Re: [Math Future] the "power law" (linear, areal, volumetric :: 1 : 2 : 3)

[Rakesh Biswas]

Thanks Kirby for sharing this amazing imagery to explain the topic. I wish most of us could make explicit what our limited cognition compels us to keep implicit. :-)

Also sharing this with other physicians who would be interested to learn more. 

best, 

rb

On Sun, Jul 17, 2016 at 11:38 PM, kirby urner <kirby...@gmail.com> wrote:

When rising through the ranks (grades) in elementary school,

I can't remember ever having it clearly expressed to me, the

clear thought that:

Given any mundane random shape, such as a sewing machine,
if we scale it up and and down, the surface area will change
exactly as a 2nd power of the change in linear inter-distances
(as between any two points in a straight line), and the volume
will change exactly as a 3rd power.

Say we have:

>>> machine = SewingMachine()

>>> machine.scale
1

>>> machine.area
17.2839238

>>> machine.volume
89.4883934

The area and volume numbers are completely out of a hat,

in whatever units this ethnic group uses for area and volume
(square-shaped for area, cube-shaped for volume, is a popular
option, though triangles and tetrahedra are topologically simpler).

Digression:

Here I'd pull away from the numbers and talk about model

railroads and tiny towns built to scale.  What's HO scale again?

Raise your hand if you've been to Disneyland.  What's the scale

there, on Main Street, anyone know?

We all have toy cars and dolls and stuff right?  Forget that

Barbie would be scary if human-sized, lets just talk about the

linear height of a Barbie, and what the linear growth factor

would need to be, to make her 5 foot 9 inches.

Teacher Notes:

Have students thinking about how plastic toy dinosaurs are

scale models and therefore "similar" in shape.  Shape may be

conveyed in terms of surface and central angles, independently

of linear units or linear dimension.  Given the angles, we get

the shape, independent of size.

Realization:

Given area changes as a 2nd power and volume as a 3rd power,

when we scale up the sewing machine by 7.569, we don't have to
do anything to recompute the area or volume from scratch.

The new area will be:

17.2839238 * 7.569 * 7.569

and the new volume will be:

89.4883934 * 7.569 * 7.569 * 7.569

More generically, given S is the scale factor:

machine.Area * S**2

machine.Volume * S**3

are the new volumes (** means "raised to the power of").

Another Realization:

Since A/V as a function of S is not a constant, i.e. we have a

way to work backwards from this ratio to figure out what the
scale factor S was.

Teacher Notes:

Even tiny changes in scale may be of interest.  In Synergetics

we have a very minute volume difference between the T and the

E modules.  Their linear dimensions are 99.9+% the same, and

yet they merit different names, despite being exactly the same

shape, having all the same angles.

http://www.rwgrayprojects.com/synergetics/s09/figs/f86548.html

(about the very fine difference between T and E)

http://www.rwgrayprojects.com/synergetics/s09/figs/f86561.html

(this is where the Koski stuff connects -- building with mods that

phi-scale up and down, i.e. S = phi)

The rhombic triacontahedron of 120 E modules precisely encloses

a unit radius ball (like a ping pong ball), whereas the RT of 120 T

modules has tiny holes in the face centers, where an itty bitty bit

of sphere surface bulges through in all 30 diamonds.

http://www.rwgrayprojects.com/synergetics/s09/figs/f86419.html

(shows RT with all faces divided into four giving 120 modules)

http://www.rwgrayprojects.com/synergetics/s09/figs/f86405.html

(show in figure B. an RT with a ball inside)

Picture the pitcher's mound on a baseball field, but with the bases
in a rhombic (diamond) pattern and much further apart so the bulge
is relatively even shallower. 

The contained sphere barely bulges at the center of each diamond.

A baseball diamond is called a diamond, but is also just a square:
https://youtu.be/2n_tfg7hico

The RT diamond has long:short diagonal ration of Phi:1.

The T module has exactly the same volume as the A and B mods

of 1/24 (vis-a-vis the tetrahedron with edges = length of ping poing

ball diameter).

http://www.rwgrayprojects.com/synergetics/s09/figs/f86161.html

(unit edge of tetrahedron)

Kirby

From the archives (distant past):
http://mathforum.org/kb/message.jspa?messageID=124546

(dialog with someone not liking the nomenclature "A and B

modules").

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