Mamikon A. Mnatsakanian

Unwrapping Curves from Cylinders and Cones

Now imagine the elliptical cross section replaced by any curve lying on the surface of a right circular cylinder. What happens to this curve when the cylinder is unwrapped? Consider also the inverse problem, which you can experiment with by yourself: Start with a plane curve (line, circle, parabola, sine curve, etc.) drawn with a felt pen on a rectangular sheet of transparent plastic, and roll the sheet into cylinders of different radii. What shapes does the curve take on these cylinders? How do they appear when viewed from different directions? A few trials reveal an enormous number of possibilities, even for the simple case of a circle. This paper formulates these somewhat vague questions more precisely, in terms of equations, and shows that they can be answered with surprisingly simple twodimensional geometric transformations, even when the cylinder is not circular. For a circular cylinder, a sinusoidal influence is always present, as exhibited in Figures

Sums of Squares of Distances in m-Space

1. INTRODUCTION. Given a finite set of fixed points in 3-space, what is the lo-cus of a point moving in such a way that the sum of the squares of its distances from the fixed points is constant? The answer is both elegant and surprising: the locus is a sphere whose center is at the centroid of the fixed points (if we allow the empty set and a single point as degenerate cases of a sphere). The answer to this question and its extension to any finite-dimensional space was given in [2] and was based on equation (3), which relates sums of squares of distances between points in m-space. This formula was used in [1], [3], and [4] to find, without calculus, the areas of cycloidal and trochoidal regions. It was also applied in [2] to problems regarding a regular simplex in k-space. This paper gives generalized formulas that lead to remarkable relations for sums of squares of distances in any finite-dimensional space. It also points out some applications to geometry.

Solids Circumscribing Spheres

1. INTRODUCTION. Although it is well known that every tetrahedron circum-scribes a sphere, the following two simple consequences apparently have not been previously recorded. First, any plane through the center of the inscribed sphere divides the tetrahedron into two smaller solids whose surface areas are equal if and only if their volumes are equal. Second, the centroid of the boundary surface of a tetrahedron and the centroid of its volume are always collinear with the center of the inscribed sphere, at distances in the ratio 4:3 from the center. This paper shows that both these and deeper results hold, not only for the tetrahe-dron or any polyhedron that circumscribes a sphere, but for more general solids called circumsolids (defined in section 4), whose faces can be curved as well as planar. The curved faces can be cylindrical, conical, or spherical. Each circumsolid circumscribes a sphere (its insphere), and all share the following property, proved in section 4:

A Fresh Look at the Method of Archimedes

1. INTRODUCTION. A spectacular landmark in the history of mathematics was the discovery by Archimedes (287–212 B.C.) that the volume of a solid sphere is twothirds the volume of the smallest 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 slices of a larger cylinder, using centroids and the principle of the lever, which were also among his remarkable discoveries. The volume ratio for the sphere and cylinder can be derived from first principles without using levers and centroids (see [5]). This simpler and more natural method, presented in sections 2 and 3, paves the way for generalizations. Section 4 introduces a family of solids circumscribing a sphere. Cross sections of each solid cut by planes parallel to the equatorial plane are disks bounded by similar n-gons that circumscribe the circular cross sections of the sphere. We call these solids Archimedean globes in honor of Archimedes, who treated the case n = 4. The sphere is a limiting case, n →∞ . Each globe is analyzed by dividing it into wedges with two planar faces and one semicircular cylindrical face. In fact, Archimedes discussed (both mechanically and geometrically) volumes of wedges of this type. Figure 1 shows the top view of examples of globes with n = 3, 4, 6, and the limiting sphere.

New horizons in geometry

New Horizons in Geometry represents the fruits of 15 years of work in geometry by a remarkable team of prize-winning authors-Tom Apostol and Mamikon Mnatsakanian. It serves as a capstone to an amazing collaboration. Apostol and Mamikon provide fresh and powerful insights into geometry that requires only a modest background in mathematics. Using new and intuitively rich methods, they give beautifully illustrated proofs of results, the majority of which are new, and frequently develop extensions of familiar theorems that are often surprising and sometimes astounding. It is mathematical exposition of the highest order. The hundreds of full color illustrations by Mamikon are visually enticing and provide great motivation to read further and savor the wonderful results. Lengths, areas, and volumes of curves, surfaces, and solids are explored from a visually captivating perspective. It is an understatement to say that Apostol and Mamikon have breathed new life into geometry.

Tangents and Subtangents Used to Calculate Areas

Thus, the subtangent determines f(x) up to a constant factor. Therefore it is not surprising that information about f(x) can be extracted from a knowledge of s(x). For example, if b is a nonzero constant, then s(x) = b if and only if f(x) = f(O)ex/b; and s(x) = bx if and only if f(x) = f(l)xl/b. In particular, the parabola f(x) = x2 has subtangent s(x) = x/2, and the hyperbola f(x) = 1/x has subtangent s(x) = -x. In fact, if r A 0 the function f(x) = xr can be defined as that function with f(l) = 1 whose subtangent is s(x) = x/r. In [1] we used subtangents as an aid to draw tangent lines. Here we use them to calculate areas in a natural and intuitive geometric fashion rather than analytically.

Figures Circumscribing Circles

1. INTRODUCTION. The centroid of the boundary of an arbitrary triangle need not be at the same point as the centroid of its interior. But we have discovered that the two centroids are always collinear with the center of the inscribed circle, at distances in the ratio 3 : 2 from the center. We thought this charming fact must surely be known, but could find no mention of it in the literature. This paper generalizes this elegant and surprising result to any polygon that circumscribes a circle (Theorem 6). A key ingredient of the proof is a link to Archimedes’ striking discovery concerning the area of a circular disk [4, p. 91], which for our purposes we prefer to state as follows: Theorem 1 (Archimedes). The area of a circular disk is equal to the product of its semiperimeter and its radius. Expressed as a formula, this becomes A = 1 Pr, (1)

A New Look at the So-Called Trammel of Archimedes

The paper begins with an elementary treatment of a standard trammel (trammel of Archimedes), a line segment of fixed length whose ends slide along two perpendicular axes. During the motion, points on the trammel trace ellipses, and the trammel produces an astroid as an envelope that is also the envelope of the family of traced ellipses. Two generalizations are introduced: a zigzag trammel, obtained by dividing a standard trammel into several hinged pieces, and a flexible trammel whose length may vary during the motion. All properties regarding traces and envelopes of a standard trammel are extended to these more general trammels. Applications of zigzag trammels are given to problems involving folding doors. Flexible trammels provide not only a deeper understanding of the standard trammel but also a new solution of a classical problem of determining the envelope of a family of straight lines. They also reveal unexpected connections between various classical curves; for example, the cycloid and the quadratrix of Hippias, curves known from antiquity.

The method of sweeping tangents

What is the area of the shaded region between the tyre tracks of a moving bicycle such as that depicted in Figure 1 ? If the tracks are specified, and equations for them are known, the area can be calculated using integral calculus. Surprisingly, the area can be obtained more easily without calculus, regardless of the bike’s path, using a dynamic visual approach called the method of sweeping tangents that does not require equations for the curves.

New Insight into Cycloidal Areas

A point on the boundary of a circular disk that rolls once along a straight line traces a cycloid. The cycloid divides its circumscribing rectangle into a cycloidal arch below the curve and a cycloidal cap above it. The area of the arch is three times that of the disk, and the area of the cap is equal to that of the disk. The paper provides deeper insight into this well-known property by showing (without integration) that the ratio 3:1 holds at every stage of rotation. Each cycloidal sector swept by a normal segment from the point of contact of the disk to the cycloid has area three times that of the overlapping circular segment cut from the rolling disk. This surprising result is extended to epicycloids (and hypocycloids), obtained by rolling a disk of radius r externally (or internally) around a fixed circle of radius R. The factor 3 is replaced by (3 + 2r/R) for the epicycloid, and by (3 − 2r/R) for the hypocycloid. This leads to several interesting consequences. For example, for any cycloid, epicycloid, or hypocycloid, the area of one full arch exceeds that of one full cap by twice the area of the rolling disk. Other applications yield (again without integration) compact geometrically revealing formulas for areas of cycloidal radial and ordinate sets.