Math Is Fun Forum

model

Member

Registered: 2011-08-10

Posts: 142

compactness and non- compactness of objects

What does mean by compactness ?

There  is image : http://imageshack.us/photo/my-images/190/42484776.png/

In the image , 1st figure shows the  compactness whose value is less may be 1 or 0.754 some thing .

Note : according to the  1st  figure as circle  presented in the image .
that shows more thickness and compact  and while in the 2nd figure , it shows less compact and more thicknesss

What does that mean actually . Compactness is based on what thing ? ?

Please clear this concept thanks

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TMorgan

Member

Registered: 2011-04-13

Posts: 25

Re: compactness and non- compactness of objects

I believe the cocept of compactness is related to what in science we would call density. In those 2 figures if you assume that the area enclosed by the 2 shapes is the same then the first manages to enclose the same amount of area in a smaller enclosure. In 2 dimensions the circle encloses the most area in the smallest length of perimeter. In 3 dimensions the sphere encloses the most volume with smallest area of surface. If there is a quantitative measure of compactness I would think that the first figure would have a value of 1 and the second would be less than 1.
  Suppose you had a bunch of items of different sizes and shapes you had to send through the Post Office. It would make sense to figure out how to arrange them to fit in the smallest box possible, since the Post Office charges more for a bigger box. You would be trying to maximize compactness.
  Have you seen the puzzles where a 3 X 3 cube is cut into 27 smaller cubes, but some of the smaller cubes are stuck together to make differently shaped pieces? The trick is to figure out how to put them back together as a 3 X 3 cube. That would be the most "compact" solution. If you don't get it right the pieces add up to something bigger than the 3 X 3 cube. I think the original puzzle was called Soma, but now it is copied all over.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: compactness and non- compactness of objects

hi,

Have a look at

http://en.wikipedia.org/wiki/Compactnes … of_a_shape

Bob

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Children are not defined by school ...........The Fonz
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