Posts tagged with symplecticity

Why is the slope of perpendicular lines flipped over and switched signs?

Oh! This one only took me 17 years or so to figure out. This was a “fact” I had committed to memory in school but never thought about why.

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From The Symplectization of Science by Mark Gotay and James Isenberg:

There are some connections to circles and homogeneous coordinates (`v/‖v‖`) but let’s leave those for another time.

Gotay & Isenberg’s exposition using the metric makes it clear that the
`/‖v‖` part of the definition of `cosine` isn’t where the right-angle concept comes from. It comes from the `v``₁ w``₁ + v``₂ w₂`.

$\large \dpi{150} \bg_white \!\!\!\! \text{Given two vectors named } {\color{Golden} \vec{v}},{\color{DarkBlue} \vec{w}} \text{ made up of } \\ \\ {\color{Golden} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} =\vec{v}} \text{ and } {\color{DarkBlue} \begin{pmatrix} w_1 \\ w_2 \end{pmatrix} =\vec{w}} \\ \\ \text{the metric \scriptsize{(which is going to encode geometric information)}\Large\ is:} \\ \\ \boxed{g = {\color{Golden} v_1} \cdot {\color{DarkBlue} w_1} + {\color{Golden} v_2} \cdot {\color{DarkBlue} w_2}}$

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So if the slope of my starting line is `m`, why is the slope of its perpendicular line `−1/m`?

First I could draw some examples.

I drew these with http://www.garrettbartley.com/graphpaper.html which is a good place to count out the “rise over run” and “negative run over rise” `Δx` & `Δy` distances to make sure they really do look perpendicular.

The length and the (affine or “shift”) positioning of perpendicular line segments doesn’t matter to their perpendicularity. So to make life easier on myself I’ll centre everything on zero and make the segments equal length.

The metric formula is going to work if let’s say my first vector `v` is `(+1,+1)` (one to the right and one up) and my second vector goes one down and one to the right. Then the metric would do:

`+1 • +1` (horizontal) `+ +1 • −1 `(vertical)

which cancels.

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What if it were a slope of 9.18723 or something I don’t want to think about inverting?

This is a case where it’s probably easier to think in terms of abstractions and deduce, rather than using imagination in the conventional way.

If I went over `+a` steps to the right and `+b` steps to the up (slope=`b/a`), then the metric would do:

`a•? + b•¿`

What is that missing? If I plugged in `(?←−b, ¿←a)` or `(?←b, ¿←−a)`, the metric would definitely always cancel.

And in either of those cases, the slope of the question marks (second line) would be `−a/b`.

So the multiplicative inverse (flipping) corresponds to swapping terms in the metric so that the two parts anti-match. And the additive inverse (sign change) means the anti-matched pairs will “fold in” to zero each other (rather than amplifying=doubling one another).

The fundamental mystery of capitalism, in my mind, is how a lot of locally zero-sum fights—over customers, over bread, over a job opening—can result in a globally positive-sum game like 2%/year economic growth over a century.

Just had a small maybe-insight into this question. Let’s take the case of a negative-sum court battle where the victim of a rollercoaster accident tries to recover damages in court. What’s being negotiated, at expense, is the transfer of wealth from one party to another—no growth here.

But, this suit constitutes a sample from the probability space of tort losses. The threat—in probability space—with low probability, of high expected loss—incents theme parks to take more precautions.

Maybe the precautions taken in response to the probabilistic threat are what causes the growth.

Agree? Disagree? Missing a wider point? On the right track?

Antifragility

Nassim Taleb wants us all to go long vol — not just be able to withstand volatility, but to actively seek it out.

He can certainly bet that way (and does — though it’s not paying off), but it’s a bad idea to make society anti-fragile.

Let me define a few words describing potential responses to volatility:

• fragile — Taleb means systems that break when catastrophic volatility is applied; he’s thinking of people who deep short volatility or at least indirectly bet on stability
• robust — like a bridge, or an earthquake-resistant building: built to withstand shocks
• agile — able to adapt to shocks
• anti-fragile — shock-loving; shock-seeking; volatility-loving; risk-avid

Taleb points out that there is no word for "the opposite of fragile”; only for “not fragile”. True.

But we really shouldn’t try to make the system break in the case of no catastrophes. Imagine a bridge that shattered only-and-always, when no cars drove on it. Or a building that toppled only-and-always, when no earthquakes were shaking it. (Those would be anti-fragile things.)

It would be stupid to build things that way. Same with the financial system — we want to be prepared for bad times but also, ready to capitalise on good times. A mouse who’s so afraid of cats that it never goes to look for food, will die.

What makes anti-fragility an especially bad idea in finance, is that people might try to sabotage, tweak, or influence the system to make their bet pay off. Let’s say some powerful crook is long volatility — that is, s/he will only get paid if some huge catastrophe happens within the next year. Maybe s/he will engineer a catastrophe. That could be truly terrible.

UPDATE: @nntaleb has clarified on twitter that he does intend “antifragility” to mean “long gamma”.

UPDATE 2: Jared (@condoroptions) suggests at minute 30 of this Volatility View podcast that @nntaleb must mean long convexity, not long gamma. I interpret that to mean buying stability, selling normal levels of volatility, and buying extreme levels of volatility. In other words things will usually stay the same, but when they change they’ll change more than people expect.
That answers the finance part. I still don’t see how to practically design real things to be antifragile without giving up normal functionality under typical circumstances.