Abstract: Equality of zonal areas on a sphere and pseudosphere is extended by elementary geometric methods to surfaces of revolution of constant total (Gaussian) curvature, and constant mean (Delaunay) curvature. Bicycle wheels are used to trace profiles of these surfaces. Surprisingly, cycloids appear as limiting cases of such profiles.

Publication: American Mathematical Monthly Vol.: 123 No.: 5 ISSN: 0002-9890

ID: CaltechAUTHORS:20160930-143148171

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Abstract: Circle and sphere properties are extended to the tractrix and pseudosphere, the catenary and catenoid, and to higher-dimensional analogs.

Publication: American Mathematical Monthly Vol.: 122 No.: 8 ISSN: 0002-9890

ID: CaltechAUTHORS:20151119-095812173

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Abstract: In earlier work ([1]-[5]) the authors used the method of sweeping tangents to calculate area and arclength related to certain planar regions. This paper extends the method to determine volumes of solids. Specifically, take a region S in the upper half of the xy plane and allow the plane to sweep tangentially around a general cylinder with the x axis lying on the cylinder. The solid swept by S is called a solid tangent sweep. Its solid tangent cluster is the solid swept by S when the cylinder shrinks to the x axis. Theorem 1: The volume of the solid tangent sweep does not depend on the profile of the cylinder, so it is equal to the volume of the solid tangent cluster. The proof uses Mamikon's sweeping-tangent theorem: The area of a tangent sweep to a plane curve is equal to the area of its tangent cluster, together with a classical slicing principle: Two solids have equal volumes if their horizontal cross sections taken at any height have equal areas. Interesting families of tangentially swept solids of equal volume are constructed by varying the cylinder. For most families in this paper the solid tangent cluster is a classical solid of revolution whose volume is equal to that of each member of the family. We treat forty different examples including familiar solids such as pseudosphere, ellipsoid, paraboloid, hyperboloid, persoids, catenoid, and cardioid and strophoid of revolution, all of whose volumes are obtained with the extended method of sweeping tangents. Part II treats sweeping around more general surfaces.

Publication: Forum Geometricorum Vol.: 15ISSN: 1534-1178

ID: CaltechAUTHORS:20150514-132136696

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Abstract: In Part I (Forum Geom., 15 (2015) 13-44) the authors introduced solid tangent sweeps and solid tangent clusters produced by sweeping a planar region S tangentially around cylinders. This paper extends Part I by sweeping S not only along cylinders but also around more general surfaces, cones for example. Interesting families of tangentially swept solids of equal height and equal volume are constructed by varying the cylinder or the planar shape S. For most families in this paper the solid tangent cluster is a classical solid whose volume is equal to that of each member of the family. We treat many examples including familiar quadric solids such as ellipsoids, paraboloids, and hyperboloids, as well as examples obtained by puncturing one type of quadric solid by another, all of whose volumes are obtained with the extended method of sweeping tangents. Surprising properties of their centroids are also derived.

Publication: Forum Geometricorum Vol.: 15ISSN: 1534-1178

ID: CaltechAUTHORS:20150514-132525329

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Abstract: Archimedes’ mechanical balancing methods led him to stunning discoveries concerning the volume of a sphere, and of a cylindrical wedge. This paper introduces new balancing principles (different from those of Archimedes) including a balance-revolution principle and double equilibrium, that go much further. They yield a host of surprising relations involving both volumes and surface areas of circumsolids of revolution, as well as higher-dimensional spheres, cylindroids, spherical wedges, and cylindrical wedges. The concept of cylindroid, introduced here, is crucial for extending to higher dimensions Archimedes’ classical relations on the sphere and cylinder. We also provide remarkable new results for centroids of hemispheres in n-space. Throughout the paper, we adhere to Archimedes’ style of reducing properties of complicated objects to those of simpler objects.

Publication: American Mathematical Monthly Vol.: 120 No.: 4 ISSN: 0002-9890

ID: CaltechAUTHORS:20130509-103547032

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Abstract: Classical dissections convert any planar polygonal region onto any other polygonal region having the same area. If two convex polygonal regions are isoparametric, that is, have equal areas and equal perimeters, our main result states that there is always a dissection, called a complete dissection, that converts not only the regions but also their boundaries onto one another. The proof is constructive and provides a general method for complete dissection using frames of constant width. This leads to a new object of study: isoparametric polygonal frames, for which we show that a complete dissection of one convex polygonal frame onto any other always exists. We also show that every complete dissection can be done without flipping any of the pieces.

Publication: American Mathematical Monthly Vol.: 118 No.: 9 ISSN: 0002-9890

ID: CaltechAUTHORS:20111111-153809155

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Abstract: We introduce a string mechanism that traces both elliptic and hyperbolic arcs having the same foci. This suggests replacing each focus by a focal circle centered at that focus, a simple step that leads to new characteristic properties of central conics that also extend to the parabola. The classical description of an ellipse and hyperbola as the locus of a point whose sum or absolute difference of focal distances is constant, is generalized to a common bifocal property, in which the sum or absolute difference of the distances to the focal circles is constant. Surprisingly, each of the sum or difference can be constant on both the ellipse and hyperbola. When the radius of one focal circle is infinite, the bifocal property becomes a new property of the parabola. We also introduce special focal circles, called circular directrices, which provide equidistance properties for central conics analogous to the classical focus-directrix property of the parabola. Those familiar with paperfolding activities for constructing an ellipse or hyperbola using a circle as a guide, will be pleased to learn that the guiding circle is, in fact, a circular directrix.

Publication: Mathematics Magazine Vol.: 84 No.: 2 ISSN: 0025-570X

ID: CaltechAUTHORS:20111005-083237390

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Abstract: The classical involute of a plane base curve intersects every tangent line at a right angle. This paper introduces a tanvolute, which intersects every tangent line at any given fixed angle. This minor change in the definition of a classical concept leads to a wealth of new examples and phenomena that go far beyond the original situation. Our treatment is based on two differential equations relating arclength functions for the base curve and its tanvolute, the tangent-length function from the base to the tanvolute, and the fixed angle. The parameters in the differential equations contribute many essential features to the solution curves. Even when the base curve is relatively simple, for example a circle, the variety in the shapes of the tanvolutes is remarkably rich. To illustrate, as a circle shrinks to a single point, its tanvolute becomes a logarithmic spiral! An application is given to a generalized pursuit problem in which a missile is fired at constant speed in an unknown tangent direction from an unknown point on a base curve. Surprisingly, it can always be intercepted by a faster constant-speed missile that follows a specific tanvolute of the base curve.

Publication: American Mathematical Monthly Vol.: 117 No.: 8 ISSN: 0002-9890

ID: CaltechAUTHORS:20110207-091141076

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Abstract: 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.

Publication: American Mathematical Monthly Vol.: 116 No.: 7 ISSN: 0002-9890

ID: CaltechAUTHORS:20091203-091743750

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Abstract: 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.

Publication: American Mathematical Monthly Vol.: 116 No.: 2 ISSN: 0002-9890

ID: CaltechAUTHORS:20090901-094318840

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Abstract: 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 a twisted cylinder. We generalize the classical focus-directrix property of conics by what we call the focal disk-director property (Section 2). We also generalize the classical bifocal properties of central conics by the bifocal disk property (Section 5), which applies to all conics, including the parabola. Our main result (Theorem 5) is that each of these two generalized properties is satisfied by sections of a twisted cylinder, and by no other cures. Although some of these results are mentioned in Salmon's treatise [6] and a not by Ferguson [4], they are not widely known, and we go far beyond these earlier treatments.

Publication: American Mathematical Monthly Vol.: 115 No.: 9 ISSN: 0002-9890

ID: CaltechAUTHORS:APOamm08

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Abstract: Using analytic functional equations, Berndt derived three reciprocity laws connecting five arithmetical sums analogous to Dedekind sums. This paper gives elementary proofs of all three reciprocity laws and obtains them all from a common source, a polynomial reciprocity formula of L. Carlitz.

Publication: Pacific Journal of Mathematics Vol.: 98 No.: 1 ISSN: 0030-8730

ID: CaltechAUTHORS:APOpjm82

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