Posts tagged with path

In contrast to copper and tin, iron is very widely spread as the great archaeologist Gordon Childe put it “cheap iron democratized agriculture and industry and warfare too”. So the jump to Iron Age technology may have impeded the development of states in Africa by making it more difficult for elite to concentrate and monopolize power.

Africa never experienced the nascent period of political centralization that Europe did during the Bronze Age, perhaps also with a path dependent legacy.

distances in the tree to the path connecting the corners in a uniform spanning tree of a 200×200 grid

by Russ Lyons

hi-res

## Death and Symplecticity

The universe is receding behind you every second. One of the lessons of special relativity is the −ct term:
$\large \dpi{200} \bg_white (- + + +)$

• you can stand still where you are,
• you can run away as fast as you can,
• you can stop and go and wander around,
• you can focus like a nail and pound deep into something,
• you can get bored or be excited,
• you can build something & raise the Lagrangian or veg & leave it low,

time is still flowing past you, that metric subtracting −ct ticks at a rate of one tick per tick.

"Your prison is walking through this world all alone"

In other words, freedom and independence, too, have a cost, perhaps exactly equal to the cost of

• or spending your “best years” raising children instead of “achieving” career-wise.

A tumbleweed sees more but also less than a tree.

If you want to think about lifetime as being a fixed length (ignoring that its length comes from a probability distribution, which itself is conditional on your choices) then you can derive my favourite equation:

$\large \dpi{200} \bg_white \text{consumption} = \text{wage} \cdot (\text{lifespan}-\text{leisure})$

the tradeoff between work, leisure, and wealth. That idea as well is symplectic. And many other such tradeoffs ∃. Symplecticity is the theoretical basis of all budget constraints. It’s another way of talking about all the tradeoffs that make choice meaningful and also unavoidable (even not-choosing is a choice). You can strain and strive as much as you want, all you will do is slide amongst alternatives and never do everything.

If you want to use a picture of the form of Christopher Alexander’s

and just substitute in names of various other things that you want—then the “metric signature”, due to time flowing over and beyond us like a river always, is − in so many of the pursuits one might like to do, such as

• making money
• learning algebraic topology
• spending time with kids
• learning to do a backflip
• travelling in Asia
• playing guitar
• writing an opera
• living so you get to Heaven after this life (ok, I said I wouldn’t bring in any probability distributions but I had to cheat on this one. It’s an interesting measure theory question, isn’t it? If there is even a finite chance of getting an infinite payoff, then unless the utility function becomes flat above a certain payoff, then the only logical thing to do is make 100% sure you get the infinite payoff. OK, /rant)
• making the sex, many times. Or, not:

Sure, sometimes one lucks out and there is a positive association between two things, like learning mathematics and being a quant—but the magnitude might be less than you expect. (Pure maths alone is insufficient and unnecessary to finance.)

In terms of the 10,000-hours-to-expertise paradigm—despite some complementarities (+)—there are only so many 10,000-hour blocks in your life. And the Type A personality who squeezes out the most 10,000-hour blocks, gets the most toys or becomes the world’s best cyclist or visits all the countries, learns the most languages, or whatever, still miss out on something.

Leaving aside that the human encyclopedia and Tony Hawk also will turn back to dust, just even evaluating only the finite path [0,1] → life , that busy body necessarily misses out on

• the down moments,
• the still time,
• the zoning out,
• the chilling,
• the doing nothing and being OK with it,
• the taking in instead of forcing out,
• and perhaps those have some value as well.

In English it sounds so obvious to be trivial: you can’t do everything, because nothing is also something and if you’re doing something you can’t be doing nothing.

But the mathematical language, in addition to sounding more exotic and smartypants, adds something real, at least for me—which is the sense of those − signs attaching me to everything. Every time I do something, I’ve lost some other opportunity. Every person I become, I drift further away from the possibilities of who else I might have been. Every commitment loses a freedom and every freedom wastes a commitment. Every nothing wastes a something and every something forgoes a nothing. Everything is receding, decaying, entropying, with or without me, until eventually the waters will cover my head and I never surface again.

Sufficiently convolved with the

$\large \dpi{200} \bg_white x+y+z + \varsigma + \xi = 100\%$

all the paths sum to a constant and that constant quantity eventually runs out.

Just playing with z² / z² + 2z + 2

$g(z)=\frac{z^2}{z^2+2z+2}$

on WolframAlpha. That’s Wikipedia’s example of a function with two poles (= two singularities = two infinities). Notice how “boring” line-only pictures are compared to the the 3-D ℂ→>ℝ picture of the mapping (the one with the poles=holes). That’s why mathematicians say ℂ uncovers more of “what’s really going on”.

As opposed to normal differentiability, ℂ-differentiability of a function implies:

• infinite descent into derivatives is possible (no chain of C¹ ⊂ C² ⊂ C³ ... Cω like usual)

• nice Green’s-theorem type shortcuts make many, many ways of doing something equivalent. (So you can take a complicated real-world situation and validly do easy computations to understand it, because a squibbledy path computes the same as a straight path.)


Pretty interesting to just change things around and see how the parts work.

• The roots of the denominator are 1+i and 1−i (of course the conjugate of a root is always a root since i and −i are indistinguishable)
• you can see how the denominator twists
• a fraction in ℂ space maps lines to circles, because lines and circles are turned inside out (they are just flips of each other: see also projective geometry)
• if you change the z^2/ to a z/ or a 1/ you can see that.
• then the Wikipedia picture shows the poles (infinities)

Complex ℂ→ℂ maps can be split into four parts: the input “real”⊎”imaginary”, and the output “real"⊎"imaginary”. Of course splitting them up like that hides the holistic truth of what’s going on, which comes from the perspective of a “twisted” plane where the elements z are mod z • exp(i • arg z).

ℂ→ℂ mappings mess with my head…and I like it.

### Topology of the United States.

At a gross resolution, just considering the land area, the United States has three disconnected parts:

• {Alaska, Hawai'i, mainland}.

The complement of the United States is a connected space with a genus of three.

At a finer resolution you would measure a much higher genus. (Does Lake Tahoe count as a “hole” in the mainland US? What about Lake Winnibigoshish?) The Aleutian islands would all register as separate from Alaska, as would the parts of Hawai’i and even Nantucket. So at a fine resolution the complement of the land area of the United States would have a genus well over 100.

For the UK & Ireland, again it depends on resolution. At a gross scale we could simply talk about two islands but that would leave off Orkney, Man, Guernsey, Jersey, the Hebrides, Skelligs, Ione, Skye, Shetlands, and many more.

According to various Ordnance Surveyors in the Daily Mail (1995):

• Our 1:625,000 scale database shows Great Britain (England, Scotland and Wales) has a total 6,289 islands, mostly in Scotland. Of these, 803 are large enough to have been ‘digitised’ with a coastline by our map-makers. The rest are recorded as point features
• The 1:250,000 scale map of Northern Ireland shows 160 islands; 57 offshore.
• Our 1:250,000 map of the Republic of Ireland has 279 offshore islands.

So, at fine resolution, the genus of the complement

• |∁ {UK}∪{Ireland}| = 6289

and at a coarser scale, the genus of the complement of the isles is 803.

(Source: Wikipedia)

There is a century-old tree at the end of my street. Right before you get to the graveyard with its wrought-iron gates. That tree saw my grandmother play in the street when she was a little girl. It saw her ride the train to the college, carry groceries in a paper sack. The tree—I don’t know its name—it saw my da walk across town—from school to that house on Broad, when they used to live there. It can see my great-grandfather’s grave right now—it’s tall enough. He built this house in 1921. They say he was a drunk. The floor slants a little and the window frames aren’t square. He built the other houses on our block, too. Before he built them, it was just this house and greenhouses. The greenhouses were filled with roses. The whole neighbourhood used to smell like roses. At some point they used to call this Rose City, even though there’s a meatpacking factory only two kilometres away.



They also say he could multiply long numbers in his head, without any paper. Now this house is holding a different kind of “family”. I can’t even say it’s a modern one. More like a gathering of moneyless relations. Ambitious failures; I sometimes wonder what the house thinks of us. It’s certainly used to the self-help books: Latin; Linux; teach yourself guitar. The trains in this town used to carry passengers. They took my grandmother to the teacher’s college. My da must have walked past this graveyard a thousand times. No, more—maybe even ten thousand. I walk in the graveyard every day. The tree sees me. My favourite is when it’s snowy. Some of the graves announce strange names. A woman named Ruby. She would be 136 now. A man named Reason. Apparently the brothers who lie beneath the massive Romanesque columns at the highest point in the graveyard invented a transport that was used massively during the War. You can see most of the town standing among those columns. Past the roads there’s a small forest, beyond that farms.

I’m thinking about my path γ(t) versus the tree’s λ(t). Neither of us can be everywhere at once. We’ve stood at or around the same spot often enough. But every time I’ve gone “adventuring”, I haven’t seen what’s happening in λ(t). Is the small-town life “worse” than the jet-setter lifestyle? It depends what functional you convolve against γ(t). I don’t like repetitiveness, but maybe what the tree has seen isn’t so repetitive. Two World Wars. The rise of feminism. A time before plastic, a time before tarmac, a time when white supremacists would parade through the streets. My grandmother recognised someone’s shoes and shouted his surname; her mother covered her mouth. The tree saw her in most stages of life.

On we go, hurtling through spacetime. The speed of γ equals the speed of λ. From a galactic perspective the tree and I are whirling in almost the same place — regardless of whether I whisk from here on the earth to there on the earth by plane. I’m bound to the ground, ultimately. The tree just recognises that. People used to wear hats here. Everybody wore hats. Now it’s practically a ghost town except for pensioners and welfare recipients. The tree’s children can’t have blown too far.

Spinning in the same spot on 360° × [−90°, +90°] = ∂(S²×[0,1]). γ torques and twists about the sphere but its length is exactly the same. Does the tree wish λ had summited a mountain at some point? Perhaps, but it would be blown down up there, and the ground is tough and nutritionless anyway. It’s suited to this life.

It bears the snow. It puts up with the heat.

I go inside after a couple hours out of doors, of course. But the tree spends all night, every night facing the elements. Maybe it likes being strong. Digging. Growing big. Drinking in sunlight like an athlete at a water fountain.

I’m more like a tumbleweed, rootless, quick to change course. Hanging out for a bit and then rolling—without announcing a goodbye. Untethered. Free, yet constrained by the same holonomy constraining the tree. One path, and one path only. The same width as all the others.

γ isn’t so much more interesting than λ. My γ is filled with magazines, airports, computer screens. Parties where people say more or less the same things, always indicating the hope that their gradient’s pointing in the right direction.

The Idea of Holonomy by Robert Bryant

from the MAA:

“Can I roll the ball from any point to any other point and have it wind up in a given orientation that we want?” Bryant asked.

If I draw a dot with this marker, can you eventually roll the ball enough times so that the dot would touch down anywhere on the table, anywhere at all? Or is the logic of the situation constrained, so that certain spots on the ball pair with certain spots on the table? The answer, he said, has consequences for fields from robotics to control theory.

To me, the idea of constrained motion sounds more like the fundamental economic dilemma.

• You can’t live in as nice of a house as you want and work as little as you want and have all the other stuff you want.
• Even if you had \$100,000,000, you still couldn’t spend the weekend fishing in Chile and attend the Davos seminar and go to your son’s art exhibition.
• There’s a direct tradeoff between how long you work on building the perfect product (say, a game console) and how soon it will be released. You might be able to achieve a little more of both by investing more money into the project … but that comes at the expense of something else.

The “optimal path" — if such a thing even exists — will never be feasible, because the choice space is fundamentally characterised by tradeoffs.