Posts tagged with noncommutative
Playing around with polaroid screens at Google.
If you shine light through
- a light polariser
- another light polariser
Bthat’s perpendicular to
ABrepresents “shine the light through
- then no light comes through (it’s black.)
If you shine light through
- a polaroid
- another polaroid
B ⊥ A
- a third polaroid
Cthat’s halfway between
then no light comes through (it’s black.)
So far, formulaically, we have:
- whence it follows
But! This is surprising to watch and surprising to see the formula.
- If you shine through
B, it’s kind of light!
Earlier in the talk Ron Garret does a two-slit experient with two mechanical-pencil leads and a laser pointer. Wave-particle duality with an at-home science kit.
16 pages for non-brainiacs on the Hopf fibration by David Lyons
- mapping from
ƒ(a,b,c,d) = (a²+b²−c²−d²,,)
- in general a linear transformation in 3-D requires 9 parameters (3×3 matrix — see general linear group)
- but a rotation only requires ≤4 parameter
- mapping from
ƒ[a,b,c,d] = [a²+b²−c²−d²,2(ad+bc),2(
- but a rotation only requires ≤4 parameters
- understanding maps from high-dimensional spheres to low-dimensional spheres is a Hard Problem
Gimbal lock, composition of rotation maps
to get the general three-angle rotation group
but this is ugly and wrong—not because there are too many trig words, but because if you play around with it enough you’ll see that—just like the North Pole and South Pole have redundant longitude coordinates—various combinations of [phi;,theta;,psi;] can overlap each other or even get caught in a Gimbal Lock.
A list of quasimetrics from real life:
- one-way street
- temperature preferences
- running distance around the track
- downside risk in a financial portfolio
- Facebook EdgeRank
- I love you, but you like me back. Ouch. (“He loves me in his own way”, however, is not on a 1-D scale against “like” — so a functional to a quasimetric would pervert the meaning.)
When I was in kindergarten, we would argue about whose dad made the most money. I can’t fathom the reason. I guess it’s like arguing about who’s taller? Or who’s older? Or who has a later bedtime. I don’t know why we did it.
- Josh Lenaigne: My Dad makes one million dollars a year.
- Me: Oh yeah? Well, my Dad makes two million dollars a year.
- Josh Lenaigne: Oh yeah?! Well My Dad makes five, hundred, BILLION dollars a year!! He makes a jillion dollars a year.
(um, nevermind that we were obviously lying by this point, having already claimed a much lower figure … the rhetoric continued …)
- Me: Nut-uh! Well, my Dad makes, um, Infinity Dollars per year!
(I seriously thought I had won the argument by this tactic. You know what they say: Go Ugly Early.)
- Josh Lenaigne: Well, my Dad makes Infinity Plus One dollars a year.
I felt so out-gunned. It was like I had pulled out a bazooka during a kickball game and then my opponent said “Oh, I got one-a those too”.
Now many years later, I find out that transfinite arithmetic actually justifies Josh Lenaigne’s cheap shot. Josh, if you’re reading this, I was always a bit afraid of you because you wore a camouflage T-shirt and talked about wrestling moves.
Georg Cantor took the idea of ∞ + 1 and developed a logically sound way of actually doing that infinitary arithmetic.
¿¿¿¿¿ INFINITY PLUS ?????
You might object that if you add a finite amount to infinity, you are still left with infinity.
- 3 + ∞ = ∞
- 555 + ∞ = ∞
- 3^3^3^3^3 + ∞ = ∞
and Georg Cantor would agree with you. But he was so clever — he came up with a way to preserve that intuition (finite + infinite = infinite) while at the same time giving force to 5-year-old Josh Lenaigne’s idea of infinity, plus one.
- ∞ + 1
- ∞ + 2
- ∞ + 3
- ∞ + 936
That’s his way of counting "to infinity, then one more." If you define the + symbol noncommutatively, the maths logically work out just fine. So transfinite arithmetic works like this:
- 1 + ∞ + 3 = ∞ + 3
- 418 + ∞ + 3 = ∞ + 3
- 1729 × ∞ + 3 = ∞ + 3
- 43252003274489856000 × 287 × 1.4142135623730954 + 3→3→64→2 × ∞ + 3 = ∞ + 3
All those big numbers on the left don’t matter a tad. But ∞+3 on the right still holds … because we ”went to infinity, then counted three more”.
By the way, Josh Lenaigne, if you’re still reading: you’ve got something on your shirt. No, over there. Yeah, look down. Now, flick yourself in the nose. That’s from me. Special delivery.
#### ORDINAL NUMBERS ####
W******ia's articles on ordinal arithmetic, ordinal numbers, and cardinality flesh out Cantor's transfinite arithmetic in more detail (at least at the time of this writing, they did). If you know what a “well-ordering” is, then you’ll be able to understand even the technical parts. They answer questions like:
- What about ∞ × 2 ?
- What about ∞ + ∞ ? (They should be the same, right? And they are.)
- Does the entire second infinity come after the first one? (Yes, it does. In a < sense.)
- What’s the deal with parentheses, since we’re using that differently defined plus sign? Transfinite arithmetic is associative, but as stated above, not commutative. So (∞ + 19) + ∞ = ∞ + (19 + ∞)
- What about ∞ × ∞ × ∞ × ∞ × ∞ × ∞ × ∞ ? Cantor made sense of that, too.
- What about ∞ ^ ∞ ? Yep. Also that.
- OK, what about ∞ ^ ∞ ^ ∞ ^ ∞ ^ ∞ ^ ∞ ^ ∞ ^ ∞ ? Push a little further.
I cease to comprehend the infinitary arithmetic when the ordinals reach up to the ∞ limit of the above expression, i.e. ∞ taken to the exponent of ∞, ∞ times:
It’s called ε₀, short for “epsilon nought gonna understand what you are talking about anymore”. More comes after ε₀ but Peano arithmetic ceases to function at that point. Or should I say, 1-arithmetic ceases to function and you have to move up to 2-arithmetic.
===== SO … WHAT COMES AFTER INFINITY? =====
You remember the tens place, the hundreds place, the thousands place from third grade. Well after infinity there’s a ∞ place, a ∞2 place, a ∞3 place, and so on. To keep counting after infinity you go:
- 1, 2, 3, … 100, …, 10^99, … , 3→3→64→2 , … , ∞, ∞ + 1, ∞ + 2, …, ∞ + 43252003274489856000 , ∞×2, ∞×2 + 1, ∞×2 + 2, … , ∞×84, ∞×84 + 1, … , ∞^∞, ∞^∞ + 1, …, ∞^∞^∞^∞^∞^∞, … , ε0, ε0 + 1, …
Man, infinity just got a lot bigger.
PS Hey Josh: Cobra Kai sucks. Can’t catch me!
"Hitting a baseball is the hardest thing to do in sports." —Ted Williams
On the subject of noncommutative things from everyday life: what’s the hardest sport? People love to debate this question. Fans with a favourite sport say their athletes are the fastest, strongest, most adept, or otherwise better than athletes from other sports. These disputes se basan the television show Last Man Standing, which pits strongmen against outdoorsmen against finesse athletes against … yoga instructors? Well, I’ve even heard the argument made that skateboarding is the most difficult sport.
One way to justify that one sport is more difficult than another is to measure how long it takes for sportsman A to master sportsman B’s sport and vice-versa.
This gives rise to a connected graph with sports at each node. Suppose an objectively hardest sport exists — i.e., it is easier to transition from Calvinball to another sport than the reverse, for every sport which is not Calvinball.
Even if such a sport exists, there’s no reason to think that the graph would look at all symmetrical, transitive, or obey other nice mathematical properties. addition or composition would be obeyed on the graph. In the scenario I drew above, the ease with which a Calvinballer can transition to another sport tells you very little about how easy it is for an athlete from Sport S to transition to Calvinball. One could measure difficulty of a sport other than Calvinball either by how hard it is for a Calvinballer to transition to it, or by many other measures (aggregate or individual).
It would be more accurate to put individual athletes at each node rather than “a sport”. That I can mathematically write down either scenario shows how varying levels of abstraction (even prejudice) can be incorporated into a mathematical model.
Whenever a government assesses “the” value of a property for tax purposes; whenever you project figures in a pitch to investors; whenever an accountant writes a single number on a line that isn’t referring to Cash Itself — a distribution has collapsed.
I put "the value" in scare quotes because one number can’t express an asset’s worth. Even if dollar-value-to-me is a 1-D concept — a dubious proposition itself — there are still many other players who value the asset differently. And those valuations could change with new laws, market conditions, reading a book and getting a crazy idea, etc. It all depends. It’s a function; it’s multi-dimensional; it has a topology; different opinions = different metrics, and so on.
Anyone who has done accounting in the real world knows intuitively that Numbers Aren’t Facts. Numbers are opinions. Numbers are estimates. Numbers are vague, but less vague than the alternative.
Let me delve into the economics of one imaginary ambiguous valuation for the more scholarly readers.
Imagine a farmer who owns land in Montana and has worked it all his life — never considering selling the farm because he loves farming, is a farmer, sees himself as a farmer, and wouldn’t know what to do with himself if he didn’t farm. He wants to pass the farm on to his kids, but the property will be subject to taxes as it passes from hand to hand.
How much should the government tax his land asset? They have to estimate the worth of the property and then charge a fair, fixed percentage that they would charge to anyone else. But there is no single worth of the property. It’s worth different amounts to different people, for different reasons.
- In the hands of his kids, and in his hands, it has sentimental value.
- In a farmer’s hands, it has agricultural value as well as the utilitarian pleasure the farmer gets from working the land.
- In a real estate developer’s hands, if the developer does everything right and sells a good number of lots, the property will be worth a lot of dollar bills.
- In a rich fool’s hands, the property is worth as much as he thinks it’s worth.
- In a rich sentimentalist’s hands, the property is worth whatever s/he has to pay to acquire that beautiful thing.
- In a conservation fund’s hands, the property’s worth will fluctuate with the carbon price, donations (which in turn will fluctuate with certain segments of the economy), grant money, the fund’s ability to obtain other nearby properties, the priorities and opinions of the Board of Directors, and the migratory patterns of endangered species.
By the way, I could also add this family farm story to my list of noncommutative phenomena in business — because the price to develop farmland into a subdivision isn’t the same as to go the reverse direction. (Similarly for conservation or any different use of the land.)
The farmer may want to live in his own universe where he works the land, his kids work the land, their own little economy — but the tax man’s assessment will definitely look at what other people outside that universe are willing to pay for the property.
A Marshallian supply & demand graph from Econ 101 makes the point. This is a 1-D chart so it doesn’t capture the full variegation of the stories I sketched above. But it shows more detail than a single number “the” value of the property.
The farmer only has one of “good X” to sell, so supply is inelastic.
In Econ 101, the market clearing price is the price which makes all units ship. In this graph there are multiple prices which would make all units ship, so there is no single market price.
In the family farmer situation, the farmer doesn’t even want to put his place up for public auction. But the ambitions and tastes of some rich guy or group of rich guys can still sway how much tax the farmer pays handing the property over to his kids. The farmer can choose whom he sells it to (and refuse to sell to a developer for instance), but that preference has no effect on the tax rate.
The family farm story isn’t the only case where a unique asset has to be assigned “a” single value.
- Selling a restaurant — will the next owner sell alcohol there?
- Shares of Bear Stearns around February 2008.
- Any of the hot potatoes the Fed scooped up during the financial crisis. (google “mark to market” or “fair market value” for illiquid securities, it’s a controversial issue)
- Any asset I buy for use in my business. Obviously I wouldn’t buy $10M of factory equipment if I didn’t think they would be worth much more than $10M to me. But their value in a liquidation would have to be lower, especially if the buyers know the liquidator is under time pressure and can’t use the stuff any other way.
Everything is connected — fortunately or unfortunately.
A circle is made up of points equidistant from the center. But what does “equidistant” mean? Measuring distance implies a value judgment — for example, that moving to the left is just the same as moving to the right, moving forward is just as hard as moving back.
But what if you’re on a hill? Then the amount of force to go uphill is different than the amount to go downhill. If you drew a picture of all the points you could reach with a fixed amount of work (equiforce or equiwork or equi-effort curve) then it would look different — slanted, tilted, bowed — but still be “even” in the same sense that a circle is.
Here’re some brain-wrinkling pictures of “circles”, under different L_p metrics:
p = ⅔
p = ½. (Think about a Poincaré disk to see how these pointy astroids can be “circles”.)
p = 3/2
The moves available to a knight ♘ ♞ in chess are a circle under L1 metric over a discrete 2-D space.