Defining the New Kilogram

Exciting news today: the kilogram is officially going to be redefined. More specifically, it’s being redefined “based on physical constants”. Also, all the other SI units are being redefined “based on physical constants” at the same time.

“Wait, how does that work? If they’re all being redefined, what’s special about the kilogram? What’s the current definition? Who’s in charge of that anyway?”

Le Bureau

Last things first: the BIPM (Bureau International des Poids et Mesures, International Bureau of Weights and Measures) has been in charge of these things since May 20, 1875. Delegates from the member states of the BIPM meet, roughly every four years, at the CGPM (Conférence générale des poids et mesures, General Conference on Weights and Measures), and the 26th CGPM just voted to adopt the new definitions next year, on May 20 (date chosen for symbolic reasons, or so I assume).

I think it’s easiest to understand the “New SI” as encompassing two principles.

The first principle, which I’ll call “universality”, is that units should be based on objects for which perfectly identical copies are already universally available — like atoms or molecules, and unlike metersticks or the planet Earth. This requires a lot of practical changes to how units are defined and measured. It’s an idea that’s been gaining ground for centuries: James Clerk Maxwell said in 1870 that “If…we wish to obtain standards of length, time and mass which shall be absolutely permanent, we must seek them not in the dimensions, or the motion, or the mass of our planet, but in the wavelength, the period of vibration, and the absolute mass of these imperishable and unalterable and perfectly similar molecules.” (Quote borrowed from PhysicsWorld.)

The current definitions of the meter and second are compliant with this principle; the kilogram isn’t, but that’s about to change.

The second principle is that units should be defined by reference to the values of physical constants, not by reference to physical measurements. This is a bit subtler, and also — I’m going to argue — purely semantic. If you do the legwork of following the first principle, you tend to end up at the second one, whereas the second principle doesn’t function very well on its own (but can be adopted whenever you like by changing the language you use).

All of the SI units are being “redefined” in May in accordance with the second principle. This doesn’t change anything in practice, but it’s sort of aesthetically appealing, so there’s that.

That’s enough metrological philosophizing. Let’s learn some history.

Defining the Meter

The SI is the International System of Units, aka the “metric system”. It’s used directly by most of the world’s population, and used indirectly even in the US; the foot is officially defined as 0.3048 meters.

Back in the ancient past (before 1960), the meter was defined as the length of a meterstick. More precisely, a particular platinum-iridium meterstick (the International Prototype Metre) kept at the BIPM.

From 1960 to 1983, the meter was defined (per Wikipedia) as “1 650 763.73 wavelengths of the orange-red emission line in the electromagnetic spectrum of the krypton-86 atom in a vacuum”. We can choose to view this in two ways: either we’re measuring a bunch of wavelengths of some light (from a specific source) and calling that our meterstick, or we’re fixing a particular physical constant, the wavelength of the orange-red emission line of Kr-86, to be equal to (1/1 650 763.73) meters exactly. The 1960 definition clearly satisfies the first principle I mentioned — the definition is based on something universally replicable — but it fails the second principle, entirely because an artifact-based phrasing was chosen instead of a constant-based one.

This is what krypton’s spectrum looks like. The orange line is the one we’re talking about. Source: Adric Riedel

These days (since 1983) the meter is defined as the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. This again follows the first principle (universality) and still technically fails the second: the distance traveled in some interval isn’t quite a physical object, but it’s a particular thing one meter in length that you can visualize.

This is being replaced in May by a new definition which uses the fixing-a-constant language: The meter will be defined by taking the fixed numerical value of the speed of light in vacuum, c, to be 299 792 458 m/s.

If you wanted to, you could even rewrite the meterstick-based definition in terms of a “constant”: before 1960, the meter was defined by the requirement that the physical constant known as “the length of the International Prototype Metre” have the value “1 meter”.

You might object that this isn’t really a universal physical constant — there’s only one International Prototype Metre, but there’s identical copies of the Kr-86 atom (and its emitted light) everywhere in the universe. This shows that the second principle (constant-oriented phrasing) is at best unappealing without the first one (universality): if we’re basing our unit on a complex, unique objects like the Prototype Metre, which can get damaged or lost or just gradually change its shape, then the “constants” involved won’t be very interesting ones.

Fun Games with Cesium

The second, by the way, is defined as 9 192 631 770 periods of the hyperfine transition of cesium. The term “hyperfine transition” requires a lecture on quantum physics, but I’ll try to summarize.

Imagine two magnets. If you place them next to each other with the north poles facing the same direction, they’ll try to squirm out of your grasp. As long as you hold them in place, there’s potential energy which is just waiting to become kinetic energy (e.g. by flinging one magnet across the room). If you flip one magnet around, the magnets will cling to each other quietly. There’s less potential energy. (In fact, negative potential energy: you need to apply some force to separate them.)

Cesium atoms are like that, but one magnet is the nucleus and the other is the electron cloud. (It’s like a really tiny electromagnet: a loop of electric current (the electrons in their orbits) generates a magnetic field.) There’s an unstable high-energy state with the nucleus and electrons pointing the same way, and a stable low-energy state where they’re pointing in opposite directions.

Here’s a diagram of the hyperfine transition, on a plaque that we strapped to the Pioneer spacecraft and flung into interstellar space. Hopefully, aliens will be able to figure out what it means, so that we can establish a common system of units with them. Source: ScienceCavern

If the cesium atom is in the high-energy state, it will rearrange itself into the low-energy state, ditching the (very tiny) excess energy by emitting a (very low-energy) photon. That photon will have a frequency of 9 192 631 770 Hz, ie, it oscillates 9 192 631 770 times every second. This is related to the energy difference by E = h ν, where h is Planck’s constant and ν (nu) is the frequency.

While we’re at it, the wavelength λ (lambda) of that photon is given by λ = c/ν, where c is the speed of light. This works out to (299 792 458/9 192 631 770) meters, or about an inch and a quarter. So if you aren’t convinced yet that the current definition of the meter can really be considered “artifact-based”, here’s an equivalent definition that’s very similar to the old krypton-89 one:

One meter is the length of (9 192 631 770/299 792 458) wavelengths of the light emitted when the little magnets inside cesium flip into proper alignment.

At Long Last, the Kilogram

Right now, the kilogram is a bit of an embarrassment. It’s defined as the mass of the International Prototype Kilogram, a platinum-iridium cylinder that lives at the BIPM, thereby failing the universality principle. But that’s about to change!

Starting next year (per today’s BIPM resolution), the kilogram will be defined by requiring Planck’s constant, h, to have the value 6.62607015×10−34 kg m2/s.

This is a constant-based definition, to match the constant-based definitions that are being adopted at the same time for the second, meter, etc. But as I’ve argued, the difference between fixing a constant and measuring a physical artifact is more or less semantic. The really important thing is that we’re picking a better artifact. So it’s interesting to ask what definition we’d be using next year if the BIPM had decided to square away the universality issue before worrying about basing things on constants.

In other words, if you interpret the new kilogram as the mass of some reference object, what object would that be?

Well, the most famous equation in physics tells us that E equals m c squared. Or, as Einstein originally wrote it, m = E/c2. (No, really.) What this means is that the presence of energy in an object always contributes a little bit (a tiny bit!) to the object’s mass. In fact, according to Einstein, that’s all mass is: internal energy by a different name. Does this mean that cesium is slightly heavier when it’s in the higher-energy state? Yup!

By how much? Combining m = E/c2 and E = h ν, we find that it’s m = h ν/c2.

Dividing “1 kg” by that expression, we find that the new definition of the kilogram is exactly equivalent to this one:

One kilogram is the extra mass present in (299 792 458)2/(6.62607015×10−34 × 9 192 631 770) cesium atoms if they’re all in the excited hyperfine state (magnets pointing the same way) instead of the ground state (magnets pointing in opposite directions).

However, I don’t recommend you try to make that measurement directly; you’d need a few trillion tons of cesium, and as soon as it all decays to the ground state, you’ll be bombarded with a kilogram’s worth of energy. (It’s enough energy to power San Francisco for a couple of years.)

But in principle, a kilogram can still be thought of as the mass of a specific defining thing. Just not a thing that can get lost or damaged, or gradually lose mass over time.

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Paper: The Stochastic Axion Scenario

I have a paper out with my advisor, Peter Graham:

preprint link (arXiv)

journal link (Physical Review D)

In a sentence: Really light QCD axions don’t have to be fine-tuned; you can avoid overclosure naturally if inflation is long enough and slow enough.

This went up on arXiv in May and was published in PRD last week. In the meantime, I’ve given talks on it at two conferences: the Patras Axion-WIMP at DESY in Hamburg*, and IDM (Identification of Dark Matter) at my alma mater, Brown University.

Talk slides (IDM edition): PDFPPT

The PPT version has a cute animation, which I left out of the PDF because my laptop can’t display PDF animations anyway.

If you’re curious about my work but aren’t a specialist in axion dark matter, let me know and maybe I’ll write a post explaining all of this with less jargon 🙂

 

*If you’re ever in Hamburg, check out the gardens (Planten un Blomen) and the model train museum (Miniatur Wunderland). And DESY, of course.

We’re Back!

I’m reactivating this blog after half a decade.

All of the posts prior to this one are random odds and ends from high school and college. Going forward, I’ll probably be using this for updates on my research and such. Maybe some fun side projects, too.

Puzzle: Forty-Seven and Four Sevens

The 4 small squares have a side length of 7; the large square has a side length of 47.

foursevens

What is the area of the grey region?

Bracket: A bracket-only programming language.

I just came up with a lambda-calculus-based language with my roommate. (Warning!: assumes familiarity with lambda calc.)

EDIT: Curly and square brackets are unnecessary and confusing. Future versions of Bracket will be nice and round:
(XY) –> (()XY)
{XY} –> (XY)
[X] –> (X)

This leads to a single elegant function (), which can take 0-3 arguments:
() is the lambda metaprocedure
(X) is an identifier
(XY) is the procedure X applied to argument Y
(MXY) is the metaprocedure M applied to arguments X and Y

This simplifies the language, and should therefore make code much more readable. It also helps me achieve my goal of creating the purest form of LISP: one in which there are no data types and no keywords — only parentheses. (Everything (must (be parentheses))).

ENDEDIT.

Here’s the syntax for Bracket:
(XY): the lambda expression λX.Y, where X is an identifier and Y is an expression.
[X]: an identifier (variable) unique to X (that is, [], [[]], [potato] are all unique identifiers.)
{XY}: the expression X applied to Y. Equivalently, {} is just used for grouping, but all lambda expressions must be explicitly grouped with anything they are applied to.

For example, ([][]) is the identity function, [x] is an identifier, and {([][])[x]} = [x].

A program is just an expression that evaluates to a list* of numbers**, which can be translated into a string of characters with
{{P([]([[]]{{C{{[]I}O}}[[]]}))}N}
(the Bracket syntax for Pλx.λy.C(xIO)yN)
Where:
P is the program
C is a function that prepends an ASCII character to a string
I is a function that increments an ASCII character once
O is the ASCII null character 0x00 or \0
N is the empty string “”

*lists are represented in Church encoding, “Higher-order function” style.
**numbers are Church numerals (see the same article)

Now that that’s out of the way, here’s a “Hello World” program: (actually prints “HW” for the sake of brevity)

{{([]([[]]([[[]]]([[[[]]]]{{[[[]]][]}{[[]]{[[[]]][[[[]]]]}}}))))([]([[]]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[][[]]}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}))}{{([]([[]]([[[]]]([[[[]]]]{{[[[]]][]}{[[]]{[[[]]][[[[]]]]}}}))))([]([[]]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[]{[][[]]}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}))}([[[]]]([[[[]]]][[[[]]]]))}}

And here it is in the new, 300% rounder version of Bracket:

(((()(())(()((()))(()(((())))(()((((()))))(((((())))(()))(((()))((((())))((((())))))))))))(()(())(()((()))((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))(((()(())(()((()))(()(((())))(()((((()))))(((((())))(()))(((()))((((())))((((())))))))))))(()(())(()((()))((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((())((()))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))(()(((())))(()((((()))))((((()))))))))

Efficient nearest-neighbor algorithm

I just thought this up last night: (math/CS warning)

Let’s say you have a large collection of points in a plane, and you need to find the closest point to point P. The obvious (and inefficient) way is to calculate the distance to every point, which requires N square roots and takes far too long to be useful in some cases (such as an early draft of Voronoid that didn’t work out because of this inefficiency). Here’s a better way I came up with:

  • For every point Q, calculate |xQ-xP| and |yQ-yP|. Find the sum and maximum of these values and store them.
  • Find the two points with the smallest sum and the smallest maximum.
  • If the same point has the smallest value for both, you win! That’s the closest point to P. Otherwise…
  • Go through the points again, using the stored values this time. For every point that has a smaller sum than that of the point with the smallest maximum and a smaller maximum than that of the point with the smallest sum, calculate the standard Euclidean distance from P: sqrt((xQ-xP)2+(yQ-yP)2). The closest point to P within this (likely very small) set of points is also the closest point overall.

I’m sure there are absurdly efficient algorithms for doing this better, but this one is very easy to implement (assuming it works, which I’ve sort of heuristically proven). In fact, I might go back and try my early version of Voronoid with this algorithm and see how it runs.

Doodling on an HTML5 canvas

I just tossed this together this morning. Next up: pretty colors!

(Note: Internet Explorer 8 or below can’t do canvases.)

“You have not discovered a potion for remembering, but for reminding”

I found an interesting quote today:

You have not discovered a potion for remembering, but for reminding; you provide your students with the appearance of wisdom, not with its reality. Your invention will enable them to hear many things without being properly taught, and they will imagine that they have come to know much while for the most part they will know nothing. And they will be difficult to get along with, since they will merely appear to be wise instead of really being so.

It could easily be about Wikipedia, Google, or the Internet in general.

Actually, it’s about a much older technology: writing. Specifically, it’s (Plato’s account of) Socrates’ (account of Thamus’) commentary on writing from Phaedrus. Full text here, via an interesting NY Times article on an AI Jeopardist.

What do you think? Was writing worth it? How about the Internet?

(Postscript: relevant comic from Chuck & Beans)

Voronoid: multiplayer puzzle game inspired by graph theory

Voronoid is my final project for CS50. It’s a colorful game based on Voronoi diagrams, with a twist — check it out!
Voronoid Screenshot

[Note: So far, Voronoid supports Chrome and Firefox, on any operating system. It may work in Opera, but probably won’t in IE or Safari.]

Firesheep is Scary! (And a Chrome/Firefox Solution)

Firesheep is a new Firefox plugin that makes it very, very, very easy for anyone to mess with your online accounts. It exploits the unbelievable lack of security in most websites.

One solution (and probably the easiest good one) is to use SSL encryption. That sounds complicated, but it’s usually as easy as entering “https://” instead of “http://” before the URL. I say “usually,” because not all websites support SSL (it’s much more complicated from their end!).

Of course, it’s a pain in the neck to type that in all the time, so here’s a quick solution for Chrome or Firefox. In either browser, it forces “s gmail.com” to redirect to “https://gmail.com/“, for example.

Chrome:
Wrench menu –> Options –> Basics tab –> Default Search section –> Manage –> Add…

Then add the following “search engine” keyword:

Name: SSL
Keyword: s
URL: https://%s

Firefox:
Bookmarks –> Organize Bookmarks –> Unsorted Bookmarks (in the sidebar)
Then Organize –> New Bookmark…

Then add:
Name: SSL
Location: https://%s
Keyword: s

Caveat:
This will force the first page you visit to use SSL, but the website may drop the security for future pages. For example, “s gmail.com” will stay encrypted, because Google is smart, but “s facebook.com” will go back to plain old HTTP after you log in, because Facebook…. isn’t that smart.

Plugins like HTTPS Everywhere (Firefox only, because of technical limitations in other browsers) will automate the process and keep your connection secure all the time.