## Frank Morgan's Math Chat |

April 6, 2000

**ANSWER** (see Figure 1). Suppose A is less than or equal to B. If A equals B,
the rectangles are identical and adjacent, a third longer in the direction
of the shared wall, as rediscovered last week by Ms. Mattuck's 8th graders
at the Richmond School using Geometry Sketchpad. As A decreases down to
B/2, the dimensions change until the larger region is square and the
smaller region is only half as wide as tall. As A decreases below B/2, the
shapes remain similar but the smaller rectangle shrinks up the side of the
larger square. Incidentally, when A decreases below about B/5.34, the
smaller region would prefer to jump into the corner of the larger square.

*Figure 1. The most efficient way to fence in two prescribed areas goes
through three types as the smaller area A decreases.*

These results were proved in a more general context by college students Christopher French, Scott Greenleaf, Brian Wecht, Kristen Albrethsen, Charene Arthur, Heather Curnutt, Christopher Kollett, Megan Barber, and Jennifer Tice. Then they took electron microscope photos of table salt and found some similar behavior, probably for similar reasons of minimizing perimeter or energy (see Figure 2). I just described these results Saturday at the Lowell AMS meeting.

**NEW CHALLENGE.** John A. Shonder of the Oak Ridge National Laboratory notes that
"engineers are often interested in the average temperature over a day,
because the amount of energy used in a building (and by extension, in an
entire city) usually correlates well with daily average temperature. The
National Weather Service reports what they call "daily average temperature"
for various locations around the U.S., but this temperature is really just
half the sum of the daily high and low temperature. At
first glance, you wouldn't think this would be a very accurate way of
approximating the average. Who ever heard of approximating the integral
f(x) over some interval by taking half the sum of the maximum and
minimum values of f(x) on the interval, and then multiplying by the length
of the interval? Yet that is exactly what the National Weather Service
does. And what's more, it's a good approximation! I have some temperature
data collected at 15-minute intervals over a three year period for a site
in Louisiana. Using this data I was able to compare the actual daily
average temperature (the average of 96 temperature measurements) with the
average of just the high and low temperatures. Over a three year period,
the standard
deviation of the difference between these two averages is less than 2
degrees Fahrenheit.

My question is, why is this such a good way of approximating daily average temperature? Is there a mathematical explanation for it?"

**QUESTIONABLE MATHEMATICS.** Dan Ullman reports that music conductors refer to
the second bar as "one bar after A," but the tenth bar as "ten bars after
A." Ullman notes that "this is clearly inconsistent. At some fuzzy point
(between 1 and 10?) the interpretation changes."

Readers are invited to send in more examples of questionable mathematics.

Send answers, comments, and new questions by email to
Frank.Morgan@williams.edu, to be eligible for* Flatland *and other book
awards. Winning answers will appear in the next Math Chat. Math Chat
appears on the first and third Thursdays of each month. Prof. Morgan's
homepage is at www.williams.edu/Mathematics/fmorgan.

THE MATH CHAT BOOK, including a $1000 Math Chat Book QUEST, questions and answers, and a list of past challenge winners, is now available from the MAA (800-331-1622).

Copyright 2000, Frank Morgan.