A 'Conjecture' - And A Quote From Arnol'd
I have been pondering this problem for a while:
Given any 2d convex shape (say a circular disc). We need to divide it into N pieces of equal area so that the length of cuts is least (in other words the sum of the perimeters of all the N equal area pieces is the least possible).
Here is a Claim (or a conjecture): For *any* convex shape to be partitioned and any positive integer N, *all* pieces (they need not be identical in appearence to each other) with minimum total perimeter must be necessarily *convex*.
If this claim is true, we could immediately have the following corollaries:
1. The lines which cut the target convex shape into the equal area
pieces with minimum total perimeter are all straight lines.
(even if the shape to be partitioned is not convex, I doubt if any of the cuts in the least total perimeter partitioning of it into N pieces can be 'non-straight'. at least I have not yet found any specific example)
2. For the case of dividing a *circular region* into two (ie. N=2) equal area pieces with minimum total perimeter, a diameter is the cut.
----
I am told this problem could be related to Plateau's problem and other problems from variational calculus. It could also be related to concepts in computational geometry such as 'minimum ink partitioning'. I am yet to find any concrete leads though...
----
To sign off this post, here is a quote from Vladimir Arnol'd:
-"Mathematics is a part of Physics. Physics is an Experimental Science, a part of Natural Science. Mathematics is that part of physics where experiments are cheap."
The article from which the above was extracted may be read here .
Given any 2d convex shape (say a circular disc). We need to divide it into N pieces of equal area so that the length of cuts is least (in other words the sum of the perimeters of all the N equal area pieces is the least possible).
Here is a Claim (or a conjecture): For *any* convex shape to be partitioned and any positive integer N, *all* pieces (they need not be identical in appearence to each other) with minimum total perimeter must be necessarily *convex*.
If this claim is true, we could immediately have the following corollaries:
1. The lines which cut the target convex shape into the equal area
pieces with minimum total perimeter are all straight lines.
(even if the shape to be partitioned is not convex, I doubt if any of the cuts in the least total perimeter partitioning of it into N pieces can be 'non-straight'. at least I have not yet found any specific example)
2. For the case of dividing a *circular region* into two (ie. N=2) equal area pieces with minimum total perimeter, a diameter is the cut.
----
I am told this problem could be related to Plateau's problem and other problems from variational calculus. It could also be related to concepts in computational geometry such as 'minimum ink partitioning'. I am yet to find any concrete leads though...
----
To sign off this post, here is a quote from Vladimir Arnol'd:
-"Mathematics is a part of Physics. Physics is an Experimental Science, a part of Natural Science. Mathematics is that part of physics where experiments are cheap."
The article from which the above was extracted may be read here .
posted by R.Nandakumar @ 5:33 AM
5 Comments:
At 1:48 AM,
Two Minds said…
No take on the Geometrical problem; you know very well of my geometrical inabilities.
I fdo take Vladimir head on: his statement is WRONG. Physics lead to, and needs, a lot of Math doesn't imply any containment relations. In what way is Number Theory is contained in Physics?
At 3:15 AM,
R.Nandakumar said…
vandamme,
as far as i can make out, arnol'd is only *reacting* to the marginalization of geometry and physical thinking in the practice of mathematics; he is trying to correct a skewed picture by applying the opposite skew transformation to it. the article is intentionally biased and polemical; but he does seem to have a point - algebra and analysis, of an overly 'manipulative' kind have been overemphasized to the near-exclusion of geometry even in our own school and college math.
At 9:34 AM,
Anonymous said…
I have no idea about your problem, but it is definitely interesting, as I study certain convex bodies as part of research.
Did you prove your conjecture?
Off-topic, I 'killed' Vishnulokam because it was taking a hell lot of time. I am trying to find more time for things, and this was one of them. I still continue to read blogs.
At 8:30 PM,
R.Nandakumar said…
vishnu,
thanks for your interest.
the conjecture is NOT true. you may want to amuse yourself by trying to find a simple counter-example :) - a case where the equal area fragments with min total perimeter are not all convex.
i shall be making a new post (within the next 48 hours or so) with pointers to a reference, with a counter example.
At 5:46 AM,
Two Minds said…
Our curriculum is aimed at making human machines for the industries. These maanipulative skills go a long way in the industry. One does look at how maipulative one is in arriving at a solution, rather than what picture (geometry) one has.
Though bad, that IS life.
Post a Comment
<< Home