Re: [Phys-L] math +- visualization etc. etc.

["Rauber, Joel" <Joel.Rauber@SDSTATE.EDU>]

I found the program for plotting hyperbolic coordinates superposed on Cartesian to be immensely useful for simplifying the drawing of x-t space-time diagrams.

It can be found at:

http://www.physics.pomona.edu/sixideas/sicpr.html

This is from Moore's "Six Ideas . . ." website, the program can be downloaded about halfway down the page in the section "For Use With Unit R".  It easily saves 60-80% of your time in drawing simple 2-D spacetime diagrams.

Joel

-----Original Message-----
From: Phys-l [mailto:phys-l-bounces@phys-l.org] On Behalf Of John Denker
Sent: Tuesday, December 17, 2013 1:34 PM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] math +- visualization etc. etc.

On 12/17/2013 11:19 AM, LaMontagne, Bob wrote:
> I envisioned a line divided into 27 parts and one divided into 17 
> parts. I could "see" that 25 parts were closer to the right end of the 
> line than 15 parts - it was all purely visual - no deliberate use of 
> formal logic.

a) I know some people who are intensely visual.

b) I also know some who are not, including some very fine physicists who find geometrical arguments annoying and just want to get to the equations as quickly as possible.

Approach (b) is guaranteed to work, if you push it hard enough.
I'm not sure that approach (a) /by itself/ is sufficient for really hard problems.

In any case, I strongly recommend using *both* approaches together.
The picture tells you what equations to write down, and the equations tell you what pictures to draw.

The grandmaster of this approach was Feynman.  He could visualize all sorts of stuff, and he also had formidable mathematical skills.
He used them both *together* not separately.  The famous "Feynman diagrams" are the perfect example of this:  each one is a picture of something happening in spacetime ... and each one corresponds to an integral that needs to be evaluated.

The "number line" approach Bob mentioned is isomorphic to the "rearrange the math" approach that I mentioned, i.e. 1 - 2/27 versus 1 - 2/25.

===========

The main problem with the visual approach is that visual communication is sometimes very expensive.
 ++ I can write down a four-vector equation in a few seconds.
 ++ I can visualize the corresponding spacetime diagram in my
  mind very quickly.
 -- Actually drawing the spacetime diagram so that somebody other
  than me can see it might take a couple of hours.  Ouch.

==================================

If you want to know how I really solved this problem....  I solved it three or four times in rapid succession.

1) At first glance, it was obvious that both numbers were close to  unity.  So write them as 1 - ε and 1 - δ.  This immediately makes  me think of Taylor series.  I'm not going to mention Taylor series  in front of high-school students, but you asked how *I* approached  it.

 I am accustomed to solving much harder problems than this, and I  like to collect /systematic/ approaches that work for more than  one problem.  The  Taylor series is overkill for this particular  problem, but I can apply the Taylor series in less time than it  takes to tell about it, so why not?

 If I'm talking to people who know about Taylor series, this answer  is super-easy to explain.  It's obviously correct.  For the more  devious approaches mentioned below, the proof of correctness is  more elaborate.

2) Then I thought of the number-line picture.

3) Then I immediately formulated that picture in terms of regrouping  the math.

4) Then I classified it as an example of a more general notion of  "regrouping", which is a tool that can be applied to a huge class  of problems.  The value of solving this particular problem is small,  whereas the value of learning the general-purpose tool is large.

 This generalization protects against the well-nigh-inevitable student  complaint that the question is not fair "because we never studied this". 
 Actually they should have studied it.  It's on the second- or third-  grade syllabus.  The educationalese term is "compensation" but that's  just jargon for any indirect, devious, outside-the-box approach.


On 12/17/2013 11:37 AM, Richard Tarara wrote:
> > I look at the percentage change of the numerator versus the
> > percentage change of the denominator.

That's clever.  That implicitly depends on the fact that the
logarithm is a monotone function.

It is characteristic of the clever solutions that they rely on
facts that might at first glance seem unrelated to the original
question, such as monotonicity of the logarithm, monotonicity of
the inverse, et cetera ... plus the not-entirely-obvious fact 
that monotone functions commute with inequalities.

This makes the solution hard to explain.  When we solve the problem,
these facts are ten layers deep in the subconscious.  It would be
an exaggeration to say we use these facts without thinking; rather 
we use these facts without /conscious/ thought.  I reckon 99.999%
of all thought is subconscious.
_______________________________________________
Forum for Physics Educators
Phys-l@phys-l.org
http://www.phys-l.org/mailman/listinfo/phys-l