Thursday, 22 January 2009

Patterns in Iso-perimetric Problems-1

Way back in my younger days I was a regional director for the Michigan Council of Math Teachers, and gave a lot of talks ... somewhere in that time I wrote some stuff about the type of optimization problems that mathematicians call iso-perimetric problems.. Often they involved a given perimeter (a farmer has 360 feet of fence).. and the object is to find the maximum area under some given conditinos (the area is a rectangle, etc..). Over the years I noticed some patterns to the types of solutions that often made the solution transparent with very little actual calculus or advanced math required. I wanted to write a couple of blogs to explain and, hopefully, organize my thoughts on them... but first, some introductory notes I wrote a long time ago about the problems for folks who are not familiar with the term isoperimetric, and to give a little history.

The word isoperimetric literally means "same perimeter". It is usually used in mathematics to refer to to figures having the same perimeter but different shapes. Mathematicians use the term Isoperimetric Problem to describe problems relating to finding which of two figures with equal perimeter have the greatest area. The problems date back at least to Zenodorus , a 2nd century BC Greek mathematician who wrote On Isometric (same measure) Figures. In it he showed that of all isoperimetric polygons having the same number of sides, the regular polygon had the greatest area. [For example of all quadrilaterals having a perimeter of 16, the 4x4 square has the greatest area.] He also showed that the area of a circle was greater than any regular polygon with the same perimeter as the circumference as the circle.

A related isoperimetric problem is called Dido's problem. Dido's problem is to find the maximum area for a figure with a given perimeter and bounded against a strait line. For example, if you had 100 feet of fence and wanted to enclose the maximum area with one side of the property along a straight river, or the side of a barn; what shape would enclose the maximum area.

Dido was a Phoenician princess in Virgil's Epic tale, THE AENEID. The story tells of her founding of Carthage. When she fled from her brother, Pygmalion, who was trying to kill her (sibling problems seemed pretty dramatic in the Greek classics) she landed along the coast of Northern Africa and tried to buy land from the local king, King Jarbas. She was told she could have as much land as could be enclosed by a bull's hide. Dido had her followers cut the hide into tiny strips which were then strung together to make a great length, and enclosed a great area against the sea. That's where the strait line on one side comes from in the problem. Dido's solution was a semicircle. (pay attention here, this is one clue about generalizing the solution of isoperimetric problems when one side is against a natural straight boundary)

Dido's problem is sometime called the Problem of Hengist and Horsa for a similar conclusion to an English story about two wizards/lords who come from Germany to defend against a Saxon invasion in the fifth century. The two turn out to be the antagonists of the young King Arthur and Merlin. Even Dido's story is related to England in a round-about way. Her brother Pygmalion would create a beautiful statue and then fall in love with it. Aphrodite brought it to life for him. This transformation of a stone statue found its way through a George B. Shaw play into the famous English musical, My Fair Lady

Perhaps one more small side-note to bring all this full circle to my interest in language, the model for Professor Henry Higgins in the play was a real live linguist named Henry Sweet who was instrumental in the development of hte International Phonetic Alphabet in 1897.

Ok, enough for now, but hopefully soon I can exaimine one of the patterns I have noticed. Maybe you can find it too... here is a guided discovery tour...

1) A farmer has 360 feet of fencing, and wants to build two congruent rectangular pens that share a common fence. Maximize the total area (this is a common textbook problem... ) and find the dimensions

2) step two is to do the same thing except one side of the area is bounded by a straight river.. so that side does not need to be fenced...

3) step three is to extend the original problem to three congruenct rectangular pens

4) Now there are two ways to arrange four pens, in a row or in a 2x2 square... try both ways...

Now look at all the answers and pay close attention to the dimensions... I will return to this soon.. so work fast...