))

(foo 'a)
 => (log a)

((D foo) 'x)
 => (/ 1 x)

The scmutils library is very elegant once you realise how it works. The definition of the scheme ‘foo’ function is just that – a scheme function. You can use it in one of two ways. You can pass a number to it – eg. (foo 5) – and it’ll evaluate it numerically – eg. 1.609. Or you can pass that same function a symbol, such as ‘a, and it’ll give you back a symbolic expression – eg. “log a”. It has a built-in simplifier too, as seen here:

(define (addaddadd x) (+ x x x))
=> #| addaddadd |#

(addaddadd 'a)
=> #| (* 3 a) |#


December 23rd, 2013

A while ago, I wrote an emacs ‘reading mode’. It highlights a single sentence at a time, fading the rest of the text into a gentle grey, and a keypress moves onto the next sentence. It retains the familiarity and consistency of normal text layout, but provides additional cues about the extent of the current sentence.

Tonight, I played with the idea of including smarter parsing into this reading mode. The Stanford Parser parses english sentences. It tells you about the grammatical structure (noun phrases, verb phrases, etc) and dependencies between words. This is just about enough to do what I had in mind – a “superfluous word” highlighter. The whole world is absolutely packed full of so many documents with wholly unnecessary words. Ideally, I’d like to just delete the pointless words. But it’s rare for a word to be completely devoid of semantic meaning. So, my compromise is just to highlight those decorative words – adjectival and adverbial modifiers – which are commonly guilty.

Here’s some examples, not completely perfect, but useful nonetheless:

I REALLY want some SUPER TASTY chocolate.
The system has been VERY CAREFULLY designed, and will cope admirably with all 
  CONCEIVABLE combinations of circumstances.
I wanted to leave my SMALL pond and see HOW I'd fare in a BIG one, with some 
  of the BEST developers in the world.
You define HOW you want your data to be structured ONCE, THEN you can 
  use SPECIAL GENERATED source code to EASILY write and read your STRUCTURED data.


September 28th, 2013

“Wald applied his statistical skills in World War II to the problem of bomber losses to enemy fire. A study had been made of the damage to returning aircraft and it had been proposed that armor be added to those areas that showed the most damage. Wald’s unique insight was that the holes from flak and bullets on the bombers that did return represented the areas where they were able to take damage. The data showed that there were similar patches on each returning bomber where there was no damage from enemy fire, leading Wald to conclude that these patches were the weak spots that led to the loss of a plane if hit, and that must be reinforced.”

– http://en.wikipedia.org/wiki/Abraham_Wald


September 14th, 2013

I was mucking about in javascript, simulating geostationary satellites orbiting around the earth. Then I started thinking about simulating moon missions – ie. properly simulating the thrust of (say) the Saturn 5 rocket at various stages of the launch, how the fuel mass decreased and so resulted in increasing acceleration at fixed thrust. There’s plenty of data available about when they lit engines and started roll programmes for the Apollo missions.

Anyhow, this lead me to realise that I’m too earth-centric in my coordinate systems. I need to know where the moon was in 1969 when Apollo 11 took off. Actually, I’m wasn’t even sure which coordinate system you’d measure that in. WGS84, used by GPS sysems, ain’t so much use if you’re flying to the moon! The ICRF is what you need.

It also made me think about a location-a-pedia. Ie. something which tells you where objects were at a certain time. Where was the moon at the instant when Apollo 11 took off? Where was it when Apollo 11 landed? Perhaps for flying sims, you might have historical data about where different commercial flights were at different times. For space sims, you need to know where the objects of the solar system were. Newton’s laws will tell you how they move, but you need a starting point.

(Update: Omg, it’s 2013 and I just used a telnet interface to an online system).

Maybe in 20 years, all objects will be reporting their coordinates (in some galactic coordinate system) to a central database. That way, if you lose your keys, you’d have an easy way to find them. Even if you were on Mars.