Posts tagged with vectors
The isomorphismes links (last three tweets) might be worth sharing with everyone. (I’ve been accused that this site is hard to browse—sorry!)
@jtc_19 Stanford Encyclopedia of Philosophy (http://t.co/g9QwMVxwfT) is the best starting point. Adding to @condoroptions: Chris Fuchs (ftp://netlib.bell-labs.com/who/cafuchs/index.html), Itamar Pitowsky, various links to other interesting perspectives within isomorphismes.tumblr.com/tagged/physics…, and don’t miss the religious & philosophical speculations from the fathers of QM: http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/sokalhoax.html @epicureandeal @munilass— protëa(@isomorphisms) April 15, 2013
Consider ℂ, the field of complex numbers, as a 1-dimensional vector space. The balanced sets are ℂ itself, the empty set and the open and closed discs centered at 0 (visualizing complex numbers as points in the plane). Contrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segment with midpoint at (0,0) will do.
As a result, ℂ and ℝ² are entirely different as far as their vector space structure is concerned.
When I strike a ball with my foot, look at the exact contact point (and pretend the ball doesn’t squish across my foot—even though it does, because my pegas are so poderosas). There’s a
tangent space at that
contact point and the exact particulars of my posture, shape of my foot|shoe, and so on determine the force vector that I use to bend the ball like Beckham.
f=ma corrected Aristotle’s theory of motion
Vacuum isn’t possible: Vacuum doesn’t occur, but hypothetically, terrestrial motion in a vacuum would be indefinitely fast.
So we know we only need that one strike and then inertia minus drag convolved with lay-of-the-land will determine a full path for the ball: till my teammate’s head contacts it—another tangent space, another vector—and we score.
That’s football for simple tangent spaces—the Aristotle-v-Newton scenario. Now onto the most poetic sport, where two holonomic dynamical systems dance within each other’s gyre, moving in and out of each other’s movements and—occasionally—osculating a tangent that links one part of me with one part of you.
All of the logic about connecting vectors head-to-tail with parallelograms is only to reason about one single strike. The whole linear algebra on force vectors is a complete examination of the possibilities of the variations and the ways to strike at that exact same point.
To talk about striking different points requires a connection (parallel transport). [And remember if you hook my cheek versus jab my forehead, those arenotparallel so even more maths is involved. Also hard tissue (bone) versus soft tissue (cheek).]
Striking the ball on the side (to spin it) or under (to chip it) would be a
connection on the S² manifold of the ball — moving the point of tangency — which is a different algebra’s worth of logic. Landing a punch on a different part of me is also a connection (that would be parallel transport of the strike vector on the surface manifold (with boundary) of my face/torso/armpit/whatever).
Keeping my torso over the ball is even yet another calculus—one that I suspect is much, much more complicated. Even though Julian James Faraway and others work on these questions of whole-body mechanics, I don’t think ∃ a complete mathematical theory of how all of the fixed
holonomic parts of bodies that have very similar morphogenetic shapes (similar ratios of forearm to humerus length, etc) interact—how soft tissue and hard tissue in the usual places interact and specific angles can make a judoist with excellent technique and little strength able to whirl the body of a weight-lifter around her fulcrum.
Or how this guy can deliver a lot of force with proper dynamical posture (“shape”) when he’s clearly weak and fat. I can start to imagine the beginnings of something like that but it doesn’t obviously fit into the tangent space points & vectors story, except in a very complicated way of vectors connected to vectors connected to vectors, with each connection (not the same as the parallel-transport connection sense of the word I used above! Back to the base namespace) holonomically constrained or even "soft-constrained" in the case of soft tissue. Same with a blow landing parallel-transported different places on the surface of my body. The transport takes care of the initial strike vector but not how those forces travel and twist through my skeleton, soft tissues, down to the floor (either through my feet or through my *rse if the blow knocks me down). And sure, you could FEM-approximate my interior with a 3-D mesh and a deep geometric logic—and even make the ribs more rigid than the liver—but that’s just going to make you want higher maths even more as you grapple with the army of ε’s you’ll convoke with all the approximations,
dx size, parameters, and so on.
Or why it’s better to swing the bat this way, or teach your body to do its swimming strokes in that motion, or various other dynamical or static particulars of human body shape and internal anatomical facts.
BTW, an amateur ultimate fighter once told me that the order of importance for winning a fight is:
- technique (brasilian jiu jitsu technique)
He didn’t mention tolerance for getting punched in the head but he seemed to put up with it remarkably. PS: judo guys have sexy bodies.
This is for my homies in maths class.
Left “times” right equals target. Each entry in the target is the result of a series of +'s and ×'s along the red and blue. A long sum of pairwise products.
- Your left hand goes across and your right hand goes up/down.
- There need to be as many
abcdefg's as there are
1234567's or else the operation can't be done.
- Also you can tell how big the output matrix will be. There can be three blue rows so the output has three rows. There can be four red columns so the output has four columns.
- This is the “inner product” because multiplying vector-shaped blocks (tall blocks) like Aᵀ•B results in an equal or smaller sized output.
(There is also an “outer product" which is a different way of combining the info from the two matrices. That gives you an equal or larger shaped result when you multiply vector/list-shaped tall blocks A∧B.)
- Try playing around with this one or that one.
Matrix multiplication is the simplest example of a linear operator, the broad class of which explains quantum mechanics and ODE’s. You can also apply different matrices at different points as in a vector field — on a flat surface or a curvy, holey surface.
Vector fields pervade. I think about them every time I throw a frisbee in wind.
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.
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.
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.
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.
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.
(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.)
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.
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.
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!