scaling laws

["John S. Denker" <jsd@AV8N.COM>]

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,....."

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).

Students oftentimes come in with severe
misconceptions.  In particular, they know that
if you make a *square* twice as large, it will
have four times the area ... but they commonly
think that squares are special in this regard.
In particular, many of them are convinced that
if you make a triangle twice as large, it won't
have four times as much area, because it's not
a square.  Maybe only three times as much area.

So I draw for them a double-sized triangle and
show that it really does contain *four* unit-sized
triangles.  They're shocked.  They think it's a
magic trick.

This is illustrated at:
  http://www.av8n.com/physics/img48/scaling.png

Continuing down this line, a double-sized hexagon
can be seen to have the same area as *four* unit-sized
hexagons.  This is one step trickier than the
squares or triangles, because you have to cut and
re-arrange the hexagons, but it's not totally
incomprehensible.

After you've done the triangle and hexagon, usually
they'll take your word for it that the idea generalizes
to arbitrary figures, even irregular figures.

For truly irregular figures, lots and lots of
cutting and re-arranging is generally necessary.
There's a whole subfield of mathematics devoted
to this, starting with Archimedes and still active
today.

For homework, have 'em sketch a triangle, a square,
and a hexagon each *three* units on a side, colored
in to show the correct number of unit-sized figures.

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

This is important physics.

In 1638 some guy published a paper called "On Two
New Sciences".  One of these new sciences was the
laws of motion.  The other was scaling laws.

The important thing about the scaling law for a
square is not that it has four corners, but that
it has spatial extent in two directions.

There are some interesting nontrivial scaling laws
in modern physics, e.g.
  http://www.nobel.se/physics/laureates/1982/press.html