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# New horizons in geometry

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## Abstract

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. ...
... Since the normal section at a given point of a surface sharing a tangent direction with the embedded curve is geodesic by definition [18] and the resultant conics being plane curves possess zero torsion, τ n,s = 0. 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). ...
... 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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Utilizing the geometrical characteristics of a simple smooth curve embedded in an orientable surface of arbitrary shape, expressions are derived in boundary-fitted curvilinear coordinates, for the normal and tangential strain rate of a fluid which contacts the surface. The derived expressions can be applied to stream-surfaces, solid walls with hydrodynamic slippage and two-fluid interfaces. The use of the expressions is discussed with specific examples and critical examination of the literature.
... Archimedes proves that the two boldly typed line segments balance each other according to the Law of the Lever 4 and concludes from it that the whole parabolic segment placed with its center of mass on the left end of the lever is in equilibrium with the light gray triangle on the right side of the equal-armed balance. Since Archimedes knows the center of mass and area of the triangle and also the center 2 If from a point on a parabola a straight line be drawn which is either itself the axis or parallel to the axis, as P V , and if QQ0 be a chord parallel to the tangent to the parabola at P and meeting P V in V , then QV D V Q 0 [6, p. 234]. 3 If from two points Q and Q 0 on a parabola straight lines be drawn which are parallel to the axis, as QV and Q 0 V 0 , if QV 0 , Q 0 V be the tangents to the parabola at Q, Q 0 respectively, and meeting each other in S , then QS D SV 0 and Q 0 S D SV . ...
... After all it is based on Mamikon's sweeping-tangent-theorem, a powerful principle that can be used for the quadrature of all kinds of curves (c.f. [2]). If we have a collection of tangent segments of a curve, which traverse or sweep out a region, Mamikon's theorem tells us that the area of this region is the same as the area of the region that we get, if we bring all the points of tangency to a common point by translating each tangent segment (c.f. ...
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The mathematical literature knows several quite different ways to determine the area of a parabolic segment. In this article they are brought together, compared and varied. Finally two new proofs are added to the collection. Each proof is displayed by an expressive figure. A colorful compilation of those figures accompanies the article as an appendix.
... After intersection, these T-Map primitives form the 3D hypersection shown in Fig. 9. It is a square elliptic globe that is inscribed in a sphere of diameter ŧ; it is one form of generalized Archimedean globe (Appendix A and Apostol and Mnatsakanian, 2012). The doubly traced cylinder from arc-segments 1 and 3 is only present as two portions of the circular edge in the e x e y -plane and the two parallel surface-lines that lie in the e x θ ′-plane and at 45° to the e x -and θ ′-axes. ...
... The square elliptical globe in Fig. 9 is formed by a dilatation in the e y -direction of an Archimedean globe, so causing the two circular cylinders to become elliptic in cross-section and the curved edges of the solid to become circles. More properties, figures, and formulae about generalized Archimedean domes and globes may be found in Chapter 5 of Apostol and Mnatsakanian (2012). ...
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For purposes of automating the assignment of tolerances during design, a math model, called the Tolerance-Map (T-Map), has been produced for most of the tolerance classes that are used by designers. Each T-Map is a hypothetical point-space that represents the geometric variations of a feature in its tolerance-zone. Of the six tolerance classes defined in the ASME/ANSI/ISO Standards, profile tolerances have received the least attention for representation in computer models. The objective of this paper is to describe a new method of construction, using computer-aided geometric design, which can produce the T-Map for any line-profile. The new method requires decomposing a profile into segments, creating a solid-model T-Map primitive for each, and then combining these by Boolean intersection to generate the T-Map for a complete line profile of any shape. To economize on length, the scope of this paper is limited to line-profiles formed from circular arc-segments. The parts containing the line-profile features are considered to be rigid.
... Even though twinning preserves the volume and hence always carries non-degenerate tetrahedra to the same, it can map rank 1 tetrahedra to rank 2 and vice versa. An examination of the patterns of natural and inverse natural parameters vanishing in rank 1 tetrahedra (Fig. 6) in fact shows that whereas twinning maps cases (4,5,6,7), (8,9,10,11) & (12,13,14,15) to themselves, applied to the cases 0, 1, 2 & 3 it produces a rank 2 tetrahedron. These rank 2 tetrahedra will be those at the lowest level of the rank 2 sub-hierarchy in Fig. 4; the other two levels in that sub-hierarchy are clearly preserved by twinning. ...
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A natural extension of Heron's 2000 year old formula for the area of a triangle to the volume of a tetrahedron is presented. This extension gives the fourth power of the volume as a polynomial in six simple rational functions of the areas of its four faces and of its three medial parallelograms, which will be referred to herein as interior faces. Geometrically, these rational functions are the areas of the triangles into which the exterior faces are divided by the points at which the tetrahedron's in-sphere touches those faces. This leads to a conjecture as to how the formula likely extends to \$n\$-dimensional simplices for all \$n > 3\$. Remarkably, for \$n = 3\$ the zeros of the overall polynomial constitute a five-dimensional real semi-algebraic variety consisting almost entirely of collinear tetrahedra with vertices at infinite distances from one another. These unconventional Euclidean configurations can be identified with a quotient of the Klein quadric by an action of a group of reflections isomorphic to \$\mathbb Z_2^4\$, wherein four-point configurations in the finite affine plane constitute a distinguished three-dimensional subset. The paper closes by noting that the algebraic structure of the zeros in the finite affine plane naturally defines the associated \$4\$-element, rank-\$3\$ chirotope, aka affine oriented matroid.
... Since all of its edges are circles of diameter t, each of its central 3D hypersections in Figs. 4(d)-4(g) is an Archimidean globe [30]. ...
Article
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... Este trabalho tem como base o livro New Horizons in Geometry [1], de Tom Apostol e Mamikon Mnatsakanian, que traz uma abordagem visual e inovadora, com métodos geométricos que requerem pouco ou nenhuma fórmula, para resolver muitos problemas clássicos do Cálculo. Como tal, grande parte dos resultados e aplicações aqui presentes são uma releitura e um detalhamento daquilo ali apresentado. ...
... Artists [33], architects [41], film makers, engineers and designers draw inspiration from visual mathematics. Well illustrated books like [21,49,85,48,34,49,10,3] advertise mathematics with figures and illustrations. Such publications help to counterbalance the impression that mathematics is difficult to communicate to non-mathematicians. ...
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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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