Posts tagged with vector field

It was the high zenith of autumn’s colour.

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We drove her car out to the countryside, to an orchard. Whatever the opposite of monocropping is, that’s how the owners had arranged things.

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The apple trees shared their slopey hillside with unproductive bushes, tall grasses, and ducks in a small pond in the land’s lazy bottom.

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Barefoot I felt the trimmed grass with my toes. A mother pulled her daughter away from the milkweeds—teeming with milkweed nymphs—because “They’re dangerous”.

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It was only walking along the uneven ground between orchard and forest that I realised that I almost never walk on surfaces that aren’t totally flat, level, hard, and constant.

 

In the Chauvet cave paintings of 32 millennia before sidewalks, the creator — rather than being hampered by the painting surface — used its unevenness to their advantage.

Photo: Horse paintings in  Chauvet Cave

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But today

  • sidewalks are completely flat in New York City; if you trip and hurt yourself because of their ill repair you can actually sue the City
  • art (not all art but a lot of painting or screen-media) is conceived on a flat surface
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  • houses are square; efficient industrial production of the straight and right-angle-based construction materials
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    and work plans
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    means it would be relatively expensive to build otherwise.
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  • yards are square
  • parks are square
  • city blocks are square
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  • (…except older cities which resemble a CW complex more than a grid)
    ComplexCity: Moscow
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In general relativity flat Euclidean spaces are deformed by massive or quick-spinning objects. 

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Due to an uneven distribution of mass inside the Earth, its gravity field is not uniform, as indicated by the lumps in this illustration.

Still image of world gravity map

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and in sheaf theory things can be different around different localities.

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The cave walls in Chauvet have been locally deformed even to the point that knobs protrude from them—and the 32,000-year-old artist utilised these as well.

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Maybe when Robert Ghrist gets his message to the civil engineers, we too will have a bump-tolerant—even bump-loving—future ahead of us.

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EDIT: Totally forgot about tattoos. 




a smooth field of 1-vectors in 3-D

a smooth field of 1-vectors in 3-D

(Source: thievess)


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I don’t know if this was the artist’s intention, but this piece gives me a feeling of musical notes and, at the same time, evokes a vector field. Like maybe a red mark could indicate the attentional direction of a goose, and the grey line could chart its flight path.
by Jorinde Voigt via triangulationblog

I don’t know if this was the artist’s intention, but this piece gives me a feeling of musical notes and, at the same time, evokes a vector field. Like maybe a red mark could indicate the attentional direction of a goose, and the grey line could chart its flight path.

by Jorinde Voigt via triangulationblog


hi-res




Vector fields pervade. I think about them every time I throw a frisbee in wind.

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In a social context, I think about vectors of intent attached to people talking at a party — vectors of flirtation, vectors of eye movement and attention, and more abstract vectors representing jokes, topics of discussion, dance moves, or songs that are playing.

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Also when I’m thinking about international trade or just the local flows of money in my community, it’s natural to use the vector-field metaphor to “see” the flows.

Electric field of 3 point charges

I also think of history (at different scales) using vector fields. Wars are like nation-states or soldiers aiming weapon vectors at each other. Commerce has many more dimensions since goods and money are both multi-dimensional. Ideas and culture also transmit in a vector-field-like way. Epidemics — well, there’s a reason mosquitoes are referred to as disease vectors.

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Information flows, thoughts, internet bits — anything that can be characterised as a vector, you can expand that thought into a more complicated vector-field thought. Turbulent versus laminar flows of ideas and culture? Maybe it wouldn’t deserve a research grant but it’s fun to think about.

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There are pretty obvious physical examples of vector fields — rivers, wind, geological eroding forces, magnetism, gravity, flying machines, bridge engineering, parachute design, weather patterns, your entire body as it does martial arts or dances. Being measurable, these are the source of most of the neat vector-field pictures you can find online.

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(Or you find programmatically simple theoretical vector fields like the above: a vector facing [−y,x] is attached to every point (x,y). So for instance the point (3,4) has a pointer going out −4 south and 3 east, which equals a total force of 5.)

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The same metaphors and visualisations, though, are open to interpretation as social or economic variables too. For example a profitable business is more of a “sink” or attractor for 1-D money flows, while a benefactor is a “source”. Likewise a blog that receives lots of links and traffic is a 2-D attractor on the graph of the web — and Google recognises that as PageRank.

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I know of at least one paper that tries to best economists’ utility theory models by imagining a person on a 1-D vector field, trying to avoid minus signs and find a path to plus signs in the space.

Lotka-Volterra-Goodwin Predator-Prey Model

There is also a game theory connection. Basins of attraction can draw you into a locally optimal place that is not globally optimal. You can imagine examples in the evolution of animals, in company policies or business practices, or in whole economic systems.

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On the one hand it may seem frivolous or crackpottical to generalise these concrete physical concepts to the social or psychological. On the other hand — that’s the power of the generality of mathematics!

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Vector fields are surfaces or spaces with a vector at each point. That’s the mathematical definition.




Technically the best business idea ever is the gift card. You take advantage of (or should I say, “leverage”) social stigma to get people to give you money. You don’t even have to deliver any products, you just get money. Maybe you will have to give up some inventory later; maybe the gift card just gets lost. Often you make even more money, if you price your cards at $20 and your merchandise at $22.99. (“You” here must generally be an incumbent brand.)

But, like, where’s the beef? That invention’s not really making the world a better or more interesting place — it’s just tweaking the vector field so that money flows toward you.

It might even make the world a worse place: think of those soul-less malls, filled with incumbent retail outlets offering pappy crap — but you’ve gotta get something and you don’t really know the receiver that well, and it’s going to be awkward if you have nothing…. Geez. Just make presents for the people you love and get everybody else chocolate or champagne.

NYT or Geico?

I have a question for every aspiring capitalist. Would you rather own a share of GEICO or the New York Times? I would rather own the Times. It doesn’t make money but it makes a difference.

I’m not saying that to put up the literary word. I would also love a share of Kentucky Fried Chicken. That sh*t is delicious. My point is that if I had tons of money, I would want to invest it in a business with a cool product. In fact, that would be probably the most fulfilling way to spend tens of millions of dollars. (Maybe this is the reason people invest in web start-ups.)

By contrast, insurance companies are practically demonic. They fight their customers. They’re propped up by statute (yes, who would buy insurance if it wasn’t required by law?). Then they try to get out of . And they suck the life out of their salespeople and claims adjusters — contrast that to journalists, who go into journalism on purpose.

Yeah, a well-run insurance company ought to make money, and serve an economic purpose to boot. A greedily run company ought to make money faster than they stuff it into their pockets. But — do you really want all that tar on your soul?