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New horizons in geometry

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The collaborative work of Tom Apostol and Mamikon Mnatsakanian has been lauded for its clarity and originality. in this volume the authors present an impressive collection of geometric results that reveal surprising connections between lengths, areas and volumes in various shapes, and allow one to compute difficult integrals, all using intuitive visual calculations. One noteworthy idea that the reader will encounter is Mamikon's Sweeping Tangent Theorem from which the authors obtain a visual derivation of the property that the length of an arc of a catenary is proportional to the area under the arc. This is one of many 'proofs without words' contained within. in addition, a variety of results are derived visually for cycloids, conic sections, and many more geometric objects. As befits a book that emphasises visual thinking, the text is beautifully illustrated. This is a book that will inspire students and enrich any geometry or calculus course.. © 2012 by The Mathematical Association of America (Incorporated) and Mamikon A. Mnatsakanian.

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... Closely-connected approaches are employed by [9][10][11]. 20 Ref. [9] uses the normal curvature in transformed coordinates to set the zero tangential stress boundary condition for a streamfunction-vorticity numerical formulation of axisymmetric flow of a Newtonian fluid. Herrera, in [10] and [12] develop and utilize a relationship between 25 vorticity components and the curvatures and torsions of a streamline. ...
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... The eccentricity of these conics are given by e = tan (θ) [20], and the radius of curvature 1/κ n,s at the vertex of any of these conics is r tan θ. These Euclidean geometry findings are readily obtainable with the Dandelin (focal) sphere construction [20] (see supplementary material for a relevant visualization ,which corresponds to θ < π/4). Using this radius for 1/κ ns in Equation (5) ...
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Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. 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 two-thirds 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. Archime-des 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 ac-complishments 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 --* o. 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.
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