A simple and useful discussion along these lines has to do
with shapes growing and shrinking. It's a good place to
start with your discussions because here the growth factor
is 1 -- you can focus on the different exponential rates
of change without any constant multipliers.
When a shape doubles in all its linear dimensions (i.e.
all linear measure twice as long), it's surface area goes
up by 2^2 (2nd power) and volume by 2^3 (3rd power). In
general, surface area is a 2nd power of linear growth and
volume a 3rd power.
Therefore the ratio of surface to enclosed volume is not
a constant (decreases linearly). An adult human has a
Body Surface Area (BSA) enclosing relatively more volume
per increment than a child human. This effects things
like heat dissipation -- why small birds need such a
high metabolism, in part, because a tiny body wraps
very little volume and therefore has a high exposure
rate in terms of surface area -- lots of heat escapes.
Scale models are also therefore not always reliable
predicators of behavior of full-scale counterparts in an
engineering sense, because of linear-surface-volume rate
changes. And a praying mantis the size of some sky
scraper (like in a science fiction movie) would probably
snap and fall into a pile under its own weight -- volume
goes up as a 3rd power which means so does weight
(assuming a constant average density e.g. same materials
and internals).
Also relevant:
Show how angles define shape, including central angles. Like
you can reduce a complicated form (e.g. sewing machine) to a
database of points with angles (surface and central). This
specifies the shape completely, but does not give the size
(relative to something else).
Relate growth the "frequency" versus "angle" (terminology)
and show how a tetrahedron growing and shrinking has a
surface and volume of F^2 and F^3 where F is the number
of edge modules (e.g. 1 inch or 1 cm increments along a
side). This demo is usually done with cubes (e.g. in
Keith Devlin's "Life by the Numbers" -- companion volume
to the PBS series of the same title) but here is your
opportunity to show how "squaring" and "cubing" as terms
for 2nd and 3rd powering are cultural, ethnic, and a
self-consistent and more topologically minimal modeling
of 2nd and 3rd powering exists in the tetrahedron.
Kirby
Further reading:
http://www.teleport.com/~pdx4d/terms.html#freq gives more
philosophical background re angle vs frequency distinction
(a part of the metaphoric language of synergetic geometry)
BSA (body surface area) is used in medicine to compute
dosage of some drugs (is a measure of the total circulatory
system and therefore blood volume, with some infusions
being a % of this quantity). A couple formulae exist,
taking patient height and weight as input. Uses log.
Fun excercise, provided calculators present (have kids
computer their own body surface area, convert to square
feet or square meters).
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