Re: [Phys-L] graphing (was: pendulum)

[John Denker <jsd@av8n.com>]

On 3/5/25 04:01, Anthony Lapinski via Phys-l wrote:
> Since length is the independent variable, it should go on
> the y-axis - plot length vs period squared

YMMV, but from my point of view it could not possibly matter
which quantity goes on which axis.

I realize that other people are of the opinion that life is
simpler for ultra-unsophisticated students if the horizontal
axis is always X and X is always the independent variable
... but that simplification comes at a terrible cost. It
will have to be unlearned, sooner rather than later, and
unlearning is always hard. I'm not persuaded that it helps
anybody to make up "rules" or "traditions" that have no
basis in reality.

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

I'm all in favor of keeping things simple in when a subject
is being introduced. The first step is the first step. Even
so, you want the first step to be in the right direction,
so here, amongst experts, it's worth discussion which way
the path has to go.

Consider for example a combination of functions: g(f(x)).
The output of f is the input to g, and you can illustrate
this nicely if you rotate the graph of g 90°. Phil Keller
wrote an introductory-level book that does this. Last time
I checked (which was a while ago), compound graphs show
up on the standardized tests.

Here's an even simpler example: Suppose you want to plot
the temperature of the atmosphere (or the speed of sound,
or the wind speed, or anything else) as a function of
height. We're talking about the actual geometrical physical
height, and it makes sense to plot that "vertically" on
the graph. Like this:
 
https://www.eoas.ubc.ca/courses/atsc113/flying/met_concepts/03-met_concepts/03a-std_atmos/images-03a/std-atmos-temperature-color.png

In this case, the vertical coordinate is clearly not a
function of the other coordinate ... but that's totally
fine.

For a /linear/ relationship, it is super-extra immaterial
which quantity goes on which axis. It's six florins versus
half-a-dozen guilders.

Having made thousands of graphs, I have some pretty strong
opinions about what's important and what's not. Consider
for example the writeup I posted a couple days ago:
  https://av8n.com/physics/ne-asteroid-groups.htm#sec-belt
Initially I plotted Q versus q, but I flipped it to become
q versus Q, simply because that fits better on a typical
screen. The was absolutely no change in meaning. Also note
that the second (more complicated) version of the diagram
has *five* different axes on a single two-dimensional plot.
  Q = aphelion
  q = perihelion
  e = eccentricity
  2a = Q+q = major axis
  T = period

In fluid mechanics it gets even more complicated. At each
point in three dimensions we have /at least/:
  temperature
  pressure
  density
  entropy density
  energy density
  three components of momentum density
  ... et cetera ............

Then there are ternary graphs, which are intrinsically
triangular, giving a two-dimensional representation of
something with three variables and one constraint. For
example:
  chemistry or mineralogy with three concentrations
  electoral politics with three candidates
  ... et cetera .....................

At some point you come to the realization that the axes,
strictly speaking, were never important. What matters
are the /contours/ of constant X, constant Y, and on
the ternary graph, constant Z. In special relativity
(not to mention general relativity) and also in
thermodynamics, very commonly the contours aren't
straight and/or aren't mutually perpendicular. This
is discussed in detail here:
  https://www.av8n.com/physics/axes.htm

You may have noticed that every graph I've created in
the last 30+ years shows the contours, not just the
axes. Some plotting packages do this for you by default;
others will do it upon request. It's OK if the contours
lines are thin and unobtrusive, but IMHO the really
ought to be there.