# New Horizons in Geometry

Tom M. Apostol
Mamikon A. Mnatsakanian
Volume: 47
Edition: 1
Pages: 528
https://www.jstor.org/stable/10.4169/j.ctt6wpwpf

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. PREFACE
(pp. ix-x)
4. INTRODUCTION
(pp. xi-xii)

Since 1997 Tom Apostol and Mamikon Mnatsakanian have co-authored 30 papers, most of which are on geometry. Their work is strikingly innovative, combining the classical with the modern. They often surprise the reader with fresh, frequently astounding conclusions that challenge the imagination. As an added attraction to the reader, only a modest background is needed to understand their work. Working together, they have won three Lester R. Ford Awards since 2005 for five papers published in theAmerican Mathematical Monthlyin 2004, 2007, and 2009 — an enviable achievement.

The citation for their 2004 Ford Award notes that they do classical...

5. FOREWORD
(pp. xiii-xiv)

This passage perfectly captures the spirit of the bookNew Horizons in Geometry, by Tom Apostol and Mamikon Mnatsakanian. In a remarkable display of mathematical versatility and imagination, the authors present us with a wealth of geometrical gems. These beautiful and often surprising results deal with a multitude of geometric forms, their interrelationships, and in many cases, their connection with patterns underlying the laws of nature. Lengths, areas and volumes, of curves, surfaces, and solids, are explored from a visually captivating perspective. The preponderance of results discussed by the authors are new, and when not new, are presented with unusual...

6. Chapter 1 MAMIKON’S SWEEPING-TANGENT THEOREM
(pp. 1-30)

Calculus is a beautiful subject with a host of dazzling applications. Anyone familiar with this important branch of mathematics would be amazed to learn that many standard calculus problems can be easily solved by an innovative visual approach that makes no use of formulas. Here’s a sample:

Problem 1.Find the area of a parabolic segment.

In Figure 1.1 the parabolic segment is the shaded region between the graph of\$y = {x^2}\$and the x axis from 0 to x. Its area was first calculated by Archimedes more than 2000 years ago by a method that laid the foundations for integral...

7. Chapter 2 CYCLOIDS AND TROCHOIDS
(pp. 31-64)

For the average lay person the word roulette means a gambling game, or perhaps a small toothed wheel that makes equally spaced perforations like those on sheets of postage stamps. In geometry arouletteis the locus of a point attached to a plane curve that rolls along a fixed base curve without slipping. A surprising number of classical curves can be generated as roulettes – the cycloid, cardioid, tractrix, catenary, parabola, and ellipse, to name just a few. If the rolling curve is a circle, the roulette is called atrochoid(from the Greek word τρoχoς for wheel). Both the cycloid...

8. Chapter 3 CYCLOGONS AND TROCHOGONS
(pp. 65-100)

In Chapter 2, Mamikon’s sweeping-tangent theorem was used to show that the areaAof the cycloidal arch in Figure 3.1 is equal to three times the areaCof the rolling circular disk,

A = 3C, (3.1)

and the significance of the factor 3 was explained.

This chapter solves the more general problem, in which the rolling circle is replaced by a regular polygon. The problem is treated by an elementary geometrical method, and the area formula for the cycloid is obtained as a limiting case. We use the formula for the area of a circular sector, but there...

9. Chapter 4 CIRCUMGONS AND CIRCUMSOLIDS
(pp. 101-134)

We begin by generalizing Archimedes’ striking discovery concerning the area of a circular disk, which for our purposes we prefer to state as follows:

Theorem 4.1. (Archimedes)The area of a circular disk is one-half the product of its perimeter and its radius.

Expressed as a formula, this becomes

A =1/2Pr, (4.1)

whereAis the area,Pis the perimeter, andris the radius of the disk. First we extend (4.1) to a large class of plane figures circumscribing a circle that we call circumgons, defined in Section 4.2. They include arbitrary triangles, all regular polygons, some irregular...

10. Chapter 5 THE METHOD OF PUNCTURED CONTAINERS
(pp. 135-168)

A spectacular landmark in the history of mathematics was the discovery by Archimedes that the volume of a solid sphere is two-thirds the volume of the smallest right cylinder that surrounds it, and that the surface area of the sphere is also two-thirds the total surface area of the same cylinder. Archimedes was so excited by this discovery that he wanted a sphere and its circumscribing cylinder engraved on his tombstone, even though there were many other great accomplishments for which he would be forever remembered. He made this particular discovery by balancing slices of a sphere and cone against...

11. Chapter 6 UNWRAPPING CURVES FROM CYLINDERS AND CONES
(pp. 169-212)

In his delightful bookMathematical Snapshots, Steinhaus [66] describes a simple, engaging construction, illustrated in Figure 6.1. Wrap a piece of paper around a cylindrical candle, and cut it obliquely with a knife. The cross section is an ellipse, which becomes a sinusoidal curve when unwrapped.

The same idea can be demonstrated with a safer instrument. Take a cylindrical paint roller, dip it at an angle in a container of paint or water color, and roll it on a flat surface. The roller prints a sinusoidal wave pattern as shown in Figure 6.2.

Now imagine the elliptical cross section replaced...

12. Chapter 7 NEW DESCRIPTIONS OF CONICS VIA TWISTED CYLINDERS, FOCAL DISKS, AND DIRECTORS
(pp. 213-242)

Conics have been investigated since ancient times as sections of a circular cone. Surprising descriptions of these curves are revealed by investigating them as sections of a hyperboloid of revolution, referred to here as atwisted cylinder. We generalize the classical focus-directrix property of conics by what we call thefocal disk-director property(Section 7.2). We also generalize the classical bifocal properties of central conics by thebifocal disk property(Section 7.5), which applies to all conics, including the parabola. Our main result (Theorem 7.5) is that the two generalized properties are satisfied by sections of a twisted cylinder, and...

13. Chapter 8 ELLIPSE TO HYPERBOLA: “WITH THIS STRING I THEE WED”
(pp. 243-266)

The title of this chapter was inspired by our modification of the well-known string construction for the ellipse. In Figure 8.1a, a piece of string joins two fixed points (the foci of the ellipse), and the string is kept taut by a moving pencil that traces the ellipse. The bifocal property of the ellipse states that the sum of distances from pencil to foci is the constant length of the string.

The same string fastened to the same points can also be used to trace a hyperbolic arc with the same foci. How is this possible? The bifocal property of...

14. Chapter 9 TRAMMELS
(pp. 267-294)

Figure 9.1a shows a line segment of fixed length whose ends slide along two perpendicular axes. It can be realized physically as a sliding ladder or as a sliding door moving with its ends on two perpendicular tracks. During the motion, a given point on the segment traces an ellipse with one quarter of the ellipse in each quadrant (Figure 9.1b), so this device is called anellipsograph, a mechanism for drawing an ellipse.

This particular ellipsograph was known to ancient Greek geometers and is often called the trammel of Archimedes [70, p. 3], but we are not aware of...

15. Chapter 10 ISOPERIMETRIC AND ISOPARAMETRIC PROBLEMS
(pp. 295-330)

Two incongruent solids with remarkable properties are shown in Figure 10.1. One is a slice of a solid hemispherical shell with inner radiusrand outer radiusRcut by a plane parallel to the equator and at distanceh < rfrom the equator. The other is a cylindrical shell with the same radii and altitudeh. The surface of each solid consists of four components: an upper circular ring, a lower circular ring, an outer lateral surface, and an inner lateral surface. The two solids share the following properties:

(a) The solids have equal volumes.

(b) The solids...

16. Chapter 11 ARCLENGTH AND TANVOLUTES
(pp. 331-374)

In Chapter 1 we used sweeping tangents to calculate area. Now we use them to find arclength. Figure 11.1 shows a close-up view of the tangent sweep introduced in Figure 1.16. Each tangent segment is attached to a curve we call thetangency curve τ. The point of tangency moves alongτin a given direction, called the positive direction, as indicated by the arrowhead in Figure 11.1. At each point ofτthe tangent line defines two rays, one in the positive direction of motion, the other in the opposite direction. It may be helpful to imagine an automobile...

17. Chapter 12 CENTROIDS
(pp. 375-400)

Archimedes introduced the concept of center of gravity. He used it in many of his works, including the stability of floating bodies, ship design, and in his discovery that the volume of a sphere is two-thirds that of its circumscribing cylinder. It was also used by Pappus of Alexandria in the 3rd century AD in formulating his famous theorems for calculating volume and surface area of solids of revolution. We can only speculate on what Archimedes had in mind when he referred to center of gravity because none of his extant works provides an explicit definition of the concept. The...

18. Chapter 13 NEW BALANCING PRINCIPLES WITH APPLICATIONS
(pp. 401-442)

The sphere and circumscribing cylinder engraved on Archimedes’ tombstone commemorate his landmark discovery that their volumes and surface areas are related by the same ratio 2/3. He discovered the volume relation and many other geometrical results by mechanical balancing.

In particular, he found the volume of a cylindrical wedge by introducing the balancing shown in Figures 13.1a and 13.1b. First, he uses the Pythagorean theorem to balance the lengths of horizontal chords of a triangle and a semicircular disk with respect to a vertical axis through its center as in Figure 13.1a. He then builds two-dimensional regions from these chords,...

19. Chapter 14 SUMS OF SQUARES
(pp. 443-472)

A point that moves in a plane so that the sum of its distances from two fixed points in the plane is constant traces an ellipse with the two points as foci.

What is the locus if the point moves so that the sum of the squares of the two distances is constant?

An elementary calculation in coordinate geometry shows that the locus is a circle with center midway between the two points.

When we ask the same question for three or more fixed points in a plane we obtain the following surprising result.

Theorem 14.1.Given n fixed points...

20. Chapter 15 APPENDIX
(pp. 473-500)

Figure 15.1 shows the parabolay=x², together with another parabola,y=(2x)², exactly half as wide as the first. Both are enclosed in a rectangle of basexand altitudex² whose area isx³. The two parabolas divide the rectangle into three regions, and our strategy is to show that all three have equal areas. Then each has area one-third that of the circumscribing rectangle, as required.

The two leftmost regions formed by the bisecting parabola obviously have equal areas, so to complete the proof we need only show that the region above the bisecting parabola has...

21. Bibliography
(pp. 501-504)
22. Index
(pp. 505-512)