Too bad we can't represent vectors with dots?

[rockyshephear]

I've always wondered this about vectors. As far as I understand if we could make it intelligible it would be ideal to visualize each infinitely small point as vectors in a field.

Since we cannot we space them out so visually so it's not an unrecognizable mess. Which is one reason that lead to my confusion about lines of flux having geometric shape etc.

It seems like since vectors are represented with arrows that have length, width and sometimes depth, they tend to cover an infinite amount of other very small points that could be also represented by vectors and give a more accurate field depiction.

This is less of an issue when dealing with a simple field where you probably can predict much of the surrounding field structure the same way you can predict the structure of a sphere if you know the how far out from the point of origin you are. But when you get into complex systems of fields, would it help to have much more accurate depictions? Even if not, it seems ideal to represent vectors with dots anyway, if it were a possiblity.

Too bad human brains cannot distinguish trilions upon trillions of colors then a single very, very small colored point would suffice to indicate a magnitude and direction such that we don't have arrows that cover other potential vector points. And like the evolution of HDTV from old television technology, vector depiction would make astounding leaps forward in visualization.

Other than, "Dude, quit wasting precious forum space with this insipid crap!"...does anyone have any thoughts on this?

 

[Tac-Tics]

Are you talking about vector fields? (And neither vectors nor vector fields have width or depth).

Liberal arts majors may disagree, but human mind is a terribly limited device. Sure, it's useful for survival and we have the ability to pass on knowledge from one generation to the next, but it's still limited. Even simple lines of reasoning are tedious and difficult unless they are phrased in special ways in which we can intuit.

I'd say that MOST concepts in mathematics are beyond direct human visualization. Take, for instance, a set. It seems really simple when they teach you it in class. A set of vectors in R^2 can be visualized in a drawing. A line is a set of points in R^2, and you can draw a line on paper. A circle is a set of points in R^2, and you can draw a circle on paper. The graph of a function is a set of points in R^2, so you can draw it on paper.

But then there are examples which aren't so nice. See, for instance, the Topologist's Sine Curve. (http://en.wikipedia.org/wiki/Topologist's_sine_curve). The left hand side is a mess. It's not clear at all what properties it has until you investigate them logically from the definitions of everything.

Koch Snowflakes (http://en.wikipedia.org/wiki/Koch_snowflake) are another example. Take any tiny piece of the snowflake. No matter how small you cut it, you have a curve of infinite length.

There are also much simpler examples. Let S be the unit interval, [0, 1]. If we graph it, it looks like a line segment. What about the open unit interval, (0, 1), with the endpoints removed? The graph looks the same on paper. What about the set consisting of the rational unit interval ([0,1], removing all the irrational numbers?) The graph looks the same on paper.

Working in physics is nice, because you can largely ignore these degenerate cases. Volumes are always finite. Functions are always analytic. If we can't solve an equation, we can fudge things, and if it gives the right answer, we use it anyway.

In math, though, you are always working from the definitions. Most things can't be visualized directly. Like I said in my other post, you usually have to try and visualize the simpler case. When you want to imagine R^n, just imagine R^2, and keep the differences between the two in your mind.

 

[rockyshephear]

Vectors or vector fields don't have width or depth but depictions of them do!