Recursive patternsAuthor(s): Lisa BrooksSource: The Mathematics Teacher, Vol. 93, No. 4 (April 2000), p. 265Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27971374 .

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Reader Reflections

Recursive patterns This year, I am teaching geometry for the first time. A few years ago, while I was student teaching, I came across an article that caught my eye. The article, "Spiral through Recursion," by Jeffrey M. Choppin (Mathematics Teacher, October 1994,504-8), discussed one of the topics that I touched on in my class this year. Mer teaching the students about recursive patterns, I gave them an opportunity to develop their own pattern or create a variation of one of the patterns given in the article. The students submitted some outstanding artwork. I have enclosed photo graphs of magnificent patterns.

I find that my students absolutely love the lessons that I teach that come from ideas that I find in the journal. In fact, they are currently working on building a Sierpinski pyramid similar to the one built by the students from Anoka, Minnesota ("Build a

Sierpinski Pyramid," by Paul Kelley, Mathematics Teacher, May 1999,384-86). It has become an across-the-curriculum

project that involves geometry students, advanced-physics

students, CAD I students, and the art and computer departments.

The Mathematics Teacher motivates my students by pre senting peers who are also doing unique things in their study of mathematics. My students and I hope that our work will inspire other high school mathematics classes.

Lisa Brooks Ridgway Area School District Ridgway, PA 15853

Arithmetic mean =

geometric mean? In "Reader Reflections: AGM inequality" (September 1999, 461), Nancy Antonellis cleverly employs the arithmetic-geometric mean inequality to find the mini mum surface area for a cylinder with a given volume. The success of this calculus-free method hinges on her contention that the arithmetic and geometric means of three quantities a, b, and c will be equal only if a = b = c.

Although this statement is true for positive numbers, the only necessary concern in dealing with physical dimensions, it is certainly not always true if that restriction is removed. For exam

ple, when a = -1, b = -1, and c = 8,

Monte J. Zerger Adams State College Alamosa,CO 81102

Antonellis responds: I had not considered negative numbers, because the context of the problem is concerned with physical dimen sions; therefore, o, 6, and c must be positive. In researching a

proof of the AGM inequality, such as the one found in I. M. Gelfand and Alexander Shen's Algebra (Boston: Birkh user Boston, 1993), I found that the assump tion that the quantities being compared are nonnegative is part

of the conjecture that

2

Thank you for pointing out this important assumption.

Circles and regular polygons with a common center The areas in an alternating series of circles and regular poly gons with a common center bear an interesting relationship.

A figure involving circles and squares is constructed by enclosing a circle in the smallest possible square, which is in turn enclosed in the smallest possible circle, which is enclosed in the smallest possible square. See figure 1 (Roof).

Fig. 1 (Roof) Circles and squares with a

common center

In this configuration, the area of any circle is twice the area of the next smaller circle, and similarly, the area of any square is twice the area of the next smaller square. For the circles,

,. na2 area ratio = -

Kb2

a2cos2(45 )

"(if = 2,

where a is the radius of the larger circle and b is the radius of the smaller one.

For the squares, simply replace n with 4 and obtain the same numerical value.

If the squares in the figure are

replaced by equilateral triangles,

the area ratio both for circles and for triangles has the value of 4.

For regular polygons having n sides, this area ratio is given by

For example, the ratio for hexa gons is 4/3. As the number of sides increases, the polygon comes closer to being a circle, thus causing the area ratio to decrease and approach the value of 1 as the limit.

Jack G. Roof jchemroof@sprintmail.com Poulsbo, WA 98370

1

Meeting the challenge Iii "Reader Reflections: Can you

meet the challenge?" on pages 626-27 of the October 1999 issue of the Mathematics Teacher, Johannas Winterink reports on a

geometry problem presented in the Washington Post. The story indicates that Gary Gruber thought that no one in the country could solve the problem.

Sorry, but this well-known problem, or a variation, is called Hansen's problem. In feet, a slightly different version appeared in Monte Zerger's article "Sharing Teaching Ideas: Hidden Treasures in Students' Assumptions,'' which appeared in the October 1993 issue of the Mathematics Teacher on pages 567-69.

This problem also appears in Challenging Problems in Geometry, by Alfred S. Posamentier and Charles T. Salkind (New York: Dover Publications, 1970-1988), on pages 30- 31 and 149 -54. The seven solutions given include both geometric and trigonometric ones. Every geometry teacher should have this delightful book. I believe that I also saw the problem in Crux Mathematicorum about twenty or twenty-five years ago.

This geometry problem is a nice one that I had fun solving many years ago.

Larry Hoehn hoehnl@apsu.edu Austin Peay State University Clarksville, TN 37044

The area of a quadri lateral--a shortcut Each year, Indiana sponsors an academic competition called the

(Continued on page 317)

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Vol. 93, No. 4 .April 2000 265

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