Re: scaling laws

[Hugh Haskell <hhaskell@MINDSPRING.COM>]

At 11:28 -0400 10/10/03, John S. Denker wrote:
> At 10:57 -0500 10/9/03, Rick Tarara wrote:
>  >>
>  >>.... " Start with some shapes for which areas or
>  >>volumes are readily calculated (squares, cubes, circles, spheres)
>  >>and have them figure out the ratios of areas, volumes, etc., for
>  >>different sizes, then extend it to irregular shapes,....."

Actually, that was me who said that.

> Yes, good.
>
> On 10/09/2003 10:15 PM, Herbert H Gottlieb wrote:
> >  *** Are your students able to figure out the ratios of volumes
> >  for irregular shapes.  I'm having a bit of trouble trying to
> >  figure it out myself. Please explain how this is done.
>
> For truly irregular figures, I doubt there is
> a simple way to do it.
>
> But perhaps we can compromise and discuss scaling
> laws for figures that are more complicated than
> perfect squares (but not arbitrarily irregular).
Since the question has been raised, let me expand a bit on my overly
brief comment. It is not hard to show that (for simplicity, I will
limit the discussion to plane surfaces), if you double every
dimension in some arbitrary plane surface, that you increase the
surface area by a factor of four--a new figure that looks just like
the old one, but occupies four times the area. If you look at
figuring out the area by dividing it up into squares, and "counting
squares," then it is relatively easy to see that the area can be
expressed as some constant, which is dependent on the shape, and some
linear dimension which is dependent on the size, squared (or in some
cases, two different linear dimensions multiplied together). In
mearuing the area by counting squares, I have my students use three
different rules for counting those squares which, by virtue of
intersecting the boundary, are incomplete. 1) count all squares, as
long as some part of it is within the figure; 2) only count those
squares that lie entirely within the boundaries; and 3) estimate the
fraction of any partially contained square within the boundary.
Clearly, methods (1) and (2) put upper and lower bounds on the
estimate of the area. Decreasing the size of the squares will reduce
the difference between (1) and (2), although at the price of a whole
lot more work.

It doesn't take much to see that repeatedly reducing the size of the
squares will reduce the uncertainty in the answer by making the
results of (1) and (2) ever closer together, and that eventually all
three methods will converge toward a single number, which we can take
to be the answer.

Of course the same method works for three-dimensional objects.
Calculating the surface area of some non-planar shapes might be quite
difficult by this method, but should be doable in principle. But it
should become clear that, even if not rigorously proved, the volume
of a three-dimensional object can be expressed at some (shape
dependent) constant times the cube of a length which is
characteristic of the object, and that the surface area of this
object can be expressed as some other constant times the square of
the same length. The immediate conclusion is that the volume of a
given shape increases faster than the area. In fact, the ratio of the
two (volume to area) increases and some new constant times the same
length. Which immediately answers why  an elephant's heart doesn't
have to beat as fast as a mouse's.

Yes, there is lots of interesting and important physics in scaling
laws (for instance, why does the power required in a radar signal
increase as the 4th power of the distance to the desired target?).
Taking time to teach about them is valuable, and will pay big
dividends later. But, it is always best enclose instruction about
scaling in real problems, like the mouse/elephant one, or why you can
see someone else's flashlight before they can see you in its light (a
variant of the radar problem), or why you can see the eyes of that
transfixed deer on the road ahead of you long before you can see the
deer itself, or how you can figure out the energy radiated from the
sun (and hence its surface temperature), by measuring the radiation
that reaches a single spot on the surface of the earth. Or even, why
is the night sky dark (which really isn't about explaining the
darkness, but about explaining why it isn't as bright as the sun, as
might be expected by area scaling)?

A slightly different problem, involving relative motion, but nicely
explainable in terms of scaling, is why two ships (or airplanes, or
cars) can be interpreted to be on a collision course if their
relative bearings from each other do not change with time (but the
apparent size of the other seems to be increasing).

I also heartily endorse John's suggestion that more than one way to
solve a problem be sought. That is especially true if one isn't well
satisfied by the first method found. If a second, independent one can
be found that gives the same result, confidence in the solution is
increased.

And this isn't true just of "solving problems." In all of science,
confidence in theoretical explanations are enhanced when approaches
from different disciplines lead to the same conclusions. Confidence
in speculative results can be enhanced if other investigators
approaching from different perspectives can come to the same
conclusions.

Finding multiple solutions to a problem is definitely a skill worth developing.

Hugh
--

Hugh Haskell
<mailto:haskell@ncssm.edu>
<mailto:hhaskell@mindspring.com>

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