Episodes in nineteenth and twentieth century euclidean geometry pdf

Trisecting the Circle: A Case for Euclidean Geometry Author(s): Alfred S. Posamentier Source: The Mathematics Teacher, Vol. 99, No. 6 (FEBRUARY 2006), pp. 414-418 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/27972006 Accessed: 09-02-2018 18:19 UTC

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DEL m er Alfred S. Posamentier

Trisecting the Circle: A Case for Euclidean Geometry

Editors' note, part 1: The paper that Alfred Posa mentier submitted to "Delving Deeper" was simply entitled "A Case for Euclidean Geometry" and con tained a number of lovely examples of problems and ideas illustrating the variety of rich mathemat ics one can encounter from a start that is "just" Eu clidean geometry. We shamelessly chopped the paper into pieces and are publishing just one of them?coincidentally, roughly a trisection of his paper. We are holding the other parts for later and are hoping, also, for contributions by other authors who find Euclidean geometry a fertile source of

This department focuses on mathematics content that appeals to secondary school teachers. It provides a forum that allows classroom teachers to share their mathematics from their work with students, their classroom investigations and projects, and their other experiences. We encourage submissions that pose and solve a novel or interesting mathematics problem, expand on connections among different mathematical topics, present a general method for describing a mathematical notion or solving a class of prob lems, elaborate on new insights into familiar secondary school mathematics, or leave the reader with a mathematical idea to expand. Send submissions to "Delving Deeper" by accessing mt.msubmit.net.

Edited by Al Cuoco, acuoco?edcorg Center for Mathematics Education, Newton, MA 02458

E. Paul Goldenberg, pgokSenberg^edcorg Education Development Center, Newton, MA 02458

mathematical ideas that go beyond here's-yet-an other-odd-theorem.

As an undergraduate mathematics major, a prospective teacher usually takes at least one geometry course. Typically, these courses

focus on non-Euclidean geometry (sometimes pre sented as Modern Geometry), or vectors, transforma tions, or topology. Instead, we at the City College of New York offer a course on more advanced Euclid

ean geometry in which prospective teachers investi gate a plethora of geometric theorems (or relation ships) that enrich their understanding of Euclidean geometry and, consequently, their teaching of it.

Encountering such a wide range of geometric theo rems provides more than breadth; the connections among them give teachers an opportunity to delve more deeply into questions that the typical high school student would ask. Our students are pretty adept at dividing a circle into two equal parts, but to divide a circle into three equal parts might be a bit more chal lenging?beyond the obvious method: the pizza cut.

Let's take a look at a few of these (see fig. 1). As we consider the task of trisecting a circle, here are just four possible methods. Each one uses some nice geometric relationships.

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Fig. 1 Four ways to cut a pizza in thirds

FOUR WAYS TO CUT A PIZZA INTO THIRDS Traditional pieces: Figure 1a Somebody who would actually take a protractor to a pizza restaurant would draw laughs from others but would be able to draw the radii that would divide the

circle into three equal parts (fig. 2), each of which is

360? a = ?= 120?

3

?if the cheese cooperates. Wrapping a piece of string around the circumference of the pizza would also help. Dividing the string into three parts would let us mark the circumference in thirds:

h_C _2nr ~3~ 3 Another way would be to take a stick and measure the radius, mark that radius off six times around the circle, and note every other marked-off point to get the three equally spaced division markers. Everybody gets an equally big slice (sector).

But there are intriguing alternatives to the traditional.

Using concentric circles to get three equal areas: Figure 1b Of course, this concentric division is hardly suit able in a restaurant; yet from a mathematical point of view, this version is quite interesting.

Without any loss of generality we can assume the radius of the initial circle to be 1. We must deter

mine the two radii rx and r2 so that the three colored

regions shown in figure 3 will be equal in area. Because the outermost circle's radius is 1, its

area is . The problem requires us to make the area of each colored region (the innermost yellow disk and the blue and red annuii) a third of that. The radius rx of the circle that bounds the yellow disk must be

Fig. 2 Partitioning a circle into three equal sections

Given: radius of the largest circle = 1 Sought after: and r2 to make the three areas equal

Solution: r1

Fig. 3 Concentric circles method

and yellow) must then be 2/3 of the area of the out ermost circle, or

2/r 3 '

so r2 itself must be

I The combined area of the inner two regions (blue

Another way to express the three radii is

Vi V2 ,S V3 V3 V3

Vol. 99, No. 6 ? February 2006 | MATHEMATICS TEACHER 415

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Or one can rationalize the denominators, but that makes this relationship somewhat less obvious:

S /6 3 ' 3 '

Using the teardrop shape: Figure 1c This curious trisection of the area begins with semicircles built on a trisection of the diameter.

Figure 4 shows how this will be divided into (semi-) circular arcs. We derive the solution by noting first that the

semicircles described by rx and r2 fit perfectly within the outer circle, and therefore the sum of their radii must equal the outer circle's radius: rx + r2 = 1. We want values for and r2 that make the area of the green teardrop the now-familiar /3. The equation we want is built from four facts:

by , we get

Area of the outer circle:

Area of the circle built on r? Area of the circle built on r2: Area of the green teardrop:

Art =-A t green ? outer 2

Dividing both sides of

^outer ~~ 7 ?

Aj = /TTj2 A2 = 7cr22

1 1 + -A, = 2 2 3

1 1 2 - ?7zt + 2 2 1 2

1 2 1 2 3

Given: radius of the outer circle = 1

Sought after: rx and r2, so that the congruent green and yellow teardrop regions are each one-third of the area of the circle

2 1 Solution: r, = ? and r, = -

1 3 3

which simplifies to

Since rl + r2 = 1, we have

r ? r = ?. 1 2 3

Combining that with rx + r2 produces

1

Because the yellow teardrop can be constructed in the same way, it too has an area of one-third of the circle, leaving one-third of the total area to the S-shaped middle portion.

Parallel lines partition: Figure 1d If one cannot find the midpoint of the pizza (see the first method), cutting parallel pieces (fig. 5) of fers us another approach, but it will become more difficult than expected.

We want each band in figure 6a, the two side blue bands and the middle yellow band, to have the same area, /3. Finding the place to cut the circle involves deriving a or d from the requirement that

we are placing on the areas of the three bands; but none of these is a shape for which we have ready made area formulas, so we need to dissect the circle differently. The area of a sector of the circle (fig. 6b) is easily stated in terms of a:

a A a a A., , = ? -A. . = ? ? = ? mue sector ?^ cacle ?^ ?

so we will pursue a to see if it leads conveniently to a solution.

The area we want, the blue band at the right side of the circle, is now easy to describe (fig. 6c):

Fig. 4 The teardrop method ^blue side "^blu it green triangle

Given: circle with radius 1

Sought after: a or ?

Solution: a ? 2.6 radians or 149.3?

Fig. 5 Parallel lines method

416 MATHEMATICS TEACHER | Vol. 99, No. 6 ? February 2006

Writing the area of the green triangle as

. a a sma _ ^ . = sin?cos? =-, it green triangle ? ? 2

a-sma

we can conclude that

Alue side

We know what we want that area to be:

a-sina 3 " 2

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^^^^^^^^ ^^^^^^^^^ ^ '^ ^^^^^^^^^ Jj^^ The plan, if only we knew The area of the blue sector is The blue area that we wish angle a or distance d from easily stated in terms of angle to set to /3 is the area of the

the center a, so we'll work with that. sector minus the area of the light green triangle.

(a) (b) (c) Fig. 6 Parallel lines method continued

This leaves only the problem of solving for a. This transcendental equation cannot be solved in

the traditional way. As an approximation, we can use a pocket calculator and get a ? 2.6053256746 (radians), which we can convert, if we like, into degrees: ? 149.274?. To make the drawing, it is convenient (though

not necessary) to go one step further and compute

d = cos- ?0.264932, 2

which, for the purposes of a drawing, is about a quarter of the radius.

We have thus divided the circle into three

equal parts in four different ways. Can you find another method for trisecting the circle? Suppose you were to construct two circles in a given cir cle, each having an area of one-third of the origi nal circle. Along with the remaining area of the original circle, would you not have then divided the circle into three equal areas? Such an exercise might get students to think about other methods and engage them in a genuine problem-solving exercise using their knowledge of geometric facts or relationships.

BIBLIOGRAPHY Altshiller-Court, Nathan. College Geometry: An

Introduction to the Modern Geometry of the Triangle and the Circle. New York: Barnes and Noble, 1952.

Coxeter, H. S. M., and S. L. Greitzer. Geometry Revisited. Washington, DC: Mathematical Association of America, 1967.

Honsberger, Ross. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Mathematical Association of America, 1995.

Johnson, Roger A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Cambridge, MA: Riverside Press, 1929.

Posamentier, Alfred S. Advanced Euclidean Geometry: Excursions for Secondary Teachers and Students. Emeryville, CA: Key College Publishing, 2002.

Posamentier, Alfred S., and Charles T. Salkind. Challenging Problems in Geometry. New York: Dover, 1988.

Editors' note, part 2: One purpose of "Delving Deeper" is to encourage articles about the mathe matics we teach, more so than the teaching of our mathematics (and also, more so than the mathe matics that bears no close connection to the high school curriculum). Another purpose is to look at that mathematics from a deeper, sometimes off beat, perspective, to wander into unusual (though presumably seldom unknown) corners of what is otherwise thought of as totally familiar territory. Posamentier's article takes that direction: a com

pletely standard problem, cutting a circle into three equal-area pieces, approached with the de liberate attempt to make the familiar a bit strange.

Brian Harvey, a close friend and colleague who has helped us in innumerable ways, including some guest-editing for "Delving Deeper," wrote to us as we were discussing this article:

When I was 12 years old, my mom was a partic ipant in a summer program for teachers about the SMSG [School Mathematics Study Group] "new math" program, and I tagged along. There was a guest presentation by George P?lya, who raised this example of trisecting

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Fig. 7 P?lya method Fig. 8 Is this interesting?

a circle [see fig. 7]. My solution was to inscribe an ellipse in the circle whose major axis is a di ameter, and whose minor axis is 1/3 the length of the diameter. Of course this is not a Euclid ean construction!

The thing that makes all of these dissections? now five of them, and a proposed sixth by the au thor?more "interesting" than, say, something like the one shown in figure 8 (or is this interest ing, too?) is how "sensible" they start out, how

much accessible mathematics might be expected to come out of them, and, perhaps, how subtle they might turn out to be. You might take on Posamentier's local challenge: What other ways are there of trisecting the circle? Or you might take on the broader challenge that inspired his original paper: Find some "ordinary" geometric idea?one, however important, that might seem prosaic and quite easy to derive or teach?and look at what else can be derived from it. One that

has long appealed to me (Paul Goldenberg) is a surprise connection between geometry and statis tics: an idea that grows out of thinking about the diagonal of a rectangle nestled against the axes in the first quadrant (or, equivalently, the radius of a circle centered on the origin), which might be represented as

and generalizing to ?-dimensions, we would use

Except for a scale factor, this is the formula for standard deviation. What connections can we

draw, or what variations on the ideas can lead us to new insights, either about balls in space, or about what standard deviation tells us about data?

One last question: Posamentier states that "we can use a pocket calculator" to approximate a solu tion to

it ? = a-sma. 3

What are some ways of doing this? We would like especially to hear from readers who apply mathe matical results or methods to the problem of obtain ing numerical approximations to equations that do not have simple algebraic solutions, oo

The three-dimensional equivalent?the diagonal of a rectangular parallelepiped with a corner at the origin and its edges along the axes (or the radius of a sphere centered on the origin)?is

To measure a four-dimensional box or ball we would use

yjx2 + y2 + z2 + w2,

ALFRED S. POSAMENTIER, asp2@

^ junoxom, is dean and professor of _ J| mathematics education at the School

of Education of the City College of New York, CUNY, New York, NY 10031. Among his many interests in mathematics and its teaching are problem solving and demonstrating the beau ty of mathematics to society at large.

418 MATHEMATICS TEACHER | Vol. 99, No. 6 ? February 2006

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Contents
p. 414
p. 415
p. 416
p. 417
p. 418
Issue Table of Contents
The Mathematics Teacher, Vol. 99, No. 6 (FEBRUARY 2006) pp. 385-464
Front Matter
Reader Reflections
ALGEBRA CHALLENGE [pp. 388-388]
LOGISTIC GROWTH [pp. 388-389]
FINDING THE X-COORDINATE OF THE VERTEX OF A PARABOLA [pp. 390-390]
PERPENDICULAR LINES [pp. 390-390]
CUBIC VERTICES REVISITED [pp. 390-390]
TYPOGRAPHICAL ERRORS [pp. 390-390]
Sound Off!
A Dialogue between Calculator and Algebra [pp. 391-393]
Discovering RELATIONSHIPS Involving BARAVELLE SPIRALS [pp. 394-400]
Connecting Research to Teaching
Good Things Always Come in Threes: Three Cards, Three Prisoners, and Three Doors [pp. 401-405]
Bugs, Planes, and Ferris Wheels: A Problem-Centered Curriculum [pp. 406-413]
Delving Deeper
Trisecting the Circle: A Case for Euclidean Geometry [pp. 414-418]
THREE BY THREE SYSTEMS: More than Just a Point [pp. 419-423]
[February Calendar] [pp. 424-429]
Teaching about Functions through Motion [pp. 430-437]